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Journal of Computational Physics ••• (••••) •••–•••

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Journal of Computational Physics

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Mimetic finite difference method

Konstantin Lipnikov a, Gianmarco Manzini a, Mikhail Shashkov b,∗a Applied Mathematics and Plasma Physics Group, T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544,United Statesb Methods and Algorithms Group, XCP-4, X-Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87544,United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 August 2012Received in revised form 14 May 2013Accepted 22 July 2013Available online xxxx

Keywords:Mimetic finite difference methodDiscrete vector and tensor calculusLagrangian hydrodynamics

The mimetic finite difference (MFD) method mimics fundamental properties of mathemati-cal and physical systems including conservation laws, symmetry and positivity of solutions,duality and self-adjointness of differential operators, and exact mathematical identities ofthe vector and tensor calculus. This article is the first comprehensive review of the 50-yearlong history of the mimetic methodology and describes in a systematic way the majormimetic ideas and their relevance to academic and real-life problems. The supportingapplications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynam-ics problems. The article provides enough details to build various discrete operators onunstructured polygonal and polyhedral meshes and summarizes the major convergenceresults for the mimetic approximations. Most of these theoretical results, which are pre-sented here as lemmas, propositions and theorems, are either original or an extension ofexisting results to a more general formulation using polyhedral meshes. Finally, flexibilityand extensibility of the mimetic methodology are shown by deriving higher-order approx-imations, enforcing discrete maximum principles for diffusion problems, and ensuring thenumerical stability for saddle-point systems.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

1.1. Design principles

Solving new and challenging problems with strong nonlinearities, discontinuities, and complex physical processes re-quire advances in the quality, accuracy and robustness of numerical algorithms. In fact, the predictions and insights gainedfrom simulations are no better than the physical models or mathematical methods used to solve them, and the numericalapproximation is often a determining factor for the reliability, accuracy, and efficiency of the simulations. Experience hasconfirmed that the best results are usually obtained when the discrete model preserves or mimics the underlying mathe-matical properties of the physical system. The objective of the mimetic finite difference (MFD) method is to create discreteapproximations that preserve important properties of continuum equations on general polygonal and polyhedral meshes.

Many algorithms used for a numerical simulation of physical problems solve discrete approximations of partial differen-tial equations (PDEs). Such PDEs are derived in the framework of a differential calculus and can be formulated in terms offirst-order differential operators that are coordinate invariant such as the gradient of a scalar or a vector, the divergence of avector or a tensor, and the curl of a vector. These PDEs express fundamental physical laws such as the conservation of mass,momentum, and total energy in fluid flows, or Faraday’s, Maxwell–Ampére’s, and Gauss’ laws in electromagnetism. Many

* Corresponding author.E-mail addresses: [email protected] (K. Lipnikov), [email protected] (G. Manzini), [email protected] (M. Shashkov).

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2 K. Lipnikov et al. / Journal of Computational Physics ••• (••••) •••–•••

of the important properties of these PDEs are better understood when the equations are reformulated by using first-orderdifferential operators. For example, the energy conservation in the gas dynamics and the coercivity of the diffusion operatorfollow immediately from the duality property of the divergence and gradient operators. To mimic these properties in adiscrete setting, a careful design of numerical methods is required. The MFD method is one of the successful examples of acompatible discretization technology.

The construction of a mimetic discretization starts with the choice of a discrete representation of scalar, vector and tensorfields on a computational mesh through suitable degrees of freedom. The degrees of freedom, whose choice is obviouslyproblem dependent, can be associated with various mesh objects, e.g., vertices, edges, faces and cells. A collection of thedegrees of freedom associated with similar mesh objects is called a grid function; hence, we have nodal grid functions, edgegrid functions, etc. The linear combinations of the grid functions of the same type (with the straightforward definition ofaddition and multiplication by a real number) form a linear space. It is worth mentioning that this correspondence betweenthe grid functions and the geometric objects forming the mesh has a very profound nature that may be well understoodusing concepts of algebraic topology, see the next subsection.

Next, we design the discrete operators that correspond to the first-order operators grad, curl, div and act on the gridfunctions. To such purpose, we first identify the connections between the most significant analytic properties of a mathe-matical model and the first-order differential operators in terms of which this model is formulated. The analytic property isoften an integral identity, which, in most cases, expresses a duality relationship between the first-order operators. Then, wechoose a discrete form for some first-order operators, which are dubbed as the primary operators (e.g., GRAD, CURL and

DIV), and construct the other operators, which are dubbed as the derived operators (e.g., GRAD, CURL and DIV), fromdiscrete analogs of the analytic properties:

GRAD = −DIV∗, CURL = CURL∗, DIV = −GRAD∗.The constructive use of discrete duality relations is one of the major design principles of the mimetic discretizations and

leads to the development of a discrete vector and tensor calculus (DVTC). The DVTC deals with grid functions and primary andderived operators. Note that the DVTC mimics only a subset of the analytic properties; therefore, it is not unique.

Let us illustrate the design concepts with a few examples. Regarding the degrees of freedom, we emphasize that theirchoice is problem dependent. In electromagnetics, it is natural to choose the normal component of the magnetic flux densityB to the mesh faces and the tangential component of the electric intensity field E along the mesh edges as a discreterepresentation of such fields. Indeed, these components of B and E are continuous at interfaces between different materials.In Lagrangian hydrodynamics, it is natural to choose the Cartesian components of the velocity u at the mesh nodes as adiscrete representation of u. Indeed, in the Lagrangian framework, the mesh nodes move with the fluid.

One of the most important analytic properties of many continuum equations is the conservation of the total energy.In fluid dynamics this property follows from the fact that the grad operator is the negative adjoint to the div operator. Inelectromagnetics, it follows from the self-adjointness of the curl operator. In the MFD method, this duality of the differentialoperators is transformed to the duality of the discrete operators. Another important analytic property is the geometricconservation law in fluid and solid dynamics. The mimetic framework ensures that the discrete divergence of the velocity isconsistent with the change of volume of a fluid parcel.

Often, it is not possible to formulate a set of primary and derived operators that mimic all the desired analytic properties.For example, in Lagrangian hydrodynamics, it is not possible to construct a discretization that simultaneously preservesenergy and entropy in smooth isentropic flows.

In addition to the conservation laws, there are other important analytic properties of the differential operators thatshould be preserved or mimicked in numerical schemes. One of them is the characterization of the null space (or kernel) of adifferential operator. Three popular examples include the exact identities: (a) div u = 0 if and only if u = curl v; (b) curl u =0 if and only if u = grad p; and (c) grad p = 0 if and only if p is a constant. When designing primary and derived operators,we pay great attention to their null spaces, e.g.

DIV CURL = 0, DIV CURL = 0.

If a discrete operator has a different characterization of its null space, nonphysical parasitic modes may pollute thenumerical solution. If the null space is too large, magnetic monopoles may appear in the electromagnetics. In the Lagrangiangas dynamics, a hourglass mesh motions may be induced. This latter phenomenon, which is responsible for numericalinstabilities, is also well-known in the finite element community and is related to the under-integration effect; hourglasspatterns can be also found in finite difference and finite volume calculations. On the other hand, if the null space is toosmall, the numerical approximation becomes stiff, a problem analogous to the well-known phenomenon of locking in thefinite elements.

The finite size of the mesh cells leads to another important consideration. While the PDEs can resolve all the scalesof the fluid motion, its numerical simulation may be more restricted. For example, in flows with high Reynolds numbers,the energy dissipation due to the molecular viscosity cannot be resolved. Since these effects of the physical viscosity arenot taken into account, we need an artificial mechanism to dissipate the correct amount of energy. In compressible flowswith shocks this mechanism is called the artificial viscosity. In fluids and gases, the forces due to the physical viscosityare isotropic. However, to effectively regularize a shock so that the flow does not depend on the computational mesh, the


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artificial viscous force needs to have the form of a (occasionally non-symmetric) second-order tensor. This is one of thereasons why we need to consider tensor quantities in the mimetic discretizations for hydrodynamics.

Another analytic property, important for verification of physical simulations, is the symmetry of physical flows. In theinertial confinement fusion program, the uncertainty of whether a non-symmetric result is due to numerical errors or thephysical design limit severely our predictive capability and ultimately our understanding of the dynamical behavior of animplosion. Here, a major numerical issue is that small violation of the spherical symmetry due to a discretization errorcan be amplified by the physics nonlinearity and produce unacceptably large asymmetries in systems undergoing strongconvergence. It turns out that numerical methods that preserve symmetries are viable for studying small perturbations ofthese symmetries. However, the development of such methods may require meshes with curvilinear edges (as opposed tostraight line segments) and the derivation of suitable discrete operators.

There are many other issues to consider that are related to the design of discrete mimetic operators. For example, acontinuum diffusion operator is symmetric and coercive. The discrete analog of this property is that the discrete diffusionoperator is a symmetric and positive definite (SPD) matrix. Such kind of matrices has important practical advantages: SPDmatrices are non-singular, thus providing well-posedness of the discrete problem, and the corresponding linear system canbe solved using efficient methods varying from a direct solver based on the Choleski factorization to the conjugate gradientmethod with a multigrid preconditioner. In the mimetic discretization, the discrete gradient and divergence operators arenegative adjoint to each other. This condition is sufficient to obtain a discrete problem with an SPD matrix.

1.2. A fifty-year long successful story

We would like to use the opportunity provided by this special issue of Journal of Computational Physics to present abrief history of the MFD method. This opportunity is especially important because the early history of this method is closelyrelated to the work that was carried out in the Soviet Union and that is not well known in the West. The references that weconsider herein are representative and by no means these historical notes pretend to be complete and objective. It is morefair to say that these notes represent the history of how the Authors became aware of mimetic/compatible discretizationsand of Author’s personal involvement in their development.

The development of mimetic discretizations can be divided approximately into four periods. The first period begins in themid-fifties. The main characteristics of the first period are:

(i) the development of the first numerical models based on discrete operators that preserve important properties of thecorresponding differential operators;

(ii) the use of orthogonal meshes, where the construction of such discrete operators is relatively simple;(iii) the use of a compatibility property of such operators to prove stability and convergence of numerical methods.

We emphasize that the common strategy adopted in this period is to build the discrete analogs of the differential operatorsindependently, and only then to prove that they satisfy an integral identity or a duality relation. To the best of our knowl-edge, the seminal paper [258] is one of the earliest work based on the concept of discrete analogs of differential operators,that satisfy discrete analogs of integral relations on orthogonal meshes. These discrete operators are suitable to derive finitedifference schemes and their mimetic properties can be used to prove the stability and convergence of such schemes. Themost comprehensive presentation of this theory is in [229,230,233,234]; see also [231], the English translation of [230].

Importance and effectiveness of compatible discretizations of differential operators have been recognized and clearly ar-ticulated in the series of papers [177–179]. There, the author introduces difference analogs of the basic operators grad, curl,and div on uniform orthogonal meshes and proves discrete versions of some fundamental relations of calculus, includingthe orthogonal decomposition theorem. Due to the simplicity of the orthogonal meshes, the discrete operators are definedthrough finite difference approximations of the first derivatives. For the Laplace equation and elliptic equations with discon-tinuous coefficients, the author proves stability and convergence of the finite difference discretizations based on the deriveddiscrete operators. Similar ideas are also used in [171] to build a discrete model for the Navier–Stokes equations in a streamfunction formulation. This discrete model satisfies a law of energy dissipation similar to the one in the continuum case.

In [172] we find a different approach based on algebraic topology. The differential equations are represented usingexterior differential forms and discrete analogs of the exterior derivative and the Hodge ∗ operator are constructed. Thisapproach is applied to the Laplace equation, the biharmonic equation, the Lame’s equations for isotropic linear elasticity,and the steady Maxwell’s equations. More details for Lame’s equations are also given in [165].

In a distinct series of paper [88–91], which are also based on the algebraic topology, discrete models of PDEs are de-veloped on orthogonal meshes. In this work, the square mesh on a plane is interpreted as a topological complex. Thecoboundary and boundary operators acting on functions of the complex and defined by the combinatorial structure, gen-erate the difference analogs of the classical differential operators of mathematical physics, such as grad, curl, div, andLaplacian. Furthermore, a discrete model for the steady Euler equations is proposed in [89]. In view of the quasi-linearity ofthese equations, it becomes necessary to introduce a suitable product between discrete differential forms; to this purpose,the Whitney product [272] is chosen. Detailed description of this approach to the construction of discrete models is in thebook of Ref. [92].



In this same period, we find a few important papers in the West that introduce elements of the mimetic methodology.In [262], strong relationships between some quantities of physical theories and basic geometric and chronometric objectsare investigated. This study leads to a classification of physical theories, where the equations of physics can be describedby a single mathematical operator, the coboundary operator, which is the exterior derivative on co-chains. In [94], a finitedifference method on simplicial meshes based on Whitney forms and a discrete Hodge theory is developed. In [274] anumerical scheme is proposed for solving time-dependent Maxwell’s equations on rectangular meshes using edge unknownsas degrees of freedom for the electric field and face unknowns for the magnetic field. This work is the foundation of anentire class of methods for computational electromagnetics, cf. [254], the finite difference time domain (FDTD) methods. In [10,11,227] mimetic methods for shallow water equations and climate modeling that preserve mass, potential enstrophy andvorticity on logically rectangular meshes are proposed. Mimetic methods with similar properties on triangular meshes arealso found in [221] and in [45]. In [122] a finite difference scheme is proposed for second-order elliptic Dirichlet boundaryvalue problems on irregular networks with the topological structure of a logically rectangular mesh. This scheme uses adiscrete divergence and a discrete gradient that can be shown to be in a duality relation. The optimal rate of convergencein a discrete energy-like norm is proved. We also mention the numerical approach proposed in [224], which preservesmass, potential enstrophy and energy preserved on hexagonal geodesic meshes, and [4], which proposes a mimetic finitedifference discretization for the incompressible Navier–Stokes equations.

It turns out that these properties are fundamental requirements for a long-term numerical integration of the equationsof incompressible fluid motion.

The second period in the development of the mimetic discretizations begins approximately in the mid-seventies. Thenew research trend is motivated by the necessity of solving more complicated PDEs with discontinuous coefficients onnon-orthogonal meshes. These issues arise naturally in modeling physical problems like the Inertial Confinement Fusion[215,266], Tokamak [156], high velocity impact dynamics [276], and shape charges [269]. These models involve domainswith complex shapes, several coupled physical processes including gas dynamics, heat conduction, and electromagnetism,and the Lagrangian approach where the mesh moves with fluid. The second period is mainly characterized as follows:

(i) the derivation of compatible discrete operators based on variational principles and integral identities and not carriedout independently for each operator as in the first period;

(ii) the choice of the local components (with respect to the mesh) of the vector variables used as degrees of freedom;(iii) conservative staggered discretizations (using cell and nodal grid functions) of the equations of Lagrangian hydrodynam-

ics.

At the beginning of this period, different variational principles are used to construct systems of mimetic operators. Oneoperator is identified as the primary operator and discretized directly; another operator is constructed through a discreteversion of a variational principle and called the derived operator. The use of variational principles in construction of mimeticdiscretizations is summarized in the books of Refs. [232,261]. This approach is successfully applied to the heat conductionequation [107,260] and the diffusion equations for the magnetic field [106,111,164].

In these papers and books, the components of the vector variables are used as degrees of freedom. For example, theheat flux and the magnetic flux density B are represented by their normal components to the mesh faces because thesecomponents are continuous across the material interfaces. Likewise, the electric field intensity E is represented by its tan-gential components on the mesh edges because these components are also continuous. For such degrees of freedom, adiscretization of integrals in a variational principle becomes a non-trivial task and leads to the development of the mimeticinner products. These inner products use discrete representations (for example, Eh and Eh) of continuum vector functions(respectively, E and E) to provide accurate approximations of integrals:

[Eh, Eh]E =∫Ω

E · E dV + O(hp) ∀Eh, Eh ∈ Eh,

where h is the characteristic mesh size and p the order of approximation. Note that similar ideas appear in the sameyears in the finite element community. These ideas are related to the development of mixed finite elements for elliptic andMaxwell’s equations, cf. [207,223].

The work of Ref. [270] introduces another approach to the discretization of the Maxwell’s equations, the finite integrationtechnique (FIT), which uses the primary mesh for the discretization of Faraday’s induction law and a dual mesh for thediscretization of the Maxwell–Ampére’s law. An interpolation of the electric field E and the magnetic field H between themeshes is needed to discretize the constitutive relations D = εE and B = μH, where ε is the electric permittivity and μ isthe magnetic permeability of the medium. Only later, it was recognized that the interpolation must satisfy special propertiesfor the method to be stable [76]. It is pertinent to note that the mimetic discretizations developed in [106,107,111,164,260]do not use a dual mesh. For further historical connections to the mixed finite element and other methods, we refer thereader to [148]; see also Section 3.1.2.

As it has been mentioned above, algebraic topology provides a natural framework to describe discrete structures. Byapplying it to the electromagnetics (see, for example, the book of Ref. [49] and the references therein) formal mathematicalstructures can be introduced such as edge elements and facet elements. Such structures correspond to the mimetic discretiza-tions of the electric and magnetic fields. The main difficulty here is in the construction of consistent adjoint operators or,



using the terminology of the algebraic topology, the discrete analogs of the Hodge ∗ operator (compare this issue with theinterpolation issue in the FIT method). Some contributions regarding this issue are in [109]. The application of the algebraictopology to the construction of a discrete analog of the metric conjugacy operator ∗ on general grids requires a complexset of mathematical tools. Moreover, this approach is natural for specific discretizations of vector fields and cannot be easilyextended to many popular discretizations including those using the nodal Cartesian components of the vector fields.

The variational principles are also actively used to construct conservative finite difference methods on staggered gridsfor gas dynamics, magneto-hydrodynamics and dynamics of deformable media, see Refs. [110,125–127,259] and Sections 6and 7 for more details.

It is worth noting that in this period the variational principles are used only in the construction of derived operatorsfrom given primary operators. This fact is reflected in several very important papers, e.g., [108,160,228,235,236] that can betruly considered as the beginning of the systematic development of the MFD method. In these papers, the design principlesfor the MFD method described in Section 1.1 are already clearly formulated. In particular, we find the notions of discretespaces equipped with inner products, primary operators, constructive way of using duality relationships to build derivedoperators, and the connection of the duality properties and related integral identities with the desired properties for adiscrete model.

In this period, these methods are not called mimetic. The closest translation from Russian into English is “support operatormethod”. This translation does not make much sense, besides the fact that the discrete operators support the derivation ofnumerical schemes for PDEs. Most of these papers were translated into English by publishers and the method was givenother names such as “basic operators” and “reference operators”.

Subsequent publications, that we list in almost chronological order, demonstrate a wide use of the mimetic approach.Axisymmetric difference operators in orthogonal coordinate systems are constructed in [162,163]. Mimetic discrete operatorsfor the Voronoi meshes are developed in [243]. Mimetic discretizations for elliptic equations on non-matching grid aredeveloped in [245]. Mimetic discretizations for Maxwell’s equations in the cylindrical geometry on an orthogonal grid aredeveloped in [93]. The approach is also extended to the heat conduction equation on the Voronoi/Dirichlet meshes [244]and to gas dynamics equations on the Voronoi meshes in the framework of free-Lagrangian methods [202,238,242]. Mimeticapproach is used for the biharmonic equation in [247].

During this period, various publications are focused on the analysis of stability and convergence properties of the mimeticdiscretizations, cf. [12–14,86,124,226]. In most of these papers, the stability and convergence results are proved in the energynorms induced by the mimetic inner products.

Mimetic discretizations are also applied to practically important problems. We mention a few representative papers:Navier–Stokes equations on the Voronoi meshes, [9]; static problems of elasticity, [161]; modeling of the Rayleigh–Taylorinstability, [118]; modeling compression of a toroidal plasma by the quasi-spherical liner, [116,117]; modeling of a controlledlaser fusion, [267]; computer simulations of an over-compressed detonation wave in a conic canal, [157]; simulation ofa magnetic field in a spiral band reel, [18,61]; calculation of viscous incompressible liquid flow with a free surface ontwo-dimensional Lagrangian meshes, [85]; modeling of a microwave plasma generator, [192]; simulation of the collapse ofa quasi-spherical target in a hard cone, [255].

The design principles of the mimetic discretizations developed during the second period are summarized in the bookreferenced in [237]. There, the author applies the support operator method to the construction of mimetic methods forelliptic and parabolic equations as well as to the Lagrangian gas dynamics. Only nodal and cell-centered discretizations ofvector and scalar functions are considered in this book. The book is accompanied by a computer disk containing examplesof codes implementing the mimetic schemes. Note also publication [239] where the support operator method is used forthe discretization of elliptic problems.

Several papers published in the second period study a different approach to mimetic discretizations; namely, compatiblediscretizations for the Lagrangian hydrodynamics, cf. [153] and [62,63]. In these papers, the differential operators are notapproximated directly, but rather compatible discretizations for the momentum and internal energy equations are derivedthrough a balance of the kinetic and internal energy that conserves the total energy. This approach, although specific to thehydrodynamics equations, is quite general. It can be applied to the case where forces of arbitrary nature (not just forcesgenerated by differential operators) are present in the momentum equation, see Section 6.6 for more details.

Finally, we mention other papers published during this period that contain mimetic ideas: [209–212,225] and [197,198].In particular, the work of Ref. [197] emphasizes the fact that a discretization of the divergence operator has to be consistentwith the change of volume of the computational cell. The same idea is used to construct a mimetic discretization in [126].

The third period in the development of the mimetic discretizations begins approximately in the mid-nineties. The maincharacteristics of the third period are:

(i) the beginning of a systematic development of the mathematical foundation of the mimetic discretizations and theDVTC;

(ii) the extension of the mimetic approach to more general meshes including polygonal, locally refined and non-matchingmeshes;

(iii) an extensive and careful testing of mimetic discretizations for many different problems.



The systematic development of the mathematical foundation for the DVTC begins in three seminal papers [145–147].In [146], natural discrete analogs (primary mimetic operators) for grad, curl and div on logically rectangular grids areconstructed. Discrete analogs of several important theorems of the continuum calculus are also proved, as, for example,that div A = 0 if and only if A = curl B. The internal structure of the primary operators is described in terms of primitivedifference and metric operators. In this paper, the terminology mimetic difference operators and mimetic discretizations isadopted for the first time, although the word mimetic had already been used in the unpublished report [144].

The discrete dual operators (the derived mimetic operators) corresponding to the primary operators are constructed in[145]. The construction of the derived operators is based on the duality principle, e.g.

[uh, GRAD ph]F = −[DIV uh,ph]C, ∀uh ∈ Fh, ph ∈ Ch,

where Fh and Ch are discrete spaces for face and cell grid functions, respectively. The inner products in discrete spaces areintroduced and corresponding SPD matrices are developed. The structure of the derived operators in terms of primitive dif-ference operators and the inner product matrices is described. The discrete analogs of major theorems of vector calculus arealso presented. The set of primary and derived discrete operators allows one to construct discrete analogs of second-orderoperators like div grad, grad div, curl curl, and the vector Laplace operator �� = grad div − curl curl, which are needed todiscretize various PDEs.

The discrete analogs of the Helmholtz orthogonal decomposition theorem for logically rectangular meshes for both faceand edge representation of vector fields is developed in [149]. Extension to general meshes is developed in Section 2.5. TheDVTC is used in [43] to construct a data transfer algorithm for divergence-free fields represented by normal components tomesh faces. Its extension to general meshes is described in Section 2.8. In [64], the mimetic approach is used to discretizethe divergence of a tensor and the gradient of a vector using two different representations of the tensor field via normalprojections to mesh faces and tangential projections to mesh edges.

In [147], the discrete operators are extended to the boundary by incorporating the boundary conditions into their def-inition. For example, on the boundary, the discrete divergence operator is equal to the normal component of its vectorargument. The inner products are to be modified to include the boundary terms, and using these new inner products, thederived gradient operator is still the negative adjoint of the (extended) divergence operator. This strategy allows us to dis-cretize Neumann and Robin boundary conditions for elliptic equations in a natural way using the framework of mimeticdiscretizations.

The mimetic inner product is usually not unique. In [151], the impact of the inner product on the accuracy of the MFDmethod is investigated. Two inner products, which correspond to different interpolations of a vector field inside a mesh cell,are compared. It is shown that the absolute error is two-three times smaller when the interpolation is based on the Piolatransformation compared to the piecewise constant interpolation. This work is the first analysis of optimal lifting operators,see the next period. It is pertinent to note that the non-uniqueness of the mimetic inner product is also analyzed in [208]and [265] to develop a unified formulation for the covolume method and the support operator method in two dimensions.

Another important paper of this period is found in Ref. [189]. There, the equations for the mimetic inner product matrixare derived from accuracy considerations, in particular, from the requirement that the discrete gradient be exact for linearfunctions. A solution to this problem is proposed for triangular meshes.

The mimetic inner products for vector functions developed in the aforementioned papers are not suitable to degeneratequadrilateral cells, i.e. cells with 180◦ angle between two edges or cells having edges with zero length, non-convex cellsand cells with hanging nodes (degenerate polygons). In [173], a new approach for the construction of inner products forgeneral polygonal cells is proposed, which is extended to the mimetic setting the mixed finite element method for polygonaland polyhedral meshes for Darcy flow calculations proposed in [174,175]. Each cell is subdivided into triangles and newtemporary unknowns are introduced on internal edges. Then, the standard mimetic inner product is defined for each triangleand, finally, the temporary unknowns are eliminated using two conditions: the discrete divergence is constant in the cell andthe inner product satisfies the stability condition (S2), see Section 2. This approach works for arbitrarily-shaped polygons.Moreover, the inner product depends continuously on the shape of the cell, for example, this is the case when a quadrilateraldegenerates to a triangle. The same construction is used in [187] for arbitrary polyhedral meshes.

A conceptually new development of the mimetic discretizations is the introduction of the local support operator methodfor diffusion problems, cf. [203], where both cell and face unknowns are used to represent the scalar variable. This ap-proach makes it possible to reformulate the discrete problem in order to use efficient algebraic solvers for the resultinglinear systems. The new approach is used to construct mimetic discretizations on triangular meshes [115], meshes withlocal refinement [183], and non-matching meshes [38]. High-order mimetic discretizations, which use a wider stencil, aredeveloped in [73,74].

During the third period convergence results for the mimetic discretizations are obtained in [36–38,152] for diffusionproblems. The second-order convergence (superconvergence) of the vector variable on smooth meshes is proved in [37].A mortar technique for mimetic discretizations on non-matching meshes is developed and analyzed in [38].

In this period, the mimetic discretizations are applied to a wide range of problem: diffusion equations with stronglydiscontinuous anisotropic coefficients, [141,143,240]; Maxwell’s equations and equations of a magnetic diffusion, [148,150];equations of the Lagrangian hydrodynamics on general polygonal meshes, [66], including an artificial viscosity, [65]; equa-tions of a solid dynamics and shallow water equations, [196]; and the Lagrangian hydrodynamics on curvilinear logicallyrectangular meshes preserving spatial symmetries, [195].



The foundation for the systematic development of conservative compatible/mimetic discretizations based on a balance ofthe kinetic and internal energy is found in [70–72] (see also Section 6.6). The reader may also be interested in the reviewpaper [159] where some other mimetic properties of numerical algorithms are discussed, as well as in the paper [221]where conservation properties of unstructured staggered discretizations are discussed.

The fourth period in the development of the mimetic discretizations begins after the IMA meeting in 2004 [96]. It isbased on the collaboration between the Los Alamos National Laboratory, USA, and a research group in Pavia, Italy. The maincharacteristics of this period are:

(i) the development of new mathematical tools for the design of mimetic discretizations of various PDEs and their conver-gence analysis;

(ii) the development of a rich parametric family of mimetic discretizations that includes many other discretization methodsas particular members;

(iii) the application of the new technology to a wide range of PDEs, the development of arbitrary-order discretizations forelliptic problems, the analysis of stability and of discrete maximum principles.

The set of new mathematical tools introduced in [55] form the foundation for a rigorous convergence theory for themimetic discretizations. The subsequent papers [57,58] develop a new approach to the construction of derived mimeticoperators on arbitrary polyhedral meshes. The key component of this construction is an accurate mimetic inner product.This inner product is built to satisfy the consistency and stability conditions that enforce the optimal convergence rateof discrete solutions and lead to independent cell-based algebraic problems. Such construction is easy to implement in acomputer code. A strategy for a systematic development of mimetic inner products for cochain spaces is discussed in [51,53].

The consistency and stability conditions have already appeared in a different form in [189], but the new approachresults in a number of important developments that are more transformational than incremental. First, the new consistencycondition can be formulated for non-convex polygonal and polyhedral cells, including cells with non-planar faces [56].Second, the consistency and stability conditions do not determine a single scheme but an entire family of schemes. Allmembers of this family share common properties such as accuracy and stability and have the same stencil size for thederived operators. Third, such family of schemes contains many well-known finite volume and finite element methods.

In [56,57], the new technology is used to develop and analyze a mimetic discretization for generalized polyhedral mesheshaving strongly non-flat mesh faces. In [185], it is applied to build a mimetic discretization for equations of the magneticdiffusion in the axisymmetric cylindrical geometry. This scheme provides an accurate treatment of the B/r2 term near theaxis of symmetry r = 0 and leads to a consistent calculation of the Joule heating on strongly distorted meshes.

The families of mimetic schemes are analyzed in [132,181,182] and sub-families of schemes with additional propertiesare found. The mimetic schemes satisfying a discrete maximum principle for diffusion problems are described in [181,182]for a class of two-dimensional and three-dimensional meshes. In [132], a new mimetic scheme for a well-studied acousticsequation is developed. This scheme has the complexity of roughly two second-order schemes but shows fourth-order nu-merical dispersion and sixth-order numerical anisotropy. The last property has never been reported for other state-of-the-artfourth-order schemes.

The mathematical tools for building accurate mimetic inner products have been extended to semi-inner products rep-resenting the energy norm. This extension allowed us to build new mimetic discretizations for primary formulations ofsecond-order PDEs. A nodal mimetic discretization on polygonal and polyhedral meshes for elliptic problems is developedin [52], where the optimal convergence estimate in the energy norm is proved. Later, this technology has been extended toReissner–Mindlin plates [33] and the Stokes equations [26,29].

It turns out that to build higher-order mimetic discretizations, it is sufficient to add more degrees of freedom andenforce a stronger consistency condition. The arbitrary-order mimetic discretizations of diffusion problems are developedand analyzed in [30]. In [24], this approach has been recasted as the virtual element method (VEM). The VEM is a finiteelement method where the discrete spaces are virtual in the sense that they are not build explicitly but are characterizedby their properties. The practical implementation of the VEM is based on the mimetic inner products.

The new mimetic discretizations have demonstrated their efficiency in solving convection–diffusion problems in bothdiffusive [69] and convection-dominated regime [25], eigenvalue problems in mixed form [67], mixed formulation of a linearelasticity [23], obstacle problem [8], modeling of biological suspensions [133], and modeling of flows in porous media [184].

A residual-based a posteriori estimators for mimetic discretizations have been developed in [7,22,31]. The estimators usea post-processing technique [68] and can be applied to an adaptive solution of elliptic problems.

In the fourth period, development, analysis, and application of the mimetic discretizations have been done in variousresearch groups in Europe and US: subsurface flows on corner-point meshes, [1]; development of mimetic discretizationsbased on a discrete calculus for fluid dynamics [154,222,249], geophysical flows, [4,45,224,257]; oil reservoir simulations[5,128,241]; seismic wave propagation on multi-GPU system [246]; viscoelastic wave modeling and rupture dynamics, [119,120]; poroelasticity problems [206]; electromagnetics, [17,190,191]; plasma physics, [220]; astrophysics, [205]; pharmaceu-tical science, [78]; general relativity, [20]; and image processing, [21]. Furthermore, a systematic comparison with othernumerical methods for solving 2D and 3D elliptic problems with strongly anisotropic diffusion tensors was carried out andpresented in the conference benchmarks [105,135].



1.3. Alternative approaches to compatible discretization

The idea of incorporating properties of the continuum calculus in the design of numerical schemes appears in variousmethods. In a series of articles published in the seventies (see, e.g., [262,263] and the references therein), it was observedthat many physical theories have a very similar formal structure from the geometrical, algebraic and analytic standpoints.For example, balance equations, continuity equations, equations of motion, and circuital equations state that one physicalquantity defined on a d-dimensional manifold is equal to another physical quantity defined on its boundary. The equationscan be reformulated in a finite framework using basic concepts from the algebraic topology such as fully discrete functions(cochains) defined on combination of grid objects (chains) rather than functions in the continuum. Going further along thisdirection, it is possible to establish a set of direct algebraic relations among geometrically-based physical variables thatis suitable to numerical applications, e.g., the cell method (CM) [199,264]. Although the CM is derived directly from theexperimental laws, thus avoiding a discretization of differential equations, its mimetic nature is evident per se. The CM isconsistent by design; however, it may result in a non-symmetric discretization for a symmetric problem. Moreover, since thestability condition is not one of the design principles, the CM may lead to an unstable discretization on a strongly distortedmesh.

A unified computational model is proposed in [75,218,219] to make a bridge between the geometry and the physicalbehavior of engineering systems. This model uses differential k-forms and their discrete representation through k-cochainsover a cell complex, a finite approximation to a manifold which abstracts only its topological properties, and the coboundaryoperator acting on cochains to represent a geometry-based differentiation process. It turns out that only a small set of theusual combinatorial operators, e.g., boundary, coboundary, and dualization, are sufficient to represent a variety of physicallaws and invariants. Cochains as a numerical discretization mechanism and connection with a finite element analysis arealso investigated in [218].

The covolume method [211] is another example of a compatible discretization method. This method was originally de-veloped for the numerical approximation of planar div–curl systems and was extended later to three-dimensional systems[212], the Navier–Stokes equations, and the Maxwell’s equations [209]. It can be viewed as a significant extension of Yee’smethod to simplicial meshes, and thus can be applied to complex geometries. Like the FDTD, the covolume method requirestwo orthogonal meshes to approximate the electric and magnetic fields. This is one of its major features but also its majorlimitation. To this purpose, the Delaunay triangulation and the corresponding Voronoi diagram are the natural choice. Everyedge of the Voronoi mesh is orthogonal to the corresponding face of the Delaunay triangulation, and viceversa. The covol-ume and MFD method use the same primary operators. However, the construction of the dual operators in the covolumemethod relies strongly on the orthogonality property of the Delaunay and Voronoi meshes.

Mimetic ideas are also found in [216,217], where finite difference approximations of differential operators on logicallyrectangular grids and weighted inner products are designed so that a summation by parts formula mimicking the integrationby parts holds. The analogy between the discrete and the continuum calculus is rather strong, as stability estimates for thesefinite difference schemes are obtained following the argument used for continuum initial and boundary value problems, in-cluding hyperbolic, parabolic, and mixed hyperbolic–parabolic systems. Further developments are found in [201,252], wherethe Euler equations are solved using an energy-stable scheme based on the fifth-order summation-by-parts operators.

The finite volume (FV) method, originally introduced in [99,100] for the heat equation and dubbed as the integratedfinite difference method, forms, perhaps, the largest class of schemes that can handle unstructured polygonal and polyhedralmeshes, nonlinear problems, and problems with anisotropic coefficients. These schemes are mimetic in the sense that theyenforce the balance equations for mass, momentum, and energy on each mesh cell. The discrete operators in the balanceequations coincide with the primary mimetic operators; e.g., the balance of fluxes corresponds to the primary divergenceoperator acting on face grid functions. The essential difference between the FV and MFD methods is in the approximationof constitutive laws. We mention here only those developments of the FV method that use duality arguments for that.

An FV approximation of the convection–diffusion equation based on a two-point flux formula is proposed in [134]. Theformulation of this scheme requires a special mesh such as the Voronoi tessellation. To overcome this limitation, a class ofFV methods, consistent by design, is proposed by introducing additional unknowns on mesh faces. Examples of such meth-ods include the hybrid finite volume method [103,104], and the mixed finite volume method [97]. These FV methods introducestabilization terms that can be connected with the stability condition of the mimetic finite difference method, see [98].A different class of FV methods is introduced by using a diamond gradient formula and suitable vertex-centered recon-structions [39–41,82,113,114,193,194]. The discrete divergence and the discrete gradient operators in these latter schemessatisfies a duality relation only approximately. Instead, an exact duality relation is satisfied by the discrete duality finite vol-ume (DDFV) method for two-dimensional diffusion problems in [95,136,137]. The resulting scheme combines two distinctFV schemes on two overlapping meshes, the primal mesh where the diamond scheme is formulated, and a dual mesh ofcontrol volumes built around the vertices of the primal mesh. The method can be reformulated in the framework of mimeticdiscretizations on staggered grids by introducing discrete divergence and gradient operators which are related by a discreteanalog of the integration by parts formula through two inner products defined on the overlapping meshes [81]. A general-ization to nonlinear elliptic equations is found in [6], to Stokes equations in [169,170], and to the div–curl problems in [84].A generalization of the DDFV method to three-dimensional problems is feasible, cf. [80]; however, the scheme formulationis quite complex as it requires three three-dimensional overlapping meshes and one additional mesh of diamond cells.



Another approach to overcome the limitations induced by the two-point flux formula comes from the multi-point fluxapproximation (MPFA) method [2,3] and similar, but developed independently, the control-volume distributed (CVD) method[101,102]. In the MPFA and CVD methods, fluxes on mesh interfaces having a common point are defined simultaneouslyfrom local consistency and continuity conditions. On general meshes, these methods produce non-symmetric schemes forsymmetric problems. A lack of a stability condition may result in numerical instabilities on strongly distorted meshes. TheMPFA method can be reformulated using the mixed finite element framework as in [271] or the mimetic framework withinner products induced by non-symmetric matrices as in [188]. The latter approach is also used to analyze convergence ofthe MPFA method in [158].

The mixed finite element (MFE) method is, perhaps, the most developed compatible discretization framework, mainly onsimplicial meshes. An overview of this method is well beyond the scope of this paper, and for this reason we just refer tothe fundamental book [54], the most recent overview provided in the book [44], and the references therein.

Although not related directly to the mimetic concepts, a wide literature has been developed in the last decade to gen-eralize the finite element method to polygonal meshes. We mention the seminal paper [268] and the most recent papers[34,83,112,204,250,251,253].

Although the reformulation of the mimetic discretizations in the framework of differential forms is beyond the scope ofthis review paper, it is worth to mention some important works in this direction. Using topological concepts, strong simi-larities between numerical methods of very different nature, such as finite volumes, finite differences, and finite elementsare outlined in [200]. The connection between the Whitney forms and the MFEs (Nedelec elements) and its application tocomputational electromagnetics are explored in a series of papers published in the nineties, cf. [46–48] and the referencestherein, and summarized in the book of Ref. [49]. An important review of basic concepts of the mimetic discretizationsand their relations with notions from the algebraic topology is found in Ref. [42]. Finite element techniques have beenrecently recasted in the framework of the Whitney forms and formalized in the finite element exterior calculus, cf. [15,16].In this respect, we also mention the work in [138,139] and the extensions proposed in [59,60]. The discrete exterior calculus[87,140] makes it possible to reproduce some well-established finite difference and finite volume methods using unifyingnotation of the differential forms. Extensions of the FDTD and finite element time domain (FETD) methods for solving tran-sient Maxwell’s equations in complex media are reviewed in [256]. A similar approach based on differential forms, algebraictopological cochains and orthogonal polynomials is also considered in [50,121,166–168], where a mimetic spectral elementmethod is developed for the discretization of the Poisson equation on curvilinear grids.

1.4. Annotated content

The paper outline is as follows. In Section 2, we introduce the basic elements of the discrete vector and tensor calculuson general polyhedral meshes. Using first principles, we derive three primary discrete operators GRAD, CURL, and DIV .

Using duality arguments, we construct the three derived operators DIV , CURL, and GRAD, which combined with theprimary operators satisfy discrete exact identities and discrete Helmholtz decomposition theorems. Similar results wereobtained for two-dimensional logically rectangular meshes in [145,146,149]. Here, they are extended to general unstructuredpolygonal and polyhedral meshes. Then, we describe an extension of the DVTC on manifolds and its application to theproblem of data transfer between unstructured meshes for a divergence-free vector field. These results are an extensionof the algorithm proposed in [43] for logically rectangular meshes. At the end of this section, we give a short descriptionof mimetic discretizations of the divergence of a tensor and the gradient of a vector, which extends the work in [64] tounstructured polyhedral meshes.

In Section 3, we summarize a few important applications of the mimetic methodology to the discretization of classical

PDEs. We start with the diffusion equation that uses DIV and GRAD operators, derive mimetic inner products, andpresent a priory convergence estimates. The presentation is based on [55,58]. Next, we describe mimetic discretizations

for Maxwell’s equations, equations of magnetic diffusion, and magnetostatics that use CURL and CURL operators. Thepresentation is based on [148,150,180]. At the end of this section, we present a mimetic discretization of the diffusionequation that uses GRAD and DIV operators. We also present mimetic discretizations for the Stokes and linear elasticityequations and equations that use mimetic semi-inner products. The presentation is based on [26,29,52].

In Section 4, we show how to analyze the family of mimetic discretizations to find a scheme with additional desiredproperties such as the discrete maximum principles for elliptic problems and a reduced numerical dispersion and anisotropyfor acoustic wave propagation problems. The presentation is based on [130,131,181].

In Section 5, we illustrate the flexibility of the mimetic methodology with two examples. First, the stability of the MFDdiscretization for saddle-point problems is ensured by adding new degrees of freedom only where it is needed on the mesh.Second, high-order mimetic discretizations are built by enforcing the consistency condition for a larger polynomial space.The presentation is based on [27,28].

In Section 6, we present the mimetic/compatible discretizations for equations of the Lagrangian hydrodynamics on two-dimensional logically rectangular meshes. Except for Section 6.7, these discretizations are based on a different set of primaryand derived operators. We introduce the notion of a general compatible discretization, specific for the Lagrangian hydrody-namics, where discrete forces in the momentum equation are defined in a compatible way to conserve the total energy. Wealso describe a tensor artificial viscosity needed for modeling flows with shocks. Finally, we present a mimetic discretiza-



tion in the axisymmetric cylindrical geometry that uses two-dimensional meshes with curvilinear edges to preserve planar,cylindrical and spherical symmetries of flow. The presentation is based on [66,186,195,196,228,237].

In Section 7, we present a mimetic discretization for the Lagrangian solid dynamics in the axisymmetric cylindricalgeometry, which requires the construction of a new set of discrete operators such as the divergence of a vector, two distinctdefinitions of the gradient of a scalar, the divergence of a tensor and the gradient of a vector. These discrete operators arederived sequentially from each other which guarantees various discrete exact identities. The presentation is based on [66,186,195,196,228,237].

In Section 8, we summarize major ideas on mimetic discretizations and discuss the possible directions for the futureresearch.

2. Elements of a discrete vector and tensor calculus

A discrete vector and tensor calculus (DVTC) is in the core of the mimetic discretization technology. It allows us toreproduce various exact identities in a discrete framework, e.g., the Helmholtz decomposition theorems (see Section 2.5),and to preserves symmetry and positivity of discrete operators. The DVTC directly deals with discrete functions and discreteoperators and can be build in a very abstract form that exploits only the duality relationship and the properties of the innerproducts. On its turn, the convergence analysis imposes a few additional limitations on the admissible inner products butleaves enough freedom to vary them for various benefits.

The definition of the primary operators of the DVTC follows naturally from the coordinate invariant definitions of gradi-ent, divergence, and curl operators and the use of the degrees of freedom associated with various mesh objects. A differentset of primary and derived operators will be considered in Section 6.

The coordinate invariant definitions of the first-order operators are given by various forms and modifications of theStokes formula:

xb∫xa

∂ p

∂τdL = p(xa) − p(xb), (2.1)

∫S

(curl u) · n dS =∮∂ S

u · τ dL, (2.2)

∫V

div u dV =∮∂V

u · n dS, (2.3)

where τ is the unit vector tangent to either a curve connecting points xa and xb or a polygonal boundary ∂ S , ∂ p/∂τ is thedirectional derivative of the scalar field p along such curves, and n is the unit normal vector to the surface S or ∂V . Thesecond formula requires the proper orientation of normal and tangent vectors (see below). Hereafter, we use bold symbolsto indicate a vector function or a tensor.

The first-order operators satisfy the duality relations that are given by various Green’s formulas (integration by partsformulas):∫

Ω

p div u dV = −∫Ω

(grad p) · u dV +∮

∂Ω

p(u · n)dS, (2.4)

∫Ω

u · divσ dV = −∫Ω

(grad u) · σ dV +∮

∂Ω

u · (σ · n)dS, (2.5)

∫Ω

u · (curl v)dV =∫Ω

(curl u) · v dV +∮

∂Ω

p(u × v) · n dS. (2.6)

The DVTC preserves the duality property by deriving one of the discrete operators from first principles (2.1)–(2.3) (theprimary operator) and the other one by using a discrete analog of an integration by parts formula (the derived operator).

PDEs have often variables coefficients. In the mimetic approach, the coefficients are embedded in the definition of thederived operator. Consider, for examples, an equivalent form of formula (2.4):∫

Ω

p div u dV = −∫Ω

K−1(K grad p) · u dV +∮

∂Ω

p (u · n)dS, (2.7)

where K is a positive definite tensor. This formula represents the duality between the two first operators div and K gradusing a weighted inner product for the vector fields (with K−1 as weight). The corresponding discrete operators will be in aduality relation with respect to such inner product.



Fig. 1. Convex and non-convex cells usable by the MFD method.

Remark 2.1. To simplify the presentation, we consider the Green’s formulas in functional spaces where the boundary in-tegrals in (2.4)–(2.7) are zeros. This implies indirectly homogeneous essential boundary conditions in the underlying PDEs.The treatment of heterogeneous boundary conditions can be done by extending the first-order operators to the boundaryand building a new DTVC, see [147] for more details. Other solutions are also possible, see Section 3.

Let us consider two discrete spaces S , S� and a primary operator D acting from S onto S� , which, in practice, may begrad, curl and div. The (unique) injective operator D� acting from S� into S is defined implicitly via the duality relationship:[

D(u), v]

S� = [u,D�(v)]

S , ∀u ∈ S, v ∈ S�, (2.8)

where the brackets denote inner products (possibly weighted by problem coefficients) in the aforementioned discrete spaces.By taking v =D(u) we obtain the inequality:[

D�(D(u)

), u]

S = [D(u),D(u)]

S� � 0, ∀u ∈ S. (2.9)

By properly choosing the discrete spaces S and S� and the primary operator D, we always get a consistent derived oper-ator D� such that (2.9) holds. This inequality leads to the symmetry and positivity of the arising discrete system if theseproperties exist in a continuum problem.

2.1. Polyhedral meshes and mesh sets

Let Ω be the computational domain where the PDEs are defined. Without loss of generality, we assume that Ω is apolyhedral domain in 3D (a polygonal domain in 2D) with a Lipschitz continuous boundary. More complex geometries canbe approximated accurately using a polyhedral domain. Let us consider an unstructured mesh Th with non-overlappingpolyhedral (polygonal in 2D) cells c that partition Ω . We allow c to be a non-convex element with volume |c|, see, e.g.,Fig. 1. We denote the faces of c by f , its edges by e, and its vertices (also called nodes) by n. Let | f | and |e| denote thearea of f and length of e, respectively. In two dimensions, the faces coincide with the edges.

Let n f be a unit normal vector to face f fixed once and for all, and e be an edge vector with a priori fixed orientation.We assume that ‖e‖ = |e|, where ‖ · ‖ denotes the Euclidean norm of a vector. We also introduce the unit tangent vectorτ e = e/‖e‖ for edge e. Let xc be the centroid of cell c. Similarly, we define centroids x f and xe for face f and edge e,respectively. Finally, xn is the coordinate vector of node n.

Extension of the mimetic technology to meshes with non-flat faces is discussed in [55]. Such meshes appear often inmoving mesh methods such as Lagrangian and ALE methods. However, the theoretical analysis of the MFD methods restrictsthe set of admissible meshes by imposing the following mesh regularity assumption. Let N� and α� be mesh independentquantities. We consider the following mesh regularity assumption:

(M1) Each polyhedron can be split into at most N� tetrahedra (triangles in 2D) t that are shape regular, i.e. for every t thereexists an inscribed ball of radius rt such that

rt > α�diam(t).

This assumption allows us to use meshes with non-convex cells, but limits the number of admissible degenerate cells.For instance, cells with extremely small faces are excluded from the theoretical analysis; however, numerical experimentsindicate that these faces do not impact the convergence rate of the MFD methods. We emphasize that this assumption doesnot require an explicit construction of a tetrahedral subgrid in a computer code, we only need to know that this subgriddoes exist.

We denote the sets of mesh nodes n, edges e, faces f , and cells c by N , E , F , and C , respectively. Let Q be one of thesesets. We define the subsets Q(c), Q( f ), and Q(e), which are formed by the mesh objects of Q that are related, respectively,



to cell c, face f , and edge e. When the argument has a higher topological dimension, the resulting set Q(c), Q( f ), or Q(e)is the collection of mesh objects that belong to the boundary of c, f , and e, respectively. For example, E(c) denotes all theedges forming the boundary of cell c. When the argument has a lower topological dimension, the resulting set Q( f ), Q(e),or Q(n) is the collection of mesh objects sharing cell c, face f , edge e, or node n, respectively. For example, C(e) denotesall cells sharing edge e. When the topological degrees are the same, we consider the subset of items that are connected insome sense to the argument. For example, E(e) is the set of edges sharing at least a node with e. Finally, the symbol |Q(σ )|where σ may be n, e, f and c is the cardinality of set Q(σ ), i.e., the number of objects that are in such set; for example,|E(c)| is the number of edges of cell c.

2.2. Primary mimetic operators and discrete spaces

Three primary mimetic operators appear naturally from the first principles that are expressed by the Stokes theoremin one, two and three spatial dimensions. These three operators are the discrete gradient operator GRAD, the discrete curloperator CURL, and the discrete divergence operator DIV .

The discrete gradient operator: GRAD : Nh → Eh . Let us consider a scalar function p that is sufficiently smooth to guar-antee the existence of its gradient and pointwise values; for example, p can be taken in the Sobolev space H1(Ω) ∩ C0(Ω).Let e be the edge oriented from vertex n1 to vertex n2. We rewrite formula (2.1) as follows:∫

e

∂ p

∂τ edL =

∫e

(grad p) · τ e dL = p(xn2) − p(xn1). (2.10)

Let us call a scalar pn at the mesh vertex n as the degrees of freedom. The vector of all pn will be denoted by ph . We saythat a vector ph is a discrete representation of a continuous function p if pn = p(xn). Eq. (2.10) suggests us to define thevalue of the discrete gradient of ph associated with the edge e as follows:

(GRAD ph)e = pn2 − pn1

|e| . (2.11)

The collection of all vectors ph (with the obvious definition of sum of two vectors and multiplication of a vector by a scalarnumber) forms the linear space Nh . The dimension of Nh equals the number of mesh vertices. Since the discrete gradientoperator GRAD acts from Nh to N �

h in accordance with the general definition (2.8), the linear space N �h must be related

to the edges of the mesh. For convenience of exposition, we rename N �h as Eh . Each component of a vector in Eh is uniquely

associated with a mesh edge, and, since Eh is the linear space formed by all the linear combinations of such vectors, itsdimension equals the number of mesh edges.

The discrete curl operator: CURL : Eh → Fh . Let us consider a vector-valued function u, which is a sufficiently regular sothat all the integrals below are well defined. Let us apply formula (2.2) to the mesh face f :∫

f

(curl u) · n f dS =∮∂ f

u · τ dL, (2.12)

where the tangential vector τ is oriented counter-clock-wise along ∂ f when looking from the tip of the normal vector n f .Let us call a scalar ue on the mesh edge e as the degree of freedom. The vector of all ue will be denoted by uh . We say thata vector uh is a discrete representation of a vector function u if

ue = 1

|e|∫e

u · τ e dL ∀e ∈ E . (2.13)

Eq. (2.12) suggests us to define the value of the discrete curl operator CURL applied to uh and related to face f as follows:

(CURLuh) f = 1

| f |∑

e∈E( f )

α f ,e|e|ue. (2.14)

Here, the factor α f ,e is either 1 or −1 and represents the mutual orientation of the vectors n f and τ e as required by thecirculation theorem.

Since the discrete curl operator CURL acts from Eh to E�h in accordance with the general definition (2.8), the linear

space E�h must be related to the faces of the mesh. For convenience of exposition, we rename E�

h as Fh . Each componentof a vector in Fh is uniquely associated with a mesh face, and, since Fh is the linear space formed by all the linearcombinations of such vectors, its dimension equals the number of the mesh faces.

The discrete divergence operator: DIV : Fh → Ch . Let us consider a vector-valued field u, which is sufficiently regular sothat all integrals below are well defined; for example, u can be taken in the Sobolev space W 1,s(Ω), s > 2, with divergencein L2(Ω). Let us apply formula (2.3) to the mesh cell c:


∫c
div u dV =∮∂c

u · nc dS, (2.15)

where nc is the outward unit normal vector. Let us call a scalar u f on the mesh face f as the degree of freedom. With aslight abuse of notation, the vector of all u f will be denoted by uh . We say that a vector uh is a discrete representation ofa vector function u if

u f = 1

| f |∫f

u · n f dS ∀ f ∈ F . (2.16)

Eq. (2.15) suggests us to define the value of the discrete divergence applied to uh and related to face f as follows:

(DIV uh)c = 1

|c|∑

f ∈F(c)

αc, f | f | u f . (2.17)

Here, the factor αc, f is either 1 or −1 and represents the mutual orientation of the vectors nc and n f .Since the discrete divergence operator acts from Fh to F�

h in accordance with the general definition (2.8), the linearspace F�

h must be related to the mesh cells and to the scalar functions. For convenience of exposition, we rename F�h as Ch .

Let us call a scalar pc in the mesh cell c as a degree of freedom. With a slight abuse of notation, the vector of all pc willbe denoted by ph ∈ Ch . We say that a vector ph ∈ Ch is a discrete representation of a scalar function p if

pc = 1

|c|∫c

p dV ∀c ∈ C. (2.18)

Each component of a generic vector of Ch is uniquely associated with a mesh cell, and, since Ch is the linear space formedby all the linear combinations of such vectors, its dimension equals the number of the mesh cells.

Remark 2.2. The three primary operators GRAD, CURL and DIV have a direct matrix representation. With a smallabuse of notation, we will denote these matrices with the same symbols of the corresponding discrete operators and wewill reinterpret the action of these operators as matrix–vector products. For example, let NN and NE be, respectively, thenumber of nodes and the number of edges of the mesh. Now, the discrete gradient operator GRAD is a rectangular matrixwith size NE × NN and the expression “vh = GRAD ph” says that the NE -sized vector vh (edge-based degrees of freedom)is given by multiplying the NN -sized vector ph (node-based degrees of freedom) by matrix GRAD. A similar interpretationholds for CURL and DIV .

Remark 2.3. The matrices representing GRAD, CURL and DIV are the adjacency matrices of the mesh up to a diagonalrescaling of their rows and columns. More precisely, matrix GRAD, once rescaled with the edge lengths, represents node-edge connections; matrix CURL, once rescaled with edge and face measures, represents edge-face connections; matrixDIV , once rescaled with face and cell measures, represents face-cell connections.

2.3. Properties of primary operators

As the definition of the primary operators follows directly from the Stokes theorem, they preserve some importantproperties of the continuous calculus. These properties are, indeed, discrete versions of the Stokes theorem incorporatednaturally in the mimetic framework.

Formula (2.1) states that the line integral between two points xa and xb does not depend on the path connecting them.Let the mesh nodes xa = x0,x1, . . . ,xn−1,xn = xb be the consecutive vertices of a mesh path consisting of mesh edges ei ,i = 1, . . . ,n. By integrating the discrete gradient of ph along this path, we obtain the discrete analog of the line integral:

n∑i=1

|ei |(GRAD ph)ei =n∑

i=1

(pi − pi−1) = p0 − pn.

Formula (2.2) states that the flux integral of the vorticity of a field u over a two-dimensional manifold S embeddedin the three-dimensional space (e.g., a surface) and bounded by a closed one-dimensional path L does not depend on theshape of S . Consider a discrete manifold Sh that consists of mesh faces f and is bounded by mesh edges e that form aclosed mesh path Lh . The definition of the discrete operator CURL applied to uh leads to the following discrete analog:∑

f ∈Sh

| f |(CURLuh) f =∑e∈Lh

|e|ue,

where we assume that the orientations of n f , f ∈ Sh , and τ e , e ∈ Lh , are consistent with the orientations of normal andtangent vectors in the Stokes formula for the surface Sh .



Similarly, the discrete analog of formula (2.3) states that a discrete version of the divergence theorem holds for any unionof cells Vh that are connected and bounded by a closed surface Sh of mesh faces:∑

c∈Vh

|c|(DIV uh)c =∑f ∈Sh

| f |u f ,

where we assume that the normals n f , f ∈ Sh , point out of Vh .Let img(D) denote the range (or image) of operator D, and ker(D) its null space. The following result is related to the

stability of a mimetic discretizations for the diffusion problem.

Lemma 2.1. The primary divergence operator DIV is surjective, i.e., img(DIV) = Ch.

Proof. We have to show that for any discrete field qh in Ch there exists a discrete field vh in Fh such that qh =DIV vh . Wecan easily build such field vh as follows. Let q ∈ L2(Ω) denote the piecewise constant function on Th that takes the value qc

on each cell c. Let ϕ be the solution of the problem defined by �ϕ = q on the computational domain Ω with homogeneousboundary conditions on ∂Ω . Let v = gradϕ , so that div v = q, and vh ∈ Fh be the discrete field collecting the degrees offreedom of v, see (2.16). We claim that qh =DIV vh . In fact,

(DIV vh)c = 1

|c|∑

f ∈F(c)

αc, f | f |v f = 1

|c|∑

f ∈F(c)

αc, f

∫f

v · n f dS = 1

|c|∮∂c

v · nc dS

= 1

|c|∫c

div v dV = 1

|c|∫c

q dV = qc,

which holds for all the mesh cells c of Th . �2.4. Derived operators

In this subsection, we introduce the derived operators GRAD, CURL, and DIV , which are obtained through a dualityrelation from the primary operators DIV , CURL, and GRAD. According to (2.8), for each primary operator D there existsan adjoint operator D� , which is defined uniquely by D and the inner products on the spaces S and S� . Let us assume thatthe spaces Nh , Eh , Fh , or Ch are equipped with inner products (their construction will be discussed in Section 2.6). Applying(2.8) for the primary operators GRAD, CURL, and DIV , we obtain the adjoint operators GRAD� , CURL� , and DIV� .As it will be clear later, it is convenient to rename the adjoint operators to reflect their nature better. More precisely, we

will identify: GRAD ≡ −DIV� , CURL≡ CURL� , and DIV ≡ −GRAD� .To define these operators formally we do not need an explicit expression for the mimetic inner products, but just to

know that such inner products exist and that a matrix representation is available for each one of them. We recall that aninner product [· , ·]Q can be represented by a symmetric and positive definite matrix MQ , where Q is one of the spacesNh , Eh , Fh , or Ch . The derivation of inner product matrices that lead to an accurate numerical method is in the heart of themimetic technology; therefore, we dedicate a special section to this topic. For the moment, we simply assume that

[uh, vh]Q = (uh)T MQ,vh, ∀uh,vh ∈ Qh. (2.19)

Let us now define the derived operators. Inserting the primary and derived operators DIV and DIV� into formula (2.8),

using the above matrix representations for the mimetic inner products and the identification GRAD ≡ −DIV� yield

[vh, GRAD ph]F := −[vh, DIV�ph]F = −[DIV vh,ph]C ∀vh ∈ Fh, ph ∈ Ch. (2.20)

As vh and ph are arbitrary vectors in Fh and Ch we readily obtain

GRAD ≡ −DIV� = −M−1F DIV T MC, (2.21)

where DIV T is the transpose of the matrix DIV , cf. Remark 2.2. From Eqs. (2.21) and (2.4) it is natural to identify DIV�

with a discrete gradient operator and rename it as GRAD.

The duality relation between the discrete curl operator CURL and its adjoint CURL ≡ CURL� implies that

[vh, CURLwh]E = [vh,CURL� wh]E = [CURLvh,wh]F ∀wh ∈ Fh, vh ∈ Eh, (2.22)

from which we obtain that

CURL ≡ CURL� = M−1 CURLT MF . (2.23)
E



Likewise, the duality relation between the discrete gradient operator GRAD and its adjoint DIV ≡ −GRAD� impliesthat

[qh, DIV wh]N = −[qh,GRAD� wh]N = −[GRAD qh,wh]E ∀qh ∈ Nh, wh ∈ Eh, (2.24)

from which we obtain that

DIV ≡ −GRAD� = −M−1N GRADTME . (2.25)

Remark 2.4. For unstructured meshes, the mimetic inner product matrices are often irreducible matrices. Thus, their inversematrices are dense and the stencil of the derived operators is non-local. Some exceptions will be considered in Sections 2.6and 6.

2.5. Exact identities and discrete Helmholtz decomposition theorems

Exact identities, which characterize the kernel of the primary and derived operators, and discrete Helmholtz decompo-sition theorems play an important role in numerical analysis of PDEs such as Navier–Stokes and Maxwell’s equations. Thepreservation of such properties in a discrete setting is one of the most important characteristics of the mimetic discretiza-tions. In this subsection, we show how the mimetic approach satisfies these relations. Let (Q)⊥ denote the orthogonalcomplement of the linear space Q and {0}Q the trivial subspace formed by the zero element.

Lemma 2.2. Let mesh Th be face-connected. Then, the null space of GRAD consists of the constant vectors of Ch.

Proof. From the definition given in (2.11) it is obvious that GRAD ph = 0 whenever ph is a constant vector, i.e., all ofits entries have the same value. From Remark 2.3 we also know that GRAD is the NE × NN -sized adjacency matrixthat expresses the topological node-edge connections. Therefore, its rank is equal to (NN − 1) and its kernel must havedimension one. This fact implies that the kernel of GRAD coincides with the subspace of the constant vectors in Nh . �

Exact identities such as curl grad = 0 and div curl = 0 are preserved by construction of primary and derived operators.In the following lemma we prove that img(GRAD) = ker(CURL) and img(CURL) = ker(DIV).

Lemma 2.3. Let the domain Ω and its mesh partition Th be simply connected. Then,

CURLvh = 0 if and only if vh = GRAD qh (2.26)

for some qh ∈Nh and

DIV vh = 0 if and only if vh = CURLuh (2.27)

for some uh ∈ Eh.

Proof. A straightforward calculation shows that CURLGRAD = 0 and DIV CURL = 0. Let us now assume thatCURLvh = 0 for some vh ∈ Eh . We consider a mesh path of consecutive edges {ek}ni

k=1 that begins at a fixed vertex n1and ends at vertex ni . Then, we consider the discrete field qh ∈ Nh such that the component associated with the vertex niis given by

qni = qn1 +ni∑

k=1

αi,k|ek|vek , (2.28)

where the value of αi,k , which can be either +1 or −1, depends on the mutual orientation of the corresponding edge ekand the mesh path. The definition of the discrete gradient operator implies that GRAD(qh) = vh . To complete this part ofthe proof, we note that qni does not depends of the mesh path since CURL(vh) = 0 and the mesh is simply connected.Thus, the value of qni depends only on the initial value qn1 in (2.28), i.e., ph is defined up to a constant vector, but this isnot a problem since the constant vectors are in the kernel of the discrete gradient operator, cf. Lemma 2.2.

Let us assume that DIV(vh) = 0 for some vh ∈Fh .The argument based on a direct construction of vh = CURLuh is simple for logically rectangular meshes, see, e.g.,

[145,146], but its extension to general meshes is not trivial. Therefore, we use a different argument that is based on theproperties of the continuum operators: for every vh , there exists a continuum vector function v satisfying (2.16). Such afunction can be built by solving the Laplace equation in each mesh cell c with a piecewise constant Neumann boundaryconditions. By construction div v = 0. Then, there exists a vector potential u ∈ (H1(Ω))3 (a scalar stream function in 2D)such that v = curl(u). The proof can be found in [123]. Applying the previous lemma, we obtain



v f = 1

| f |∫f

v · n dS = 1

| f |∫f

(curl u) · n dS = 1

| f |∑

e∈E( f )

α f ,e

∫e

u · τ f dL = (CURLuh) f ,

where the components of the vector uh are the mean values of u · τ f along the mesh edges. �Lemma 2.4. The null space of the derived operator GRAD consists of the zero vector of Ch.

Proof. Let GRAD ph = 0 for some ph ∈ Ch . From (2.21) it follows that −M−1F DIV T MC ph = 0, which implies that

DIV T MC ph = 0, or that MC ph ∈ ker(DIV T ) because MF is a non-singular inner product matrix. Now, a standard al-gebraic relation and the fact that DIV is a surjective operator yield:

ker(DIV T )= (img(DIV)

)⊥ = (Ch)⊥ = {0}Ch ,

which implies the lemma’s statement. �This result may seem strange at a first glance as one would expect that the kernel of a discrete gradient operator consists

of constant vectors, as it is true for the primary operator GRAD, cf. Lemma 2.2. The considered DTVC assumes that theboundary integrals in formulas (2.4)–(2.6) are zeros and this lemma reflects this fact. Indeed, on an orthogonal mesh, weobtain:

(GRAD ph) f ={

(pc2 − pc1)/d12 if f = F(c1) ∩F(c2),

−pc1/d1 f if f = F(c1) ∩ ∂Ω,(2.29)

where d12 is the distance between the centroids of c1 and c2, and d1 f is the distance between the centroid of c1 and facef . Let GRAD ph = 0. Eq. (2.29) implies that all pc are equal and pc is zero if a face of ∂c is also a boundary face. Therefore,

only ph = 0 belongs to the kernel of GRAD.

Lemma 2.5. The derived discrete gradient, divergence, and curl operators satisfy the following exact relationship:

CURLvh = 0 if and only if vh = GRAD ph (2.30)

for some ph ∈ Ch and

DIV vh = 0 if and only if vh = CURLuh (2.31)

for some uh ∈Fh.

Proof. The “if ” part of the lemma follows immediately from the matrix definitions of GRAD, CURL, and DIV givenin (2.21), (2.23), and (2.25), respectively, and the results of Lemma 2.3. In fact, a straightforward calculation shows that

CURLGRAD = −M−1E CURLTMFM−1

F DIV TMC = −M−1E (DIV CURL)TMC = 0,

and

DIV CURL = −M−1N GRADTME M

−1E CURLTMF = −M−1

N (CURLGRAD)TMF = 0.

To prove the “only if ” part of the lemma, let CURLvh = 0 for some vh ∈Fh . From (2.23) it follows that −M−1E CURLT ×

MF vh = 0, which implies that CURLT MF vh = 0, or that MF vh ∈ ker(CURLT ) because ME is a non-singular innerproduct matrix. Now, a standard algebraic relation and the result of Lemma 2.3, see (2.27), yield:

ker(CURLT )= (img(CURL)

)⊥ = (ker(DIV))⊥ = img

(DIV T ),

from which it immediately follows that there must exist a vector qh ∈ Ch such that MFvh = DIV T qh . As MF is a non-singular matrix, we can introduce the vector ph = −M−1

C qh , so that MFvh = −DIV T MCph , and using (2.21) we obtainthat

vh = −M−1F DIV T MCph = GRAD ph.

Likewise, let DIV vh = 0 for some vh ∈ Eh . From (2.25) it follows that −M−1N GRADT ME vh = 0, which implies that

GRADT ME vh = 0, or that ME vh ∈ ker(GRADT ) because MN is a non-singular inner product matrix. Now, a standardalgebraic relation and the result of Lemma 2.3, see (2.26), yield:

ker(GRADT )= (img(GRAD)

)⊥ = (ker(CURL))⊥ = img

(CURLT ),



from which it immediately follows that there must exist a vector wh ∈ Fh such that MEvh = CURLT wh . As MF is anon-singular matrix, we can introduce the vector uh = M−1

F wh , so that MEvh = CURLT MFuh , and using (2.23) we obtainthat

vh = M−1E CURLT MFuh = CURLuh.

This proves the assertion of the lemma. �We can combine primary and derived operators to form second-order mimetic operators as, for example, CURLCURL

and DIV GRAD, and two discrete analogs of the vector Laplace operator �� = grad div − curl curl, which are given by��Eh = GRAD DIV − CURL CURL and ��Fh = GRADDIV − CURLCURL. These operators make it possible to designthe mimetic discretizations of PDE problems in a very immediate and elegant way. This is the topics of Section 3. Let usnow investigate the properties of the kernels of such operators.

Lemma 2.6. The product DIV GRAD is a full rank matrix, i.e., ker(DIV GRAD) = {0}Ch .

Proof. Let qh ∈ Ch such that DIV GRAD qh = 0. The definition of the derived gradient operator given in (2.20) yields

0 = [DIV GRAD qh,qh]C = −[GRAD qh, GRAD qh]F ,

from which we obtain that GRAD qh = 0. According to Lemma 2.4, this holds when qh = 0. �The matrix DIV GRAD corresponds to a cell-centered discretization of the Laplace operator with essential boundary

conditions. This operator is self-adjoint with respect to the inner product in space Ch . A nodal discretization of the Laplaceoperator with natural boundary conditions can be derived by using the primary gradient operator and the derived diver-gence operator, DIV GRAD. This operator is self-adjoint with respect to the inner product in space Nh . Likewise, ��Eh and��Fh are, respectively, an edge-based and a face-based discretization of the vector Laplace operator ��, and both operatorsare symmetric semi-negative definite with respect to the inner products in Eh and Fh .

Lemma 2.7. Let the domain Ω and its mesh partition Th be simply connected. Then

CURL CURLvh = 0 if and only if vh = GRAD qh (2.32)

for some qh ∈Nh, and

CURL CURLvh = 0 if and only if vh = GRAD qh (2.33)

for some qh ∈ Ch.

Proof. The “if ” part of (2.32) is a consequence of Lemma 2.3. To prove the “only if ” part, let us consider a vector field

vh ∈ Eh such that CURL CURLvh = 0. Using (2.22) yields

0 = [vh, CURL CURLvh]E = [CURLvh, CURLvh]F ,

from which it follows that CURLvh = 0. Lemma 2.3 (cf. (2.26)), implies the existence of a field qh ∈ Nh such that vh =GRAD qh . Assertion (2.33) is proved using the same argument and Lemma 2.5. �

The discrete operator CURL CURL corresponds to an edge discretization of the curl curl operator, it is self-adjointwith respect to the inner product in space Eh , and can be used to solve the Maxwell’s equations for the electric field.

Similarly, the discrete operator CURLCURL corresponds to a face discretization of the curl curl operator, it is self-adjointwith respect to the inner product in space Fh , and can be used to solve the equation of magnetic diffusion. These twodiscrete operators are also in the definition of the edge- and face-based vector Laplace operators, which are self-adjoint andwhose kernel properties are stated in the following lemma.

Lemma 2.8. The null space of the vector Laplace operators ��Eh and ��Fh are the trivial subspaces {0}Eh and {0}Fh , respectively.

Proof. (i) To prove that ker( ��E ) = {0}E , we consider a vector vh in Eh such that ��E vh = 0. It follows that:
h h h



0 = [vh, ��Eh vh]E (use the definition of ��Eh )

= [vh,GRAD DIV vh]E − [vh, CURLCURLvh]E(use (2.24) and (2.22)

)= −[DIV vh, DIV vh]C − [CURLvh,CURLvh]F ,

from which we have

DIV vh = 0 and CURLvh = 0. (2.34)

These two conditions imply that vh = 0. In fact, in view of Lemma 2.5 the first equation in (2.34) implies that vh = CURLuh

for some field uh ∈ Fh . From the second equation in (2.34) we obtain that CURL CURLuh = 0, and from Lemma 2.7

(cf. (2.33)) that uh = GRAD qh for some qh ∈ Ch . Re-substituting back such expression and applying again Lemma 2.5 yield

vh = CURLuh = CURL GRAD qh = 0

regardless of qh .(ii) To prove that ker( ��Fh ) = {0}Fh , we consider a vector vh in Fh such that ��Fh vh = 0. It follows that:

0 = [vh, ��Fh vh]F (use the definition of ��Fh )

= [vh, GRADDIV vh]F − [vh,CURL CURLvh]F(use (2.20) and (2.22)

)= −[DIV vh,DIV vh]C − [CURLvh, CURLvh]F ,

from which we have

DIV vh = 0 and CURLvh = 0. (2.35)

These two conditions imply that vh = 0. In fact, in view of Lemma 2.3 the first equation in (2.35) implies that vh = CURLuh

for some field uh ∈ Eh . From the second equation in (2.35) we obtain that CURL CURLuh = 0, and from Lemma 2.7(cf. (2.32)) that uh = GRAD qh for some qh ∈Nh . Re-substituting back such expression and applying again Lemma 2.3 yield

vh = CURLuh = CURL GRAD qh = 0

regardless of qh . �We present two discrete versions of the Helmholtz decomposition theorem. We prove only the first theorem as the

second one can be proved using similar arguments.

Theorem 2.1. Let domain Ω and mesh Th be simply-connected. Then, for any vh ∈ Fh there exists a unique qh ∈ Ch and a unique

uh ∈ Eh with DIV uh = 0 such that

vh = GRAD qh + CURLuh. (2.36)

Proof. Let us show that the terms in the right-hand side of (2.36) are orthogonal to one another. We apply the definitionof the derived gradient operator and Lemma 2.3 to obtain

[CURLuh, GRAD qh]F = −[DIV CURLuh,qh]C = 0.

By applying the primary divergence operator to both sides of (2.36) we obtain that

DIV vh = DIV GRAD qh. (2.37)

In view of Lemma 2.6, DIV GRAD is a full rank operator, and, thus, it is non-singular. Therefore, given vh ∈Fh , a solutionqh ∈ Ch to (2.37) exists and is unique.

By applying the derived curl operator to both sides of (2.36), we obtain

CURLvh = CURL CURLuh. (2.38)

Let us analyze the necessary conditions for the existence and uniqueness of the solution uh to this problem. The solutionuh exists if and only if

CURLvh ∈ img(CURL CURL) = (ker((CURL CURL)T ))⊥.



Thus, we need to prove that CURLvh is orthogonal to the kernel of the transpose of the combined operator CURL CURL.

Practically speaking, we need to show that any wh that belongs to ker((CURL CURL)T ) is such that (CURLvh)T wh = 0.

Let us consider a discrete field wh in ker((CURL CURL)T ). Note that

(CURL CURL)T = CURLT CURLT = CURLT MF CURLM−1

E . (2.39)

This relation implies that wh = ME GRAD qh for some qh ∈ Nh . In fact, let us multiply equation (CURL CURL)T wh = 0on the left by (M−1

E wh)T and use (2.39) and the definition of the inner product in Fh to obtain:

0 = wTh M

−1E CURLT MF CURLM−1

E wh = [CURL(M−1

E wh),CURL

(M−1

E wh)]

F .

This relation implies that CURL(M−1E wh) = 0 and the existence of a field qh such that M−1

E wh = GRAD qh follows byapplying Lemma 2.3, cf. (2.26). Using the argument from the proof of Lemma 2.7, we can show that this is the only

characterization of the null space of (CURL CURL)T . Now, to prove the orthogonality let us observe that

(CURLvh)T wh = (CURLvh)

TME GRAD qh[use (2.19) with Q = E

]= [CURLvh, GRAD qh]E

[use (2.24)

]= [DIV CURLvh, qh]N

[use (2.31)

]= 0.

To prove the uniqueness of uh under the assumption that DIV uh = 0, let us consider a second discrete field u′h such that

CURL CURLu′h = CURLvh and DIV u′

h = 0. Clearly, CURL CURL (uh −u′h) = 0, and Lemma 2.7, Eq. (2.32), implies that

there exists a discrete field qh ∈Nh such that uh − u′h = GRAD qh . Now, let us observe that

[GRAD qh,GRAD qh]E = [uh − u′h,GRADuh

]E = [DIV

(uh − u′

h

),qh]= 0,

from which it follows that GRAD qh = 0, i.e., uh = u′h . �

The result of this theorem can be easily verified for simple polyhedral meshes. The dimension of space Fh should beequal to the dimension of space Ch plus the dimension of space Eh minus the dimensions of the null spaces of the operators

GRAD and CURL. The dimension of the first null space is equal to zero and the second one is characterized by the imageof operator GRAD. Thus, it holds that

dim(Fh) = dim(Ch) + dim(Eh) − (dim(Nh) − 1).

Moreover, for a single polyhedron it holds that dim(Ch) = 1 and we obtain the famous Euler’s polyhedron formula.The second theorem concerning with the discrete Helmholtz decomposition follows below. We omit the proof as it is

based on arguments similar to those of Theorem 2.1.

Theorem 2.2. Let domain Ω and mesh Th be simply-connected. Then, for any vh ∈ Eh there exists a discrete field qh ∈ Nh, which isdefined up to a constant field, and a unique discrete field uh ∈Fh with DIV uh = 0 such that

vh = GRAD qh + CURLuh. (2.40)

The Helmholtz decomposition (2.36) is illustrated in Fig. 2 for a vector function vh ∈ Fh generated from the continuousfunction v = (x − y, x + y)T [149]. We show a smooth computational mesh, the field v at the nodes of this mesh, and, then,

we visualize the vector fields GRAD qh and CURLuh . This visualization is based on a lifting of a two-dimensional vectorat mesh nodes, which is based for each node n on the projections along the tangent and normal directions associated withthe mesh edges in E(n).

2.6. Mimetic inner products

The construction of the matrices of the mimetic inner products, e.g., MQ where Qh is one of the aforementioned spaces,is done locally, in a cell-wise way. We use the subscript c to denote the restriction to cell c of the mimetic inner product,its matrix representation and its vector arguments:

[uh, vh]Q =∑

[uh,c,vh,c]Q,c, [uh,c,vh,c]Q,c = (uh,c)TMQ,cvh,c . (2.41)

c∈Th



Fig. 2. Helmholtz decomposition (2.36): a 16 × 16 mesh (left-most panel) and the vector fields vh , GRAD qh and CURLuh (from the second panel tothe right-most panel).

In (2.41), MQ ,c is a symmetric and positive definite matrix acting on the vectors uh,c and vh,c , which contain the degreesof freedom of uh and vh related to cell c.

The crucial requirement for the inner product matrix is to provide an accurate discretization of volume integrals like

[uh,c,vh,c]Q,c ≈∫c

u · v dV , (2.42)

where u and v may be scalar or vector fields defined on c, uh,c and vh,c are their discrete representation and the dot symbol“·” is contextually defined. This requirement can be suitably modified to include scalar or tensor coefficients, thus leading tothe formulation of weighted inner products, which will be denoted by the same notation for simplicity of exposition. Thiscase is clearly problem dependent, and we will indicate in the various situations how the mimetic inner product is defined.For example, to discretize the diffusion problem in mixed form instead of (2.42) we require that

[uh,c,vh,c]Q,c ≈∫c

u · K−1v dV , (2.43)

where uh and vh are defined by (2.16). The accuracy of this approximation is directly connected to the accuracy of themimetic scheme. The derivation of the inner product is a non-trivial task especially when the degrees of freedom are thenormal or tangential components of vector-valued functions.

Let us discuss this issue in more details by beginning with the easy case of the inner product for the spaces Ch and Nh .The construction of the inner product matrix MC,c is trivial. Indeed, the restriction of a vector ph ∈ C to cell c is a number;therefore, the cell matrix MC,c is a number. There is only one quadrature rule for the volume integral that uses a singlequadrature point:

[ph,c,qh,c]C,c = |c|pcqc, (2.44)

i.e. MC,c = |c|. This leads to a diagonal global matrix MC . This inner product matrix integrates exactly a pair of functions pand q if one of them is piecewise constant on the mesh and we know the cell averages of the other function.

Let us consider the space Nh . We can consider different diagonal inner product matrices MN with different approxima-tion properties. The first matrix is assembled from scalar local matrices MN ,c = |c|/|N (c)| I, where I is the properly-sizedidentity matrix and |N (c)| is the number of nodes of cell c. This matrix is used in Section 6. The construction of the secondmatrix is based on the formula for the centroid of a polyhedral cell c in terms of the vertex coordinates:

xc =∑

n∈N (c)

ωc,nxn

where ωc,n is the weight of vertex n. Using these weights, the local inner product in Nh,c takes the form

[ph,c,qh,c]N ,c =∑

n∈N (c)

ωc,n pnqn.

When cell c is star-shaped with respect to xc , the weights ωc,n are positive. Since these weights are assembled on the maindiagonal of the cell matrix MN ,c , it is immediate to see that this matrix is diagonal and positive definite. The global matrixMN is also diagonal and positive definite. The quadrature rule associated with the weights ωc,n is second-order accurate;therefore, the inner product matrix MN integrates exactly the product pq when this function is piecewise linear on Th .

The derivation of a mimetic inner product for the remaining spaces Fh and Eh requires more work. The basic idea is tointroduce consistency conditions (exactness for polynomial functions) that lead to algebraic restrictions on the local innerproduct matrix MQ,c . These algebraic requirements are expressed as

MQ,c Ni = Ri, i = 1,2, . . . ,m, (2.45)



where Ni and Ri suitably defined column vectors and m is the number of restrictions. Usually, m is less than the size ofmatrix MQ,c ; thus, we obtain a family of admissible local mimetic inner products, each one of which corresponds to anadmissible mimetic scheme. We emphasize the fact that all the mimetic schemes in the family shares the same convergenceand stability properties.

Let Nc and Rc be two rectangular matrices that contain the vectors Ni and Ri , respectively, as columns. Precise formulasfor N and R will be presented in the next sections for various spaces Q; however, it turns out that the product matrixRT

c Nc is always symmetric and positive or semi-positive definite. Let us re-write the m algebraic equations (2.45) in thematrix form

MQ,cNc = Rc . (2.46)

Now, the following matrix

M0Q,c = Rc

(RT

c Nc)−1

RTc (2.47)

solves (2.46) provided that the inverse matrix does exist. A general solution takes the form: MQ,c = M0Q,c + M1

Q,c , where

M1Q,c Nc = 0.

Lemma 2.9. Let RTc Nc be an SPD matrix. Furthermore, let the columns of Dc form a basis for ker(NT

c ). Then, the matrix

MQ,c = Rc(RT

c Nc)−1

RTc +DcUcD

Tc (2.48)

is SPD for any SPD matrix Uc .

Proof. Let qh be a non-zero vector in Q. As RTc Nc is symmetric and positive definite, we have that

qTh,cMQ,cqh,c = (RT

c qh,c)T (

RTc Nc

)−1(RT

c qh,c)+ (DT

c qh,c)Uc(DT

c qh,c)� 0, (2.49)

where the equality may occur only if RTc qh and DT

c qh are simultaneously zero for some qh,c �= 0. Let us show by contradic-tion that this can never occur.

Let DTc qh,c = 0 and RT

c qh,c = 0. The first equality implies that qh,c is orthogonal to the columns of matrix Dc . By hy-pothesis, these columns form a basis for ker(NT

c ). Since ker(NTc ) = (img(Nc))

⊥ , qh,c is a linear combination of the columnsof matrix Nc , i.e. there exists a vector a �= 0 such that qh,c = Nc a. Inserting this expression in (2.49) and using DT

c qh,c = 0we obtain:

0 = qTh,cRc

(RT

c Nc)−1

RTc qh,c = aTNT

c Rc(RT

c Nc)−1(

RTc Nc

)a = aTRT

c Nca > 0.

This contradiction proves the assertion of the lemma. �Let k be the size of matrix MQ,c . Then, Uc is a symmetric square matrix with size k −m and is defined by (k −m + 1)×

(k − m)/2 independent parameters. It has been experienced that these parameters may vary of several orders in magnitudewith a minor impact on scheme accuracy. Later, we will pose a question of their optimal selection.

Corollary 2.1. A special member of the mimetic family (2.48) is provided by the following formula:

MQ,c = Rc(RT

c Nc)−1

RTc + λQ,c

(I−Nc

(NT

c Nc)−1

NTc

)(2.50)

with the scalar factor

λQ,c = 1

2trace

(Rc(RT

c Nc)−1

RTc

). (2.51)

Proof. Formula (2.50) is easily derived from (2.48) by choosing:

Uc = λQ,c(DT

c Dc)−1

, (2.52)

where we set λQ,c in the form given by (2.51) to take into account the proper scaling of the two matrix addends in (2.48).In fact, the columns of the m × m-sized matrix [Nc,Dc] forms a basis for the space of the m-sized real vectors and thecolumns of Dc are orthogonal to the columns of Nc . Thus, the orthogonal projectors on the span of the columns of Nc andDc satisfy:

Dc(DT

c Dc)−1

DTc +Nc

(NT

c Nc)−1

NTc = I. (2.53)

Formula (2.50) is obtained by substituting (2.52) into (2.48) and expressing the term Dc(DTc Dc)

−1DTc through (2.53). �



Remark 2.5. The scaling factor 0.5 in formula (2.51) is problem-dependent. For the diffusion problem considered in Sec-tion 3.1, it leads to a diagonal matrix MQ,c for any square cell c and any scalar diffusion coefficient.

The consistency condition can also be written with respect to the inverse of the mass matrix and takes the form

WQ,cRc = Nc . (2.54)

The general solution of Eq. (2.54) is given by

WQ,c = Nc(NT

c Rc)−1

NTc + DcUcD

Tc , (2.55)

where the columns of matrix Dc form a basis for ker(RTc ) and Uc is an arbitrary SPD matrix. We find a formula similar

to (2.51) by setting

Uc = λQ,c(DT

c Dc)−1

with λQ,c = 1

2trace

(Nc(NT

c Rc)−1

NTc

). (2.56)

Using a similar arguments on the projectors of the span of the columns of Dc and Rc , we obtain

WQ,c = Nc(NT

c Rc)−1

NTc + λQ,c

(I−Rc

(RT

c Rc)−1

RTc

). (2.57)

Remark 2.6. The matrices WQ,c in (2.55) can be interpreted as the inverses of the matrices MQ,c considered in Lemma 2.9in the sense that the inverse of each matrix MQ,c provided by (2.48) can be written as in (2.55) through a suitable choiceof the matrices Dc and U.

Remark 2.7. In general, matrix WQ,c in (2.57) is not the inverse of matrix MQ,c given by (2.50) with λc given by (2.51).

Finally, it is worth noting that the material properties, e.g., the diffusion coefficients, are incorporated into the derivedoperators through the inner product matrices. Various examples will be provided in Section 3.

2.7. DVTC on manifolds

The DVTC can be extended to manifolds. Here we consider only a planar manifold embedded in a three-dimensionalspace. To ease the exposition, we assume that the manifold coincides with the x–y plane. We denote the unit vectororiented along the z axis by nz .

Let Th be a polygonal mesh. The spaces Nh and Ch are naturally related with the vertices n and the (polygonal) cells c asbefore. In two-dimensions, we may establish the isometry Fh = Eh since edges and faces coincide. However, it is convenientto use the space Fh only when the degrees of freedom are the normal components of the vector functions. Likewise, it isconvenient to use the space Eh only when the degrees of freedom are the tangential components of the vector functions.

Since the divergence and circulation theorems also hold in two-dimensions the definitions of the primary operators arealmost identical to those introduced in Section 2.2, for example:

GRAD : Nh → Eh, (GRAD ph)e = 1

|e| (pn2 − pn1), (2.58)

where n2 and n1 are the end-points of edge e. Similarly,

CURL : Eh → Ch, (CURLuh)c = 1

|c|∑

e∈E(c)

αc,e|e|ue, (2.59)

and

DIV : Fh → Ch, (DIV uh)c = 1

|c|∑

f ∈F(c)

αc, f | f |u f . (2.60)

The construction of mimetic inner products and derived operators is similar to the construction of the corresponding three-dimensional objects.

In two-dimensions there exists a continuous curl operator that is applied to the scalar fields like A and is given by:

curl A =(

∂ A

∂ y,−∂ A

∂x

)T

. (2.61)

Here, we use the same notation as for the discrete curl operator that normally applies to the vector-valued fields as thetwo different definitions can be immediately distinguished from the context. Using the connection between the normal andtangential derivatives, we obtain:



Fig. 3. Left panel: orientation of unit vectors gives αc, f = −1 and α f = 1. Right panel: degrees of freedom for Nh (dots) and Fh (squares).

Fig. 4. Illustration of vectors Bh ∈ Fh (narrow arrows) and B ∈ Fh (wide arrows).∫e

curl A · ne dL =∫e

grad A · τ e dL = A(xn2) − A(xn1), (2.62)

where the unit vector τ e is obtained by a counter-clock-wise rotation of ne when we look from the tip of nz . In viewof (2.62), it is natural to assume that the new primary CURL operator must use the nodal values of A. Let Ah ∈ Nh; then,we define CURL :Nh →Fh by

(CURLAh) f = α f1

| f | (An2 − An1), (2.63)

where α f is either 1 or −1 depending on the mutual orientation of τ e , ne and nz (see Fig. 3). We use here the samenotation of the discrete curl operator in (2.59). There is no ambiguity in this notation as the CURL operator in (2.59) actson the edge vectors in Eh and the CURL operator in (2.61) acts on the vertex vectors in Nh .

For discrete operators, there hold the same identities of Lemma 2.3 that we state below for completeness of exposition.The proof is omitted as it is based on the same arguments.

Lemma 2.10. Let domain Ω ⊂ �2 be simply-connected and its polygonal partition Th be face-connected. Then,

CURLvh = 0 if and only if vh = GRAD qh (2.64)

for some qh ∈Nh and

DIV vh = 0 if and only if vh = CURLAh (2.65)

for some Ah ∈Nh, where GRAD, CURL, and DIV are the primary discrete operators defined in (2.59), (2.60), and (2.63).

2.8. Data transfer between meshes

In this section we will demonstrate how the discrete vector calculus can be used for the data transfer, also refereed to asremapping, between two distinct mesh partitions Th and Th of the same computational domain (see Fig. 4).

Our goal is to remap a discrete vector Bh ∈ Fh defined on the original mesh Th to a new discrete vector Bh defined onthe new mesh Th . We assume that DIV Bh = 0 and impose the constraint that the remapped vector be divergence-free, i.e.DIV Bh = 0. For clarity of exposition, we do not make any notational distinction between the discrete operators defined onTh and Th , keeping in mind that their definitions are strictly mesh-dependent. The constrained remapping problem is relevantwhen we deal with magnetic fields or incompressible velocity fields, because in such cases the divergence-free conditionhas an important physical meaning.

Let us first clarify the connection between the orientation of edges and normals and the constants αc, f and α f thatare used in the definition of the discrete divergence and discrete curl operators introduced in Section 2.7. If we choose the



Fig. 5. Left panel: two paths from point P to point Q . Right panel: spanning tree starting from the left-bottom node.

counter-clock-wise orientation for ∂c and the edge e and the normal vector ne as shown in Fig. 3, left panel, then, it holdsthat αc, f = −1 and α f = 1.

Since Bh is divergence-free, Lemma 2.10 implies the existence of a nodal vector Ah ∈Nh such that

Bh = CURLAh. (2.66)

The component-wise form of Eq. (2.66) reads as

B f = α f

| f | (An2( f ) − An1( f )) ∀ f ∈ F, (2.67)

where n1( f ) and n2( f ) denote the end-points of face f . The algorithm for the constrained remap described in [43] hasthree steps:

(i) Recovery: given a vector Bh ∈Fh , build a vector Ah ∈Nh that satisfies (2.66).(ii) Interpolation: using Ah , construct an interpolation field aIh (x, y), which is defined for every point (x, y) ∈ Ω .

(iii) Reconstruction: using aIh , define a new vector Ah ∈ Nh through the nodal projection:

An = aIh (xn, yn) ∀n ∈ N ,

and set

Bh = CURL Ah.

The interpolation step (ii) can be carried out in many different ways. Since a particular interpolation method is not impor-tant for the current presentation, we assume that one is available and we refer the readers interested in more details to[43].

Let us now focus on the recovery stage. In view of Eq. (2.67), we can conclude that

An2 ( f ) = An1 ( f ) + | f |B f . (2.68)

Therefore, for any fixed node nQ we can write the following formula

AnQ = AnP +∑

f ∈path

| f |B f , (2.69)

where “path” stands for a set of connected faces that lead from node nP to node nQ . The existence of a path is ensured ifwe consider a simply connected mesh. It is worth noting that, in general, there are many possible paths from nP to nQ .However, the final result does not depend on the choice of the path. Indeed, let us consider two different paths, namely,path and path′ , that connect nodes nP and nQ , and let AnQ and A′

nQbe the two values calculated by (2.69) (see illustration

on Fig. 5). It holds that

AnQ − A′nQ

=∑

f ∈path

| f |B f −∑

f ∈path′| f |B f =

∑enclosed cells

|c|(DIV Bh)c = 0, (2.70)

and the two values AnQ and A′nQ

are equal.This strategy yields an explicit procedure for the recovery of vector Ah on the original mesh Th that depends only on

the single parameter AnP . The recovery step requires to reach each node of the mesh from the node nP in an efficientway, e.g., by visiting each node only once. This task is accomplished by the construction of a spanning tree, which alwaysexists for a simply-connected mesh in two-dimensions but is not unique. For a logically rectangular mesh, a spanning treeis determined easily as shown in Fig. 5, right panel.



The value at the initial node nP can be set arbitrary, e.g., AnP = 0. The choice of AnP clearly affects all values in vectorAh . Nonetheless, it does not affects the face values B f which depend only on the difference of the nodal values of Ah . As tothe vector B, some restriction on the remapping must be imposed first. Eq. (2.69) can be rewritten in the vector form:

Ah = AnP e + A(0)

h , e = (1, . . . ,1)T . (2.71)

Now, the minimal requirement on the remapping from the old mesh Th to the new mesh Th is that it must be exact forconstant vectors, which implies that

Ah = AnP e + A(0)

h . (2.72)

Since CURLe = 0, the vector Bh = CURL Ah does not depend on the choice of AnP .A detailed description of the overall constraint remapping process for logically rectangular grids can be found in [43].

This paper includes a detailed description of an interpolation algorithm for the auxiliary vectors A(0)

h and discuss a special

procedure to minimize the discrepancy of the L2 norms of Bh and Bh that represent the energy of the magnetic fields Bhand Bh .

2.9. Primary and derived tensor operators

We introduce primary and derived operators for tensor fields by using the same arguments that we used in the previoussubsections for the vector fields, and, with a slight abuse of notation, we will make use of the same symbols.

Let u be a sufficiently smooth vector-valued function defined on the computational domain Ω and Th a mesh partition ofΩ , which we assume to be simply connected as before. We select the values of each component of u at the mesh vertices asthe degrees of freedom that represent such field in the discrete setting. The vector of all the degrees of freedom is denoted,as usual, by uh and the collection of all such vectors (with the obvious definitions of sum of two vectors and multiplicationof a vector times a scalar number) forms the node-based linear space N d

h , where d = 2,3 is the space dimension. Thedimension of N d

h is equal to the number of mesh vertices times d.Now, let us observe that Eq. (2.10) holds for each component of the vector function u. This fact suggests us to define the

discrete gradient operator GRAD :N dh → Ed

h such that its value associated with edge e is given by:

(GRADuh)e = un2 − un1

|e| ∀e ∈ E . (2.73)

This is nothing else but a multiple application of the gradient operator (2.11) to all the components of its vector argument.The image of this operator is associated with mesh edges and tangential components of tensor functions.

The inner product matrices in the tensor-product spaces can be defined as block-diagonal matrices; however, a muchricher family of admissible matrices does exist and is described in Section 2.5. We keep the same notation for the innerproducts and the ambiguity is resolved contextually by the nature (vector or tensor) of the arguments. From the primaryoperator GRAD we can readily define the derived divergence operator by duality:

[DIV σ h,uh]N = −[σ h,GRADuh]E , ∀uh ∈ (N )d,σ h ∈ (Eh)d. (2.74)

The definition of the primary divergence operator DIV :Fdh → Cd

h for tensors is similar to that of vectors:

(DIV σ h)c = 1

|c|∑

f ∈F(c)

αc, f | f |σ f , σ f = 1

| f |∫f

σ · n dS.

Again, we can define the inner product matrices in the tensor-product spaces as block-diagonal matrices and employ theduality argument to obtain the derived gradient operator:

[σ h, GRADuh]F = −[DIV σ h,uh]C . (2.75)

These discrete gradient and divergence operators will be used in Section 6 in the construction of an artificial viscosity.

3. Application to classical PDEs

3.1. Diffusion equation: primary divergence, derived gradient

In this section, we derive a family of mimetic finite difference methods for the linear diffusion problem in mixed form,e.g., the Darcy problem. This problem consists of the mass balance equation and Darcy’s law that relates the scalar field p(pressure) to the vector field u (velocity):



Fig. 6. Degrees of freedom of the mimetic finite difference method for the diffusion problem in mixed form: the cell-based unknown pc associated withthe polygonal cell c (left panel) and the face-based unknowns u f that refers to the faces f ∈ F(c) (right panel).

u = −K grad p in Ω (constitutive equation), (3.1)

div u = b in Ω (conservation equation), (3.2)

where K is the symmetric and strictly positive definite diffusion tensor.We supplement Eqs. (3.1)–(3.2) with boundary conditions, which may be of Dirichlet or Neumann type, or even of mixed

type, e.g., Robin, on different subsets of the boundary ∂Ω . For the sake of exposition, we focus on the case of homogeneousDirichlet boundary conditions.

To discretize the Darcy problem, we first introduce two discrete fields that represent p and u (see Fig. 6):

• the cell-based discrete field ph ∈ Ch , whose components pc approximate the cell averages of p, cf. (2.18);• the face-based discrete fields uh ∈ Fh , whose components u f approximate the face averages of the normal component

of u, cf. (2.16).

Then, we replace the differential operator div by the primary discrete divergence operator DIV given in (2.17) and

the differential operator K grad by the discrete derived gradient operator GRAD given by (2.21). As pointed out in Sec-tion 2, these discrete operators act on the discrete fields in Ch and Fh , and allows us to immediately derive the mimeticapproximation of (3.1)–(3.2) as

uh = −GRAD ph (mimetic constitutive equation), (3.3)

DIV uh = bh (mimetic conservation equation). (3.4)

The linear system arising from (3.3)–(3.4) reads:

uh −M−1F DIV TMC ph = 0, (3.5)

DIV uh = bh, (3.6)

where the vector bh ∈ Ch contains the cell averages of the forcing term b. The system is symmetrized by multiplying thefirst equation by matrix MF and the second one by −MC .

The matrices MC and MF are built by assembling the local mimetic inner product matrices MC,c and MF ,c respectively,in accordance with the general definition (2.41). The matrix MC,c is given by (2.44). The matrix MF ,c is built below usingLemma 2.9. From a theoretical standpoint, this matrix must satisfy the consistency and stability conditions:

• (S1) (consistency). Let uh,c and vh,c be the discrete representations of the vector-valued functions u and v on cell c.Furthermore, let u be a constant vector, and v be such that div v is constant in c and flux v · n f is constant on eachface f of c. Finally, let Kc be a constant diffusion tensor. Then, the local mimetic inner product returns the exact valueof the integral of u and v (with K−1

c as weight):

[vh,c,uh,c]F,c =∫c

v · K−1c u dV . (3.7)

• (S2) (stability): there exist two positive constants C� and C� independent of the mesh size such that

C�|c|vTh,cvh,c � vT

h,cMF,cvh,c � C�|c|vTh,cvh,c ∀vh,c. (3.8)



Assumptions (S1) and (S2) have a direct correspondence with the two matrix terms in the right-hand side of (2.48).The first assumption is the exactness requirement on linear polynomials (also known as the P0 compatibility condition, see[55,58]) and leads to the first matrix term. Therefore, this term is common to all compatible discretization methods thathave the aforementioned exactness property. The second assumption requires that the mimetic inner product be uniformlycoercive, and, thus, its representing matrix be non-singular. Note that both constants C� and C� are independent of meshsize h but depend on the eigenvalues of Kc .

Remark 3.8. For any discrete vector vh,c = (v f ) f ∈∂c we can find a vector-valued function v that satisfies the assumptionsin (S1). In fact, v can be chosen as the (non-unique) solution of the diffusion problem with Neumann boundary conditions:

div v = (DIV vh)c in c,

nc, f · v = α f ,c v f for all f ∈ ∂c,

where we recall that nc, f is the unit vector orthogonal to f and pointing out of c and α f ,c = n f · nc, f .

Let us use the consistency condition to derive the matrices Rc and Nc needed in formula (2.48). Since K−1c u is a constant

vector, we can express it as the gradient of a linear function q with the zero mean value on c. We substitute K−1c u = grad q

in the right-hand side of (3.7) and integrate by parts. Since div v is constant, the volume integral disappears. We also splitthe boundary integral into integrals over faces f ∈ ∂c and use the assumption that the flux of v is constant on each face:∫

c

v · K−1c u dV =

∫c

v · grad q dV = −∫c

(div v)q dV +∫∂c

qv · nc dS

=∑

f ∈F(c)

αc, f (v · n f )

∫f

q dS.

This development allows us to express consistency property (3.7) only in terms of data located on the faces of c:

[vh,c,uh,c]F,c =∑

f ∈F(c)

αc, f (v · n f )

∫f

q dS =∑

f ∈F(c)

αc, f v f | f |q(x f ), (3.9)

where we recall that x f is the centroid of face f . Let xc = (xc, yc, zc)T . We obtain three algebraic exactness conditions by

taking q1 = x− xc , q2 = y − yc , and q3 = z − zc . Let u( j)h,c be the local vectors representing the vector functions u j = Kcgrad q j

for j = 1,2,3. Then, formula (3.9) can be re-written in the equivalent form:

vTh,cMF,cu( j)

h,c = vTh,cr( j)

h,c, (3.10)

where the components of vector r( j)h,c depends on q j and on the cell geometry. In view of Remark 3.8, vector vh,c is arbitrary

and we can cancel it out, thus obtaining the algebraic requirements (2.45) with u( j)h,c in place of N j and r( j)

h,c in place of R j .

Let us now introduce the matrices Nc = [u(1)

h,c,u(2)

h,c,u(3)

h,c] and Rc = [r(1)

h,c, r(2)

h,c, r(3)

h,c], which allows us to reformulate theconsistency condition (3.9) as

MF,cNc = Rc. (3.11)

Matrix MF ,c is provided by the construction of Lemma 2.9, or, alternatively, by Corollary 2.1, if we prove that NTR is an SPDmatrix. To this end, we have first to give the explicit expression for N and R. For convenience, let us enumerate the facesof c by an index running from 1 to m, m being the number of faces of c and denote the components of the normal vectorn f i by (nx fi

,ny fi,nz fi

) and the components of the face centroid x f i by (x fi , y fi , z fi ). Let us note that the i-th component of

vector u(1)

h,c is given by (2.16) applied to face f i . Since grad q1 = e1 = (1,0,0)T and the integrand is a constant quantity overf i , a straightforward calculation yields

u(1)

h,c|i = 1

| f i|∫f i

n f i · u1 dS = 1

| f i|∫f i

n f i · Kc grad q1 dS = n f i · Kce1,

that is, vector n f i times Kce1, the first column of matrix Kc . A similar relation holds for the i-th component of the other

vectors u(2)

h,c and u(3)

h,c , and the i-th row of N, which corresponds to face f i , is thus given by(u(1)

h,c

∣∣i,u(2)

h,c

∣∣i,u(3)

h,c

∣∣i

)= nTfi

Kc = (nx fi,ny fi

,nz fi)Kc.

Collecting all the matrix rows yields:



Nc =

⎛⎜⎜⎜⎜⎝nx f1

ny f1nz f1

nx f2ny f2

nz f2

......

...

nx fmny fm

nz fm

⎞⎟⎟⎟⎟⎠Kc .

To determine the form of matrix Rc we proceed in a similar way by noting that the i-th component of r(1)

h,c|i , whichcorresponds to face f i , is given by setting vh,c = ei in (3.10), or, equivalently in (3.9). Using this latter equation and thecorrespondence between u1 and q1 yields:

r(1)

h,c

∣∣i = [ei,u(1)

h,c

]F,c = αc, f i | f i|q1(x f i ) = αc, f i | f i|(x fi − xc),

that is, αc, f i | f i | times the first component of the row vector (x f i − xc)T . A similar relation holds for the i-th component of

the other vectors r(2)

h,c and r(3)

h,c , and the i-th row of R, which corresponds to face f i , is thus given by(r(1)

h,c

∣∣i, r(2)

h,c

∣∣i, r(3)

h,c

∣∣i

)= αc, f i | f i|(x f i − xc)T

= (αc, f i | f i|(x fi − xc),αc, f i | f i|(y fi − yc),αc, f i | f i|(z fi − zc)).

Collecting all matrix rows yields:

Rc =

⎛⎜⎜⎜⎝αc, f1 | f1|(x f1 − xc) αc, f1 | f1|(y f1 − yc) αc, f1 | f1|(z f1 − zc)

αc, f2 | f2|(x f2 − xc) αc, f2 | f2|(y f2 − yc) αc, f2 | f2|(z f2 − zc)

......

...

αc, fm | fm|(x fm − xc) αc, fm | fm|(y fm − yc) αc, fm | fm|(z fm − zc)

⎞⎟⎟⎟⎠ .

Lemma 3.1. For any polyhedral cell c, we have

NTc Rc = Kc|c|. (3.12)

Proof. Without loss of generality, we place the origin of the coordinate system into the cell centroid, i.e. xc = (0,0,0)T . Wedenote the i-th spatial coordinate by x(i) , i.e., x = (x(1), x(2), x(3))T . Let ei be the three-sized vector whose i-th entry is 1 andall other entries are 0. Clearly, it holds that ei = grad x(i) . We write the (i j)-th entry of the right-hand side of (3.12) usingthe vectors ei and e j = grad x( j):

(Kc)i j|c| = ei · Kce j|c| = ei · Kc grad x( j)|c| =∫c

(Kcei) · grad x( j) dV .

Integrating by parts and noting that the volume integral is zero, we obtain

(Kc)i j|c| = −∫c

(div Kcei)x( j) dV +∫∂c

(Kcei · nc)x( j) dS =∫∂c

(Kcei · nc)x( j) dS.

We split the last integral in the face contributions and re-arrange the terms of the scalar product to obtain

(Kc)i j|c| =∑

f ∈F(c)

αc, f (Kcei · n f )

∫f

x( j) dS =∑

f ∈F(c)

αc, f (Kcei · n f )| f |x( j)f = (NTR

)i j .

This proves the assertion of the lemma. �The lemma implies that matrix NT

c Rc is symmetric and positive definite, so that Corollary 2.1 can be used for thepractical construction of matrix MF ,c . Under assumption (M1), it is possible to show the stability condition (S2) for thismatrix.

Proposition 3.1. The inner product matrix MF ,c given by (2.50) satisfies the stability condition (S2) with constants C∗ and C∗ thatdepend only on the space dimension d, the two mesh regularity constants N� and α∗ , and the ellipticity constants that bound thespectrum of Kc .

Proof. In view of Lemma 3.1, by inserting relation RTc Nc = |c|Kc into formula (2.50), we obtain:

MF,c = M0 + λF,cM1 , M0 = |c|−1RcK−1

c RTc , M1 = I−Nc

(NT

c Nc)−1

NTc .
F,c F,c F,c F,c



As Kc is strongly elliptic, there exist two positive constants κ∗ and κ∗ such that

κ∗‖ξ‖2 � ξ T Kcξ � κ∗‖ξ‖2 ∀ξ ∈ �d,

where ‖ξ‖2 = ξ T ξ . Assumption (M1) implies that all geometric objects of cell c have bounded measures:

1

γ�

hd−1c � | f | � γ�hd−1

c ,1

γ�

hdc � |c| � γ�hd

c ,

where hc is the diameter of c and γ� is a constant depending only on N� and α∗ . Using these results, we an show a numberof intermediate estimates as

‖Ncw‖2 =∑

f ∈F(c)

(nT

f Kcw)2 � N�

(κ∗)2‖w‖2 ∀w ∈ �m. (3.13)

The definition of λF ,c and the formulas for the coefficients of matrix Rc give

λF,c = 1

2|c| trace(RcK−1

c RTc

)= 1

2|c|∑

f ∈F(c)

| f |2(x f − xc)T K−1

c (x f − xc).

Since the distance between the centroids xc and x f is greater than the radius rt of the inscribed ball (see assumption (M1))and smaller than hc , we obtain the lower and upper bounds:(

κ�)−1

α2�γ

−4∗ |c| � λF,c �1

2N�(κ�)

−1γ 4∗ |c|.A similar argument can be used to derive the following upper bound:

∥∥RTc vh,c

∥∥2 =d∑

i=1

∣∣(r(i)h,c

)Tvh,c∣∣2 � d‖vh,c‖2

∑f ∈F(c)

| f |2‖x f − xc‖2 � dN�γ4∗ |c|2‖vh,c‖2.

Finally, we need a special lower bound for the euclidean norm of RTc vh,c . Let us now decompose vh,c = vh,N + vh,⊥ , where

vh,N ∈ span(N) and vh,⊥ ∈ (span(N))⊥; moreover, ‖vh,c‖2 = ‖vh,N‖2 + ‖vh,⊥‖2. Using (3.13), we obtain:∥∥RTc vh,N

∥∥= ∥∥RTc Ncw

∥∥= |c|‖Kcw‖� |c|κ∗√N�κ∗ ‖Ncw‖ = |c|κ∗√

N�κ∗ ‖vh,N‖.

With the above estimates, it is easy to obtain the upper bound in the stability estimate (S2) with a constant independentof mesh:

vTh,cMF,cvh,c �

1

|c|κ∗∥∥RT

c vh,c∥∥2 + λF,c‖vh,c‖2 � |c| (2d + 1)N�γ

4∗2κ∗

‖vh,c‖2.

The low bound requires a little bit more work. We recall the following vector inequality:

−2a · c � ε‖a‖2 + 1

ε‖c‖2 ∀ε > 0.

Using this inequality, we obtain

vTh,cMF,cvh,c �

κ∗|c|∥∥RT

c (vh,N + vh,⊥)∥∥2 + λF,c‖vh,⊥‖2

� κ∗|c|∥∥RT

c vh,N

∥∥2(1 − ε) + κ∗

|c|∥∥RT

c vh,⊥∥∥2(

1 − 1

ε

)+ λF,c‖vh,⊥‖2.

If we take ε < 1 and apply the above inequalities, we get the following estimate:

vTh,cMF,cvh,c � (1 − ε)|c|β1‖vh,N‖2 +

((1 − 1

ε

)β2 + β3

)|c|‖vh,⊥‖2,

where the positive constants βi depend only on α� , N� , κ∗ and κ∗ . The lower bound follows for ε = β2/(β2 + 12 β3). This

proves the assertion of the proposition. �Under assumptions (S1) and (S2), the mimetic finite difference method has optimal convergence properties. A priori error

estimates are derived in [55] using the mesh-dependent norms ‖|vh‖|F = [vh,vh]F and ‖|qh‖|C = [qh,qh]C induced by themimetic inner products. We report the main convergence result in the theorems below.



Theorem 3.3. Let assumption (M1) hold and u ∈ H(div,Ω) and p ∈ H2(Ω) be the solution of problem (3.1)–(3.2) with the homoge-neous boundary condition. Furthermore, let uI ∈ Fh and pI ∈ Ch be the discrete fields representing u and p. Finally, let uh ∈ Fh andph ∈ Ch be the mimetic solution of the discrete problem (3.3)–(3.4) under assumptions (S1)–(S2). Then,∥∥∣∣uI − uh

∥∥∣∣Fh

+ ∥∥∣∣pI − ph∥∥∣∣Ch

� Ch‖p‖H2(Ω),

where the constant C is independent of the mesh.

Theorem 3.4. Let b ∈ H1(Ω), the domain Ω be convex and the assumptions of Theorem 3.3 hold. Then,∥∥∣∣pI − ph∥∥∣∣Ch � Ch2(‖p‖H2(Ω) + ‖b‖H1(Ω)

),

where the constant C is independent of the mesh.

The linear convergence rate for uh has been confirmed experimentally on more general meshes than the ones satisfyingassumption (M1). A sharper superconvergence result for ph can be found in [67] where the regularity condition on b is nolonger required.

3.1.1. Related developments(i) A post-processing technique developed in [22,68] can be used to construct a discontinuous piecewise linear approxi-

mation to the scalar solution p. From the mimetic solution uh,c , we can calculate the constant vector field

g∗c = −K−1

c

|c| RTc uh,c . (3.14)

The piecewise constant vector function g∗ such that g∗|c = g∗c is a first-order approximation of the exact solution gradient

grad p on Ω , i.e., ‖g∗ − grad p‖L2(Ω) = O(h). Let χc(x) be the characteristic function of cell c, e.g., χc(x) = 1 for x ∈ c andχc(x) = 0 for x /∈ c. The piecewise linear function

p∗(x) =∑c∈Th

p∗c (x)χc(x) where p∗

c (x) = pc + g∗c · (x − xc)

is a second-order accurate approximation of the exact solution p, i.e., ‖p∗ − p‖L2(Ω) = O(h2). This post-processing methodwas successfully applied to develop a posteriori error estimators for the mimetic method described here and to implementadaptive mesh refinement strategies [22,31].

(ii) As to solving the saddle-point system (3.3)–(3.4), an efficient implementation is possible through an equivalentreformulation that is based on the hybridization technique [54]. Such hybridization introduces a new set of auxiliary faceunknowns and the coefficient matrix of the resulting system for such unknowns is sparse, symmetric and positive definite.Thus, efficient iterative Krylov solvers, such as the preconditioned conjugate gradient method, can be applied to solve itnumerically. Another advantage of this reformulation is that heterogeneous essential boundary conditions can be applieddirectly to the auxiliary unknowns.

(iii) A high-order mimetic finite difference method is derived using a more accurate discrete representation of u, whichintroduces new additional degrees of freedom on mesh faces. The design of such a method proceeds as usual by defininga family of suitable inner products and discrete mimetic operators. Formally, we have to enrich the consistency conditions(S1) by requiring that the exactness property holds for a larger polynomial space, see Section 5.2.

3.1.2. Connection with mixed finite elementsLet us multiply Eq. (3.3) by vh ∈ Fh by using the mimetic inner product of Fh , and use the duality relation between

DIV and GRAD. Let us multiply Eq. (3.4) by qh ∈ Ch by using the mimetic inner product of Ch . The mimetic methodgiven by Eqs. (3.3)–(3.4) can be reformulated as follows: find uh ∈Fh and ph ∈ Ch such that

[vh,uh]F − [DIV vh,ph]C = 0 ∀vh ∈ Fh, (3.15)

[DIV uh,qh]C = bh ∀qh ∈ Ch. (3.16)

Eqs. (3.15)–(3.16) provides a mimetic formulation that is very similar to the following mixed finite element variationalformulation of the Darcy problem (3.1)–(3.2).

Let Vh and Wh denote two finite element spaces for vector and scalar field, that are to be properly chosen for stabilityreasons, cf. [54]. The mixed finite element method reads as: find the finite element approximations uh ∈ Vh and ph ∈ Wh suchthat


∫Ω
K−1uh · vh dV −∫Ω

p div vh dV = 0 ∀vh ∈ Vh, (3.17)

∫Ω

q div uh dV =∫Ω

f q dV ∀q ∈ Wh. (3.18)

If we compare Eqs. (3.17)–(3.18) with the mimetic approximation (3.15)–(3.16), the similarity between the two formulationsis evident. We can interpret the mimetic inner products [·, ·]Fh and [·, ·]Ch as specific realizations of the integrals in themixed finite element method. This interpretation is supported by the evidence that on meshes of simplices, the MFD methoddescribed in (3.1) and the RT0 −P0 mixed finite element method have the same degrees of freedom and the same expressionfor the divergence, and the same exactness property holds for constant velocity fields. Moreover, if in Eq. (2.48) we select

Uc = 1

d2(d + 1)|c|∑

f ∈F(c)

(x f − xc)T K−1

c (x f − xc) and Dc =

⎛⎜⎜⎝| f1|| f2|...

| fm|

⎞⎟⎟⎠ ∀c ∈ Th, (3.19)

the RT0 mass matrix coincides with the mimetic matrix MF ,c [68]. Thus, we can conclude that the RT0 − P0 mixed finiteelement method is a member of the family of the MFD methods.

3.2. Maxwell’s equations: primary and derived curl operators

3.2.1. Time-dependent problems and energy conservationIn this subsection, we consider two time-dependent problems governed by the Maxwell’s equations. The first problem

for the magnetic flux density B and the electric field E is given by:

∂B

∂t= −curl E,

∂D

∂t= curl H in Ω, (3.20)

where H = μ−1B is the magnetic field, D = εE is the dielectric displacement, μ is the magnetic permeability, and ε is theelectric permittivity. We assume that the magnetic permeability and the electric permittivity, which can be tensor fields, donot depend on time; however, these fields may be discontinuous at the interface between different media. The solutions ofEqs. (3.20) are also required to satisfy the divergence-free conditions:

div B = 0, div D = 0 in Ω. (3.21)

We consider the homogeneous boundary condition n × E = 0 on ∂Ω that describes a perfect conductor.The second problem is the magnetic diffusion and is governed by the Maxwell’s equations:

∂B

∂t= −curl E, E = σ−1curl H in Ω, (3.22)

where the conductivity σ is a symmetric positive definite discontinuous tensor. We consider again the homogeneous bound-ary condition n × E = 0.

In electromagnetism, the tangential component of E and the normal component of B are continuous across media dis-continuities [155,176,248]. Thus, they are the natural choice for the discretization of the electric field intensity and themagnetic flux density. Let us introduce

• the discrete field Eh ∈ Eh whose components approximate the edge averages of the tangential component of the vectorfunction E on mesh edges, cf. (2.13);

• the discrete field Bh ∈Fh whose components approximate the face averages of normal component of the vector functionB on mesh faces, cf. (2.16).

Since we do not use discrete analogs for D and H, we reformulate (3.20) as follows:

∂B

∂t= −curl E,

∂E

∂t= ε−1curlμ−1 B in Ω. (3.23)

The primary CURL operator is the discrete analog of the curl operator in the first equation of (3.23), while the derived

CURL operator is the discrete analog of the differential operator ε−1curlμ−1 that includes all the material properties.Likewise, we reformulate problem (3.22) as

∂B = −curl E, E = σ−1curlμ−1 E in Ω, (3.24)
∂t



and introduce the primary CURL operator as the discrete analog of the curl operator in the first equation and the derived

CURL operator as the discrete analog of σ−1curlμ−1. Later, we focus on the first problem, since the numerical treatmentof the second one is similar.

For a perfectly conducting medium, the duality relationship of the first-order operators curl and ε−1curlμ−1 followsfrom (2.6):∫

Ω

curl E · μ−1B dV =∫Ω

εE · (ε−1curlμ−1B)

dV . (3.25)

As we mentioned earlier, the problem coefficients lead to modify the definition of the mimetic inner products to take intoaccount the material properties as weights. In this case, the modified inner products for spaces Eh and Fh are such that

[Eh, Eh]E ≈∫Ω

E · εE dV and [Bh, Bh]F ≈∫Ω

B · μ−1B dV . (3.26)

In the mimetic method, the adjoint relation between CURL and CURL is formulated with respect to the inner products:

[vh, CURLwh]E = [CURLvh,wh]F ∀wh ∈ Fh,vh ∈ Eh.

This is the discrete analog of Green’s formula (3.25). The derived operator CURL includes the media properties ε and μ−1.The conditions div D = divεE = 0 are discretized by defining discrete analogs of the operators div and divε. To this

purpose, we modify the integral identity (2.4) by introducing ε as a weight:∫Ω

u divεE dV = −∫Ω

grad u · εE dV .

This suggests us to define the discrete analog of divε as the negative adjoint of the primary discrete operator GRAD usingthe first inner product in (3.26) for space Eh .

Remark 3.9. The derived operators CURL and DIV are different from those introduced in Section 2 as they include thematerial properties. Nonetheless, they are consistent and a discrete analog of the fundamental relation div curl = 0 holdstrue

DIV CURL = −M−1N GRADTME M

−1E CURLT MF = −M−1

N (CURLGRAD)TMF = 0.

We use the derived operators CURL and DIV in combination with the primary operators CURL and DIV of Sec-tion 2.2 to build a mimetic approximation of the Maxwell’s equations. More precisely, let E0

h denote a proper subspace of Eh

consisting of the vectors whose components are zero for the boundary edges. The mimetic discretization reads: find Eh ∈ E0h

and Bh ∈Fh such that

∂Bh

∂t= −CURL Eh,

∂Eh

∂t= CURL Bh (3.27)

and

DIV Eh = 0, DIV Bh = 0. (3.28)

Likewise, we define a derived operator CURL that approximates the differential operator σ curlμ−1 (instead ofεcurl μ−1) and the mimetic semi-discretization of problem (3.24) reads: find Eh ∈ E0

h and Bh ∈Fh such that

∂Bh

∂t= −CURLEh, Eh = −CURLBh,

again with the divergence-free conditions (3.28).The important property of these mimetic methods is that the divergence-free condition at the initial time t = 0 is

preserved at any subsequent instant. Indeed, using the second equation in (3.27), yields

∂

∂t(DIV Eh) = DIV

∂Eh

∂t= DIV CURLBh = 0 (3.29)

in view of Remark 3.9 (see also Lemma 2.5). Likewise, using the first equation in (3.27) and Lemma 2.3, we obtain

∂(DIV Bh) = DIV

∂Bh = −DIV CURLBh = 0. (3.30)
∂t ∂t



Another important property of the mimetic method is the conservation of the electromagnetic energy, which is definedas

E = 1

2

(∫Ω

E · D dV +∫Ω

B · H dV

). (3.31)

The energy conservation for a conducting medium is strictly connected with the fundamental mathematical property thatthe operator curl is self-adjoint. Let us multiply the first equation in (3.20) by μ−1B, the second one by εE, sum them up,and integrate the result over the computational domain Ω . In the left-hand side we easily recognize the time derivative ofthe electromagnetic energy, while the right-hand side is zero due to (3.25):

2∂E

∂t=∫Ω

E · ∂D

∂tdV +

∫Ω

B · ∂H

∂tdV = −

∫Ω

curl E · H dV +∫Ω

E · curl H dV = 0.

We define the discrete analog of the electromagnetic energy (3.31) as

Eh = 1

2

([Eh,Eh]E + [Bh,Bh]F). (3.32)

The conservation of the discrete electromagnetic energy Eh is stated by the following theorem.

Theorem 3.5. The discrete electromagnetic energy Eh given by (3.32) is conserved in the mimetic method.

Proof. Let us take the product of Bh with both sides of the first equation in (3.27) using the inner product in Fh and theproduct of Eh with both sides of the second equation in (3.27) using the inner product in Eh:[

∂Bh

∂t,Bh

]F

= −[CURLEh,Bh]F ,

[∂Eh

∂t,Eh

]E

= [CURLBh,Eh]F .

By adding these two equations and using definition (3.32) and the duality property (2.22), we obtain:

2∂Eh

∂t= −[CURLEh,Bh]F + [CURLBh,Eh]F = 0.

This proves the assertion of the theorem. �Eventually, a fully discrete method can be obtained by introducing a suitable time-stepping scheme for the time deriva-

tive, which can be implicit, semi-implicit, or explicit. The effectiveness of these mimetic discretizations is shown bynumerical experiments on logically rectangular meshes [142].

3.2.2. MagnetostaticsIn this subsection, we consider the mimetic approximation of the magnetostatic problem in div–curl form, which is

characterized by different degrees of freedom and different mimetic inner products. The div–curl problem for the vectorpotential A and a suitable Lagrange multiplier field p is expressed by the following system of equations:

curl(μ−1curl A

)+ grad p = J in Ω, (3.33)

div A = 0 in Ω, (3.34)

A × n = 0 on ∂Ω, (3.35)

where J is a given current vector.Let N 0

h denote a proper subspace of Nh consisting of vectors whose components are zero for boundary nodes. Themimetic discretization of this problem uses the following discrete representations of p and A:

• the node-based discrete field ph ∈N 0h whose components pn approximate the value of the scalar field p at mesh nodes

n;• the edge-based discrete field Ah ∈ E0

h whose components Ae approximate the value of the integrals of the vector fieldA along the mesh edges e, cf. formula (2.13).

We represent the differential operators curl and curlμ−1 that appear in Eq. (3.33) by the primary discrete operator

CURL defined by (2.14) and the derived operator CURL of the previous subsection. We represent the gradient operatorin (3.33) by the primary discrete operator GRAD defined in (2.11) and the divergence operator in (3.34) by the adjointoperator DIV of the previous subsection. Using these operators, the mimetic discretization of (3.33)–(3.35) reads: findAh ∈ E0 and ph ∈N 0 such that
h h



CURL CURLAh + GRAD ph = Jh, (3.36)

DIV Ah = 0, (3.37)

where Jh = ( Je)e∈E represents the vector function J in Eh and the degrees of freedom Je are defined by (2.13). The linearalgebraic formulation follows immediately by using the definition of primary and derived operators:

M−1E CURLTMF CURLAh + GRAD ph = Jh, (3.38)

M−1N GRADT MEAh = 0. (3.39)

Since M−1E can be dense on an unstructured mesh, a computationally tractable system is obtained by multiplying the first

equation by ME . A symmetric system is obtained by multiplying the second equation by MN .The derivation of matrix ME follows the path outlined in the previous section. First, we introduce consistency and

stability conditions for local matrices ME,c . Then, we derive formulas for the matrices Rc and Nc , see [180] for details.Finally, we apply Lemma 2.9 to obtain a family of admissible matrices ME,c .

The well-posedness of the mimetic method is stated by the following theorem. Its proof can be found in [180].

Theorem 3.6. Let Th be a simply connected mesh. Then, problem (3.38)–(3.39) admits a unique solution.

The accuracy of the mimetic method is illustrated numerically in [180] on a set of academic problems and a realisticengineering problem.

3.3. Diffusion and elasticity equations: mimetic semi-inner products

3.3.1. The diffusion equationConsider again the diffusion problem (3.1)–(3.2) and discretize it using a different pair of primary and dual operators.

The discrete unknowns are given by

• the node-based field ph ∈N 0h whose components pn approximate the nodal values of p.

• the edge-based field uh ∈ Eh whose components ue approximate the tangential components of u on mesh edges e, cf.(2.13).

With this selection of unknowns, it is natural to select GRAD as the primary operator and DIV as the derived operator:

uh = −GRAD ph,

DIV uh = bh, (3.40)

where bh ∈Nh . Using (2.25), we obtain the discrete problem

M−1N GRADT ME GRAD ph = bh

or, equivalently,

AN ph = MN bh, AN = GRADT ME GRAD.

Note that heterogeneous essential boundary conditions can be imposed by setting prescribed values (instead of zeros) tothe components of ph at the boundary nodes and eliminating the corresponding equations from the global system.

The discrete gradient operator acts from Nh onto a proper subspace of Eh . This subspace is much smaller than Eh andonly the action of matrix ME in this subspace is needed to define AN . Consider vh = GRAD qh and uh = GRAD rh anddefine the following semi-inner product on Nh:

[vh,uh]E = [GRAD qh,GRAD rh]E ≡ [[qh, rh]]N .

The semi-inner product [[·, ·]]N represents the energy semi-norm in space Nh . Mimicking the additivity of integration,we assume that the semi-inner product can be assembled from cell-based contributions:

[[qh, rh]]N =∑c∈Th

[[qh, rh]]N ,c . (3.41)

The bilinear form [[·, ·]]N ,c defined in (3.41) is required to satisfy an exactness property stated by the modified consistency,(S1a), and stability, (S2a), conditions.



• (S1a) (consistency). Let qh be the discrete representation of a linear function q, rh be the discrete representation of acontinuous function r that can be integrated exactly on each face f of F(c) using only the nodal values at the verticesof f , and Kc be a constant diffusion tensor on c. Then, the local mimetic semi-inner product returns the exact value ofthe integral of grad q and grad r (with Kc as weight):

[[qh,c, rh,c]]N ,c =∫c

Kc grad q · grad r dV . (3.42)

• (S2a) (stability). Let GRADc be the restriction of the discrete gradient operator to cell c. Then, there exist two positiveconstants C� and C� independent of the mesh size such that

C�|c|‖GRADcqh,c‖2 � [[qh,c,qh,c]]N ,c � C�|c|‖GRADcqh,c‖2 ∀qh,c . (3.43)

Let Kc be a constant approximation of the diffusion tensor K on cell c. The volume integral can be reduced to a surfaceintegral by applying the Green’s formula:∫

c

Kc grad q · grad r dV = −∫c

r div (Kc grad q)dV +∫∂c

Kc grad q · nr dS

=∫∂c

Kc grad q · n r dS

=∑

f ∈F(c)

αc, f Kc grad q · n f

∫f

r dS. (3.44)

The next logical step is to clarify the requirements on the continuous function r. The convergence theory can be built ifwe assume that (3.42) is exact for a function r that can be integrated exactly for each face f using a quadrature rule withquadrature points at mesh vertices and weights ω f ,n:∫

f

r dS =∑

n∈N ( f )

ω f ,nr(xn) =∑

n∈N ( f )

ω f ,nrn. (3.45)

For example, this assumption is satisfied by the functions that are linear on ∂c up to an additive bubble function that haszero average on c.

Let AN ,c be the matrix representing the local semi-inner product. Inserting formulas (3.45) and (3.44) into (3.42) andre-arranging the summation terms, we obtain:

rTh,cAN ,cqh,c =

∑f ∈F(c)

αc, f Kc grad q · n f

∑n∈N ( f )

ω f ,nrn ≡ rTh,csh,c,

where the components of vector sh,c depend on the cell geometry, the material properties, the quadrature weights, and thelinear function q. The key point, is that sh,c can be easily computed for any given function q. The function r can be selectedsuch that it is one at any vertex of c and is zero in the other vertices. This leads to the fact that rh,c can be an arbitraryvector. Canceling it out, we get a system of linear equations with respect to the entries of the unknown matrix AN ,c :

AN ,cqh,c = sh,c(q), ∀q ∈ P1(c). (3.46)

Due to the linearity of this system, it is sufficient to consider only a set of linearly independent functions q. In threedimensions, there are only four such functions: q1 = 1, q2 = x, q3 = y, and q4 = z. Since grad q1 = 0, we obtain immediatelythat sh,c(1) = 0, i.e. the constant vector is in the null space of AN ,c .

Let m be the number of vertices of cell c and si(q) be the i-th component of vector sh,c . Let us define two rectangularmatrices Nc and Rc with m rows and four columns:

Nc =

⎡⎢⎢⎣1 x1 y1 z11 x2 y2 z2...

......

...

1 xm ym zm

⎤⎥⎥⎦ , Rc =

⎡⎢⎢⎣0 s1(x) s1(y) s1(z)0 s2(x) s2(y) s2(z)...

......

...

0 sm(x) sm(y) sm(z)

⎤⎥⎥⎦ .

This leads to a simplified form of Eq. (3.46):

AN ,cNc = Rc, (3.47)

which is similar to Eq. (2.46) studied in Section 2.6, where the role of matrix MQ,c is played by AN ,c . For matrices Rc andNc there holds a lemma similar to Lemma 3.1.



Lemma 3.2. The matrices Nc and Rc satisfy

NTc Rc =

[0 00 |c|Kc

].

Proof. Let r be a linear function in (3.42). Since, the quadrature rule (3.45) is exact for linear functions, we obtain

qTh,c sh,c(r) = qT

h,c AN ,c rh,c =∫c

Kc grad q · grad r dV = Kc grad q · grad r|c|.

Taking various pairs of linear functions q and r, we prove the assertion of the lemma. �Since NT

c Rc is a singular matrix, we need to generalize the results of Lemma 2.9 and Corollary 2.1. It is not difficult toverify that the solution to (3.47) is given by

AN ,c = Rc(RT

c Nc)†RT

c +DcUcDTc , (3.48)

where D and U have the same definitions given for Lemma 2.9, and (RTc Nc)

† denote the pseudo-inverse:

(RT

c Nc)† =

[0 00 (|c|Kc)

−1

].

As in Corollary 2.1, a one-parameter family of mimetic inner product is provided by the formula:

AN ,c = Rc(RT

c Nc)†RT

c + λN ,c(I−N

(NTN

)−1NT ), (3.49)

where

λN ,c = trace{Rc(RT

c Nc)†RT

c

}.

The matrix (3.49) satisfies the consistency condition (S1a) by construction. The stability condition (S2a) can be shown byadopting the arguments used in the proof of Proposition 3.1.

Remark 3.10. Matrix AN ,c coincides with the stiffness matrix of the linear Galerkin finite element method when c is atetrahedral or a triangular cell.

Let mesh Th satisfy the regularity assumptions formulated above. Let ‖| · ‖| be the semi-norm induced by (3.41). Then,the following first-order convergence estimate is proved in [52].

Theorem 3.7. Let Ω be a bounded Lipschitz polyhedron and K be a W 1,∞(Ω) symmetric tensor. Furthermore, let the sequence ofdecompositions Th satisfy assumption (M1), and the energy inner product satisfy assumptions (S1a) and (S2a). Finally, let pI ∈N 0

h bethe discrete field collecting the values of the exact solution, and ph be the solution of (3.40). Then,∥∥∣∣pI − ph

∥∥∣∣N � Ch

(‖b‖0,Ω + |p|1,Ω + |p|2,Ω

),

where C depends only on the mesh regularity constants and K.

3.3.2. The elasticity equationThe construction of the semi-inner products can be generalized to other second-order PDEs. Consider, for example, the

equation of linear elasticity:

−div(Cε(u)

)= f,

where u is the displacement, C is the symmetric positive definite fourth-order tensor, f is the external load, and ε(u) is thesymmetrized gradient:

ε(u) = (grad u + (grad u)T )/2.

Repeating arguments behind (3.42) and (3.44), we formulate the consistency and stability conditions for the local mimeticsemi-inner product.



• (S1b) (consistency). Let qh ∈N 3h and rh ∈N 3

h be the discrete representations of a linear vector-valued function q and ofa continuous vector-valued function r that can be integrated exactly on each face f of F(c) using only the nodal valuesat the vertices of f . Let Cc be the constant tensor in c. Then, we require that

[[qh,c, rh,c]]N 3,c =∫c

Cc ε(q) : ε(r)dV . (3.50)

• (S2b) (stability). Let hc be the diameter of cell c. Let qh,c be any discrete vector except for the vectors representing rigidbody motion or rotation. Then, there exist two positive constants C� and C� independent of the mesh size such that

C�

|c|h2

cqT

h,cqh,c � [[qh,c,qh,c]]N 3,c � C� |c|h2

cqT

h,cqh,c ∀qh,c.

Let AN 3,c be a matrix representing the semi-inner product. The consistency condition leads to a set of algebraic equa-tions with respect to this matrix. The integration by parts in (3.50) gives

[[qh,c, rh,c]]N 3,c =∫∂c

(Cc ε(q) · n

) · r dS

=∑

f ∈F(c)

αc, f(Cc ε(q) · n f

) · ∫∂c

r dS =∑

f ∈F(c)

αc, f(Cc, ε(q) · n f

) · ∑n∈N ( f )

ω f ,n rn.

The right-hand side can be calculated for any given linear vector function q. We denote the result as rTh,csh,c . It is not

difficult to show that rh,c can be an arbitrary vector. Then, we obtain the following algebraic equations:

AN 3,cqh,c = sh,c, ∀q ∈ (P1(c))3

.

In three-dimensions there are only 12 linearly independent linear vector functions qi . They define 12 equations with respectto the unknown matrix AN 3,c . We can write these equations in a matrix form resembling (3.47) with 3m × 12 rectangular

matrices Nc and Rc in place of Nc and Rc , respectively, where m = |N (c)|.Let us split qi into two groups. The first group of six functions correspond to different rigid body motions and rotations.

Since the symmetrized gradient is zero for these functions, we obtain sh,c(qi) = 0, i = 1, . . . ,6.

Lemma 3.3. Let Cc be a constant tensor in cell c. Then, matrices Nc and Rc satisfy

NTc Rc =

[O O

O |c|A0

],

where O is the 6 × 6 zero matrix. The entries of the 6 × 6 matrix A0 are calculated as follows:

(A0)i j = Ccε(qi+6) : ε(q j+6)|c|.

The proof of this lemma follows closely the proof of Lemma 3.2 and is, therefore, omitted. Finally, an analog of formula(3.48) also holds true.

This semi-inner product has been used in [26,29] to discretize the Stokes problem. Optimal convergence error estimateshave been proved there under assumptions (S1b)–(S2b).

4. M-adaptation

In this section, we show how to select a member of the family of the mimetic schemes that provides additional prop-erties such as a discrete maximum principle or reduced numerical dispersion. The selection is based on a special choice ofthe parameters that define the local inner product matrix MQ,c (where we recall that Q may be one of the discrete spacesNh , Eh , Fh , and Ch).

4.1. Monotone mimetic schemes

Let us consider the diffusion problem (3.1)–(3.2). Maximum principle and solution positivity are two of the most impor-tant properties of the continuum solutions.

Theorem 4.8 (Maximum principle). Let us suppose that the scalar field p satisfies

−div(K grad p) � 0 in Ω



Fig. 7. Degrees of freedom of the mimetic finite difference method for the diffusion problem in mixed form: the cell-based scalar unknown pc associatedwith the polygonal cell c (left panel), the face-based flux unknown u f (middle panel) and the face-based scalar unknown p f (right panel) that refer to thefaces f ∈ F(c).

where K and Ω satisfy the assumptions considered in Section 3.1. Then,

maxx∈Ω

p(x)� maxx∈∂Ω

(0, p(x)

).

Theorem 4.9 (Monotonicity property). Let us suppose that p satisfies

−div(Kgrad p)� 0 in Ω

and p � 0 on the Dirichlet boundary, n · K grad p � 0 on the Neumann boundary where K and Ω satisfy the assumptions consideredin Section 3.1 for problem (3.1)–(3.2). Then, p � 0 in Ω .

The numerical analogs of these properties, the discrete maximum principle (DMP) and the discrete monotonicity prop-erty, are among the most difficult properties to achieve in numerical methods, especially when the computational mesh isdeformed to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic.Violation of the DMP may lead to numerical instabilities such as oscillations and to nonphysical solutions such as heat flowfrom a cold material to a hot one.

In [182], a set of sufficient conditions are devised to ensure the monotonicity of the mimetic finite difference methodon two- and three-dimensional meshes. To describe the main results, we first re-formulate the method in an algebraicallyequivalent form. The degrees of freedom for the scalar variable p are associated with the mesh cells c and the mesh faces f(see left and right panels of Fig. 7). The cell-based degrees of freedom are denoted by pc and represent the cell averages(see (2.18)). The new face-based degrees of freedom are denoted by p f and represent face-averages:

p f ≈ 1

| f |∫f

p dS. (4.1)

We introduce two degrees of freedom per interior mesh face for the vector variables u denoted by uc′, f and uc′′, f , wherec′ and c′′ are two cells with the common face f . They represent the face averages of u · nc, f (see (2.16)), where nc, f is thenormal vector to face f that point out of cell c (see the central panel of Fig. 7). We impose the trivial continuity condition:

uc′, f + uc′′, f = 0. (4.2)

The goal of introducing additional degrees of freedom is to discretize the duality relationships independently in eachmesh cell. Let the boundary of a cell c be formed by the m faces f i , i = 1, . . . ,m, with measures | f i |. We consider the localvector uh,c that collects the values of uc, f i on the faces f i . Let uh be the vector collecting all local vectors uh,c . The mimeticdiscretization of the mass balance equation (3.2) reads

(DIV uh)|c = 1

|c|∑

f ∈F(c)

uc, f | f | = bc|c|, (4.3)

where bc is the cell average of the loading term b.The mimetic discretization of the constitutive equation (3.1) uses the local mimetic inner product. Using the definition

of the flux field, i.e., u = −K grad p, and applying the Green formula, we obtain:∫K−1u · v dV = −

∫(grad p) · v dV =

∫p div v dV −

∮p(v · n)dS ∀v.

c c c ∂c



Let vector uh,c collect the unknowns uc, f . The discrete analog of the above formula is

[uh,c,vh,c]F,c = −[GRAD ph ,vh]C,c = pc (DIV vh)|c|c| −∑

f ∈F(c)

p f vc, f | f | ∀vh,c .

Summing up these equations for all the mesh cells, we obtain the discrete duality Green formula used in the definition ofthe derived gradient operator. This is one of the steps in proving the equivalence of the mimetic formulations considered inSection 3.1 and here.

We define WF ,c = M−1F ,c and rewrite the above equation in the equivalent form:⎛⎜⎜⎝

uc, f1

uc, f2

...

uc, fm

⎞⎟⎟⎠= −WF,c

⎛⎜⎜⎝| f1|(p f1 − pc)

| f2|(p f2 − pc)

...

| fm|(p fm − pc)

⎞⎟⎟⎠ . (4.4)

Remark 4.11. If matrix WF ,c is diagonal, then uc, f i = −Wc,ii(p fi − pc), where Wc,ii is the i-th diagonal entry of WF ,c .The flux continuity across face f i allows us to eliminate the face-based pressures and obtain a classical two-point fluxapproximation scheme:

u fi = −T fi (pc − pc′), T fi = Wc,iiWc′,iiWc,ii +Wc′,ii

,

where c and c′ are the two cells sharing face f i .A multi-point flux approximation scheme can be derived through the mimetic framework by controlling the sparsity

structure of matrix WF ,c and allowing it to be non-symmetric, see [188] for details. The new sparsity structure allowsus to collect face-based pressure unknowns into small independent groups around mesh points. The elimination of thesescalar unknowns leads to an MPFA-type scheme. Also, the MFD framework leads to a new family of cell-centered schemesof MPFA-type for the case when four or more faces in a cell meet at a corner. The MFD framework has also been used toanalyze some MPFA schemes, see [158].

Summarizing, the mimetic discretization in the hybrid form reads: find the cell-based values pc , the face-based values p f anduc, f that satisfy the three Eqs. (4.2)–(4.4) and the boundary conditions.

One of the advantages of the hybrid formulation is that essential and natural boundary conditions are easily imposed.The Dirichlet boundary conditions are imposed for face-based values p f . The Neumann boundary conditions are imposedfor uc, f . A combination of these variables makes it possible the implementation of Robin boundary conditions.

As pointed out at the end of Section 2.6, matrix WF ,c = (wij)mi, j=1 can be constructed directly. This is the SPD matrix,

and, thus, it is non-singular and its diagonal terms are strictly positive, i.e., wii > 0. The two following assumptions on thepattern of this matrix, labeled as (A1) and (A2), are crucial for obtaining a monotone mimetic scheme, cf. [182].

Assumption 1. Let WF ,c = (wij)mi, j=1 . We assume that

(A1) The matrix WFc satisfies the geometric constraint:

wii | f i | +∑j �=i

wij| f j| � 0 ∀i,

and the inequality is strict for at least one matrix row.(A2) The matrix WF ,c is a Z-matrix, i.e., wij � 0 for i �= j.

We summarize the main theoretical results concerning the monotone mimetic discretizations below; the proofs arein [182]. The first result states the discrete maximum principle that is the discrete analog of Theorem 4.8.

Theorem 4.10 (Discrete maximum principle). Let (pc)c∈Th and (p f ) f ∈Fhbe the solution of the hybrid mimetic method and Th be a

face-connected mesh. Furthermore, let each matrix WF ,c satisfy assumptions (A1) and (A2). Finally, let bc � 0 for all cells c. Then,

maxc∈Tc

pc � maxf ∈∂Ω

p f .

Corollary 4.2 (Discrete monotonicity property). Let conditions of Theorem 4.10 hold true. Furthermore, let the Dirichlet and Neumanndata be defined by nonnegative functions. Then, pc � 0 and p f � 0 for any f and any c.



A minimum principle can be also stated for non-positive input data, i.e., the loading term b and the boundary functionsdefining the Dirichlet and Neumann boundary conditions.

On simplicial meshes and for a particular choice of the mimetic inner product, the MFD method coincides with thelow-order Raviart–Thomas mixed-hybrid finite element method. Thus, in this case the conventional angle conditions guar-anteeing the discrete maximum principle are recovered.

On general meshes, Corollary 4.2 leads to a constructive way for designing a monotone MFD method. The sufficient con-ditions for that are the cell-based inequalities in Assumptions (A1)–(A2). Efficient solutions are found for meshes consistingof parallelograms, parallelepipeds, and orthogonal locally refined elements as those used in the Adaptive Mesh Refinement(AMR) methodology. Moreover, on meshes of parallelograms a connection is established with a similar monotonicity con-dition proposed for the MPFA method [214]. The numerical experiments shown in [182] confirm the effectiveness of theconsidered monotonicity conditions.

4.2. Reduction of numerical dispersion and anisotropy

We consider a second-order (in space and time) scheme for the linear acoustics equation. Selection of parameters in themimetic inner products can be driven by physics requirements such as the reduction of numerical dispersion and numericalanisotropy.

Let us write in parallel continuous and semi-discrete acoustics equations using primary gradient and derived divergenceoperators:⎧⎪⎪⎪⎨⎪⎪⎪⎩

∂u

∂t= gradp,

p = q,

∂q

∂t= c2 div u,

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∂uh

∂t= GRADph,

MN ph = MN qh,

∂qh

∂t= DIV uh,

(4.5)

where p is pressure, c is the speed of sound, ph,qh ∈ Nh are vectors of nodal degrees of freedom, uh ∈ Eh and matricesMN ,MN represent two different mimetic inner products in space Nh . The matrix MN was introduced above and isdiagonal. The sound speed is absorbed in the definition of the derived divergence operator.

The second equation in the right panel of (4.5) is the key to understanding the method. This is an L2-projection in thesame discrete space endowed with two different inner products. Although it is unnecessary in the continuous form, it isessential for the discrete equations. The inner product matrix MN is not unique as well as the inner product matrix MEappearing in the formula for the derived divergence operator, which gives enough flexibility to find a scheme with newproperties.

Eliminating uh,qh and using the definition of the discrete divergence operator in (2.24), we obtain

MN M−1N MN

∂2ph

∂t2= −GRADTME GRAD ph. (4.6)

Let AN = GRADTME GRAD. We consider the explicit scheme with the centered second-order time discretization:

pn+1h − 2pn

h + pn−1h

�t2= −M−1

N MNM−1N AN pn

h.

We design matrix MN to have the same sparsity structure of AN . Since MN is diagonal, the complexity of this scheme isroughly equal to two matrix–vector products or twice the complexity of a scheme with MN = MN .

As shown in [77], the mass matrix representing the identity operator should generate a quadrature rule that is exact forpiecewise constant functions in order to get a second-order accurate scheme in space and time. To translate this consistencystatement in the mimetic language, we first note that

vTh MN uh =

∑c∈Th

vTh,cMN ,c uh,c ∀vh,uh ∈ Nh.

Then, we recall that a similar additivity property for matrix MN leads to the elemental matrices MN ,c . Thus, the mimeticconsistency condition is that each matrix MN ,c must satisfy

MN ,ce = |c|e, e = (1, . . . ,1)T .

For a quadrilateral, we obtain a six-parameter family of matrices that satisfy this condition.The dispersion analysis is typically used to characterize a numerical scheme for wave equations. Let us consider a rect-

angular mesh with �x × �y cells and assume that all matrices MN ,c are the same. Substituting a plane wave solution ofacoustics equation into the numerical scheme, we can calculate the relationship between the numerical angular frequencyωh and the wave vector k = (κ1, κ2)

T :



Fig. 8. Dispersion curves for various directions of a plane wave on a rectangular mesh: the state-of-the-art modified quadrature method [275] (left panel)and the optimized MFD method (right panel).

2(1 − cos(ωh�t)

)= �t2(v∗M−1N ,cMN ,cM

−1N ,cv

)(v∗AN ,cv

), v =

⎡⎢⎢⎣1

eiκ2�y

e−iκ1�xe−iκ2�y

eiκ1�x

⎤⎥⎥⎦ .

In [130,131], the optimal values of the parameters are derived to minimize the numerical dispersion, i.e., the difference

between the numerical angular frequency ωh and the exact angular frequency ω = c√

κ21 + κ2

2 . Note that the separation of

parameters that define MN ,c and AN ,c leads to a bilinear optimization problem.For a square mesh, the resulting scheme has the 4th-order numerical dispersion (in terms of |k|�x) and the 6th-order

numerical anisotropy:

|ωh − ω| = C(|k|�x

)4 + O((k1�x + k2�x)6),

where C is a discretization-independent constant. The state-of-the-art of non-mimetic schemes [275] has also 4th-ordernumerical dispersion but only 4th-order numerical anisotropy. For a rectangular mesh with �x = 2�y, the optimized MFDscheme has the 4th-order numerical dispersion, while the scheme in [275] drops down to the second-order (see Fig. 8).Thus, for both meshes, we were able to find a new scheme for a well studied acoustics problem.

For an unstructured mesh, the optimization strategy leads to a global problem. Due to its bilinear structure, its solutionis computationally feasible and will be studied in the future.

5. Flexibility and extensions of MFD technology

We illustrate the flexibility of the mimetic discretization framework with two examples. First, the stability of an MFDdiscretization for a saddle-point problem can be ensured by adding new degrees of freedom only where they are needed.Second, a high-order scheme can be built by enforcing the consistency condition for a larger polynomial space.

5.1. Enforcing stability for Stokes equations

A mimetic discretization of the two-dimensional Stokes equations leads to a saddle-point problem and must satisfy astability condition as the inf–sup (or LBB) condition in the finite element method. This condition implies a balance betweenthe discrete spaces for the velocity u and the pressure p. Consider the case where the degrees of freedom for velocityand pressure are at mesh nodes and cells, respectively, i.e. uh ∈ N 2

h and ph ∈ Ch . The uniqueness of the solution to thesaddle-point problem requires the null space of DIV T to be trivial. The stability condition is a stronger requirement andguarantees the optimal convergence of the mimetic method. Such condition states that there exists a positive constant β

independent of the mesh resolution factor h such that

infqh∈Ch

supvh∈N 2

h /{0}[DIV vh,qh]C

‖vh‖N � β. (5.1)

The pair of spaces N 2h and Ch is known to violate the stability condition. However, it was shown in [27] that a slight

enrichment of the velocity space is sufficient to prove the inf–sup condition on arbitrary polygonal meshes. This enrichment



Fig. 9. Two examples of macro-elements. The original and stabilizing degrees of freedom are marked with solid and empty circles, respectively.

Fig. 10. The location of stabilizing degrees of freedom is indicated by dots (roughly 52% and 25% of total edges for triangular and square meshes, respec-tively).

is done through additional degrees of freedom at selected mesh edges, which represent the normal component of thevelocity at the edge mid-points; see Fig. 9. The local mesh configuration shown on the right panel leads to a stable method.The other configuration requires one additional velocity unknown.

To identify the edges where the additional degrees of freedom have to be added, the following results from [27] canbe used. First, we assume that the maximum number of vertices in a polygonal cell is bounded from above by a constantindependent of the mesh resolution. Second, we assume that each cell is shape-regular.

The algorithm for introducing the additional degrees of freedom is recursive. On each iteration, we may have the lo-cal configurations shown in Fig. 9 without (left) and with (right) additional degrees of freedom. These configurations areconsidered in Lemmas 5.1 and 5.2, respectively.

Given an internal mesh node n, we consider the sets C(n) and E(n) of the cells and edges sharing the node n. We splitE(n) into two complimentary subsets, Eb(n) and Ex(n), of edges with and without stabilizing degrees of freedom. Let k, kb ,and kx = k − kb be the number of elements in the sets E(n), Eb(n), and Ex(n), respectively.

Lemma 5.1. Let n be any internal mesh node and the only internal node in C(n). Furthermore, let either (a) kx < 3 or (b) kx = 3 and allthe three angles naturally defined by the three edges in Ex(n) be less or equal than π . Then, the stability constant β does not dependon the mesh.

A stronger result follows easily from this lemma. We call a frontier e any collection of at least two adjoint mesh edgesthat separate two mesh cells, e = ∂c ∩ ∂c′ , with c, c′ ∈ C(n). By our assumptions, Lemma 5.1 considers only superelementsC(n) with no frontiers. In a general case, all the internal boundaries among cells in C(n) can be divided into frontiers andregular (single) edges. The next result shows that frontiers can be essentially ignored.

Lemma 5.2. Let n be any internal mesh node. Furthermore, let kx indicate the number of edges of Ex(n) which are not a part of afrontier. Let either (a) kx < 3 or (b) kx = 3 and all the three angles naturally defined by the three respective edges be less or equalthan π . Then, the stability constant β does not depend on the mesh.

The above result covers, for example, the case of a mesh with convex elements where all the nodes have the property ofbeing the junction of at most 3 edges. For example, the Voronoi meshes satisfy this property and in such a case the resultshows that no stabilizing degrees of freedom are needed.

For a logically square mesh, we need to add the stabilizing degrees of freedom approximately to every fourth edge, seeFig. 10. This is sufficient to kill spurious pressure modes. In a more general case, it is sufficient to add the stabilizing degrees



Fig. 11. Degrees of freedom in the higher order MFD method for the diffusion problem in the mixed form: the cell-based scalar unknown pc (left panel)and the face-based flux unknowns (u0

f ,u1f ) on the faces f ∈ F(c) (right panel).

of freedom where needed, as indicated in Lemma 5.2. On polygonal meshes, a small number of additional unknowns is oftensufficient for the stability of the method.

Remark 5.12. The stabilization strategy developed for the Stokes equations can be generalized to other PDEs with a saddle-point structure. This will be the topic for future research.

5.2. High-order MFD schemes

As stated by Theorem 3.3, the mimetic method for the Darcy equations (3.1)–(3.2) provides a first-order approximationof the velocity u. The first-order accuracy is due to the constant representation of u · n on mesh edges, see (2.16). Thus,a second-order approximation is expected by using a linear representation. Let h f be the characteristic length associatedwith f , e.g. h f = | f | in two dimensions and h f =√| f | in three dimensions.

Let u f and u1f represent the mean and the slope of u · n on the mesh face f . The two-dimensional case is illustrated on

the right panel in Fig. 11. Using the degrees of freedom, we can define a linear function of each mesh face:

u f (x) = u0f + x − x f

h f· u1

f ∀x ∈ f . (5.2)

We define the vector uh as the collection of all the degrees of freedom. Let Fh be the corresponding linear space of suchvectors.

The MFD method reads again as in (3.3)–(3.4), but with different primary, DIV , and derived, GRAD, mimetic operators.The divergence theorem stated in (2.15) suggests us to define the discrete divergence operator on each cell c as follows:

(DIV uh)c := 1

|c|∑

f ∈F(c)

αc, f

∫f

u f (x)dS. (5.3)

We substitute (5.2) into the above equation and note that the second integral is zero since x f is the centroid of face f :

|c| (DIV uh)c =∑

f ∈F(c)

αc, f

∫f

u0f dS +

∫f

x − x f

h f· u1

f dS =∑

f ∈F(c)

αc, f u0f | f |.

Therefore, the divergence operator has the same structure as in (2.17) but its matrix representation has more zero entries.

The mimetic inner product in Fh , required to define the derived operator GRAD, must satisfy the consistency and stabilityconditions. The stability condition is given by (3.8) and guarantees well-posedness of the discrete problem. The consistencycondition is similar to condition (S2) and guarantees additional accuracy for the velocity approximation. It reads

• (S1c) (linear consistency). Let uh,c and vh,c be the discrete representations of two vector-functions u and v in cell c.Furthermore, let u be a linear function, div v be constant in c and fluxes v · n f be constant on the faces of c. Finally, letKc be a constant diffusion tensor. Then, the local mimetic inner product uh and vh on each cell c ∈ C must be exact forthe weighted integral of u and v:

[vh,uh]F,c =∫c

v · K−1c u dV . (5.4)



The inner product in (5.4) can be expressed using an SPD matrix MF ,c . Let us transform the right-hand side of (5.4). Sinceu is a linear function, so is K−1

c u. Thus, there exists a quadratic polynomial q such that grad q = K−1c u. Now, an integration

by parts yields:∫c

(K−1

c u) · v dV =

∫c

grad q · v dV = −∫c

q div v dV +∫∂c

qn · v dS. (5.5)

The restriction imposed on the function v implies that its divergence coincides with DIV vh . Indeed, by using the divergencetheorem, splitting the boundary integral into face contributions, and applying definition (5.3), we obtain:

|c|div v =∫c

div v dV =∫∂c

n · v dS =∑

f ∈F(c)

∫f

v f (x)dS = |c|(DIV vh)c .

Similarly, because v · n = v f (x) on f , we can express the last boundary integral in (5.5) in terms of our degrees of freedom.The exactness relation (5.4) takes the equivalent form:

[uh,vh]F,c = −∫c

(DIV vh)cq dV +∑

f ∈F(c)

∫f

v f (x)q dS, (5.6)

or, equivalently,

vTh,cMF,cuh,c = vT

h,cwh,c,

where wh,c depends on q and the cell geometry. Taking various linearly independent linear functions u, we build alge-braic requirements (see, for example, (2.45)) for the matrix MF ,c . Construction of the matrix MF ,c is completed throughLemma 2.9.

Remark 5.13. When tensor K is not constant on c, the consistency condition (5.4) still holds and so is (5.6), but nowvector uh must be the discrete representation of Π1

c (K grad q), where Π1c is the orthogonal projector onto the linear vector-

functions defined on cell c, see [28,32,129].

The second-order convergence of the flux approximation can be theoretically proved when the error is measured in themesh-dependent norms ‖| · ‖|F = [·, ·]1/2

F . The main result is summarized by the following theorem; the proof is in [28].

Theorem 5.11. Let pair (u, p) be the exact solution of (3.1)–(3.2) with the homogeneous Dirichlet conditions and p ∈ H3(Ω). Fur-thermore, let pair (uh,ph) be the solution of the MFD methods based on (S1c)–(S2). Then,∥∥∣∣uh − uI

∥∥∣∣F � Ch2‖p‖H3(Ω),

where uI ∈ Fh is the discrete representation of u and the constant C is independent of h.

6. Lagrangian hydrodynamics

In this section, we consider the mimetic (compatible) discretizations of the equations of the Lagrangian hydrodynamics:

dρ

dt+ ρ div u = 0, ρ

du

dt= −grad p, ρ

dε

dt= −p div u, p = EOS(ε,ρ), (6.1)

where u is the velocity, p is the pressure, ρ is density, ε is the internal energy and EOS(·) represents a suitable equation ofstate.

In the Lagrangian hydrodynamics the mesh is moving with fluid and the mass of each cell does not change in time. Thefact that the mesh follows the fluid suggests us to discretize the velocity at the mesh nodes, un = (un, vn). Let uh ∈ N 2

h bethe vector of all velocity unknowns. For such discretization of the velocity, it is natural to use a cell-centered pressure, pc ,when we discretize the pressure gradient in the momentum equation. Let ph ∈ Ch be the vector of all pressure unknowns.The density ρ is defined as the mass of the cell, mc , divided by its volume V c . Hence, its discrete analog is also associatedwith the cell c, i.e., ρc = mc/V c . This and the equations of state suggests us to use a cell-based discretization of the internalenergy, εc .

This approach is called the staggered discretization (see Fig. 12) and is used in most production hydrocodes [35]. Insuch codes, a non-conservative form of the energy equation is used often, see (6.1). The staggered approach facilitates thediscretization of the internal energy equation by using a discrete divergence operator that acts from nodes to cells.

The mimetic discretization of the equations of the Lagrangian hydrodynamics must conserve mass, momentum and totalenergy to model the motion of a compressive fluid accurately. The form of the internal energy equation implies that the



total energy is conserved if the discrete gradient and divergence operators are negative adjoint to each other. This leads toa new DVTC based on discrete duality relations like that developed in Section 2.

There are two ways to define a discrete divergence operator DIV : N 2h → Ch using different fundamental continuum

relations. The first way is based on the divergence theorem. Note that this is not the operator defined in Section 2.2,because the unknowns now are the nodal velocities un rather than the averaged normal components of the velocity onmesh faces. This leads to a larger null space for the operator DIV . When parasitic null space modes are triggered, theyproduce the nonphysical hourglass motion of a quadrilateral mesh. The second way is based on the connection between thedivergence operator and the change of volume of the mesh cells. It is a remarkable fact that these two distinct strategieslead to the same operator DIV in the Cartesian coordinate system.

To ensure the adjointness property, the discrete gradient operator, GRAD : Ch → N 2h , is defined through a duality

relation. The derived gradient has also a non-trivial null space. For example, on a square mesh, the spurious mode in thenull space has the checker-board pattern.

It is worth mentioning that there exist methods where a discrete gradient operator is chosen as the primary operator anda discrete divergence operator is derived by duality. These operators are still negative adjoint to each other by construction,which guarantees the conservation of the total energy. Unfortunately, the primary gradient is defined on a dual mesh, andthe explicit construction of this mesh may be a difficult task, especially in three dimensions.

To summarize, we consider a new pair of dual discrete operators DIV and GRAD. A complete formulation of a discretecalculus for these operators is still an open research topic. Despite some shortcomings, these operators are widely used inthe Lagrangian hydrocodes; therefore, analysis of their properties, and in general mimetic discretizations of the equations ofthe Lagrangian hydrodynamics, is of great interest to practitioners. The presentation in Sections 6.1, 6.2, and 6.3 is based on[70,237].

More general equations of the Lagrangian hydrodynamics may use forces of different nature that either come fromcoupled physical models, such as the magneto-hydrodynamics, or are introduced for numerical stability, such as artificialviscosity forces [72] or forces that control artificial grid motion [71]. In some cases, these forces cannot be interpretedas discrete analogs of continuous operators. This fact leads to a more general concept of compatible discretizations forthe Lagrangian hydrodynamics: instead of the discrete operators we use a force-based compatible discretization of themomentum and the internal energy equations that conserve the total energy. The generic form of such a discretization isdescribed in [70]. A discretization of the momentum equation is given by

mndun

dt= −

∑c∈C(n)

fc,n, (6.2)

where mn is mass of the node, which does not depend on time, and f nc is the corner force, which expresses the force from

cell c to node (vertex) n of this cell. The conservation of the total momentum is based on the third Newton law and isformulated in cell c as follows:∑

n∈N (c)

fc,n = 0. (6.3)

A discretization of the internal energy equation compatible with (6.2), in the sense that the total energy is conserved, takesthe form

mcdεc

dt=∑

n∈N (c)

un · fc,n. (6.4)

More details about compatible discretizations are presented in Section 6.6. In Section 6.7 we describe a frameworkfor developing the mimetic artificial viscosity. The presentation is based on material published in [186]. In Section 6.8we consider another mimetic property of the discretization of the Lagrangian hydrodynamics equations, namely, the exactpreservation of the spherical symmetry by working with cylindrical r, z coordinates on a logically rectangular curvilineargrid. The presentation of this subsection is based on [195].

6.1. Continuous equations of Lagrangian hydrodynamics

Let us consider the system of hydrodynamics equations in Lagrangian coordinates that describes the motion of a com-pressible gas. The trajectory equation is given by

dx

dt= u, (6.5)

where x is the coordinate vector, and u is the material velocity vector.In the Lagrangian hydrodynamics, we consider the motion of an elemental fluid particle δV (t), whose mass is fixed in

time to ensure the mass conservation law:



d

dt

∫δV (t)

ρ dV = 0. (6.6)

The corresponding differential equation is the first equation in (6.1). The conservation of the momentum is expressed by thesecond equation in (6.1). In most Lagrangian hydrocodes, the equation for the internal energy ε, the third equation in (6.1),is discretized in a non-conservative form. The system of equations is finally closed by the equation of state, p = EOS(ε,ρ).

Let us discuss which conservation laws we need to preserve in the discrete setting and how these conservation lawsdepend on the properties of the differential operators. First, let us consider the geometric conservation law, which connectsthe change of the volume of a fluid particle δV (t) moving with the velocity field u:

d

dt(δV ) = d

dt

∫δV (t)

dV =∮

∂(δV (t))

u · n dS, (6.7)

where n is the unit vector orthogonal to ∂V (t) and pointing out of V (t).Eq. (6.7) can be obtained from the mass conservation equation. In the Lagrangian methods the density is defined as

ρ = δm/δV . Inserting this expression into the first equation in (6.1) and using the fact that the mass δm does not change intime, we obtain

d

dt(δV ) = δV div u. (6.8)

Therefore, to be consistent with the conservation of mass, the discrete divergence has to mimic the following equation:

div u = 1

δV

d

dt(δV ). (6.9)

On the other hand, by comparing (6.8) and (6.7), we conclude that to satisfy the geometric conservation law a discretedivergence operator has to be defined through a discrete analog of the following equation

div u = 1

δV

∮∂(δV (t))

u · n dS. (6.10)

Hereafter, we refer to an operator having the property (6.10) as the operator given in the divergence form. Let us show thatin the Cartesian coordinate system it is possible to define a discrete divergence operator that satisfies discrete analogs ofboth (6.10) and (6.9).

The integral form of the conservation of momentum ρu reads:

d

dt

( ∫V (t)

ρu dV

)= −

∮∂V (t)

pn dS. (6.11)

Formally, it can be derived from the second equation in (6.1) by integrating it over V (t), applying the Reynolds transporttheorem, and using the definition of the gradient operator∫

V (t)

grad p dV =∮

∂V (t)

pn dS. (6.12)

The conservation of momentum follows from the divergence property of the gradient operator.The conservation of the total energy E = ρ(ε + |u|2/2) reads:

d

dt

( ∫V (t)

ρ

( |u|22

+ ε

)dV

)= −

∮∂V (t)

p (u · n)dS. (6.13)

We can derive (6.13) by multiplying the momentum equation by u to obtain the rate of change of ρ |u|2/2 and by adding theresulting equation to the internal energy equation, integrating over V (t), and using the duality property for the divergenceand gradient operators:∫

V (t)

grad p · u dV +∫

V (t)

p div u dV =∮

∂V (t)

p (u · n)dS. (6.14)

The properties (6.10) and (6.12) can be obtained from (6.14) as special cases. Indeed, if we set u to the coordinate vectors(1,0) and (0,1) in (6.14) and take into account that the divergence of such vectors is zero, we obtain (6.12). If we set p = 1and take into account that the gradient of a constant vector is zero, we obtain (6.10).

Therefore, if the divergence is chosen as primary operator, it must be zero on constant vectors to provide the discreteconservation of momentum. If the gradient is chosen as primary operator, it must be zero on constant functions to guaranteethe discrete geometric conservation law. A thorough discussion of these issues can be found in [237, Chapter 5].



Fig. 12. Location of unknowns in a computational cell c and other notations.

6.2. The primary divergence operator

To ease the presentation, we focus on the two-dimensional case. We choose the divergence DIV : N 2h → Ch as the

primary operator. As pointed out in the previous subsection, we have two distinct relations (6.9) and (6.10) that can beused to define this operator. These relations are obviously equivalent in the continuum setting, but in principle they mightlead to two different definitions of DIV . We will derive DIV using both relations and will show their equivalence.

Let us consider (6.10). If we choose cell c as the elementary volume we have:

(DIV uh)c := 1

V c

∑f ∈F(c)

un1( f ) + un2( f )

2· S f (6.15)

where n1( f ),n2( f ) are the endpoints of face f , S f = S f n f with S f being the area of face f , and n f is the unit vector tof pointing out of c, see Fig. 12. The discrete divergence operator is zero when uh corresponds to a constant vector because∂c is a closed contour.

We rearrange the summation terms in (6.15) by collecting the coefficients related to the same node to obtain a differentform for the discrete divergence operator. To do so, we need a few additional notation shown in Fig. 12. Let n− and n+be the vertices of c that precede and follow the node n in the counter-clock-wise order. The definition of nodes n± clearlydepends on the cell to which we refer; therefore, we will often write n±(c). Also, let Sn1,n2 be the vector orthogonal to theface f with endpoints n1 and n2 and length equal to S f . Using such notation, definition (6.15) can be reformulated as

(DIV uh)c = 1

V c

∑n∈N (c)

un · Sn−,n + Sn,n+2

. (6.16)

Let us consider the alternative definition of the discrete divergence operator (6.9). Since V c depends on all the nodalvectors xn = (xn(t), yn(t)) for n ∈ N (c) and each nodal velocity un has components un = dxn(t)/dt and vn = dyn(t)/dt , wehave:

(DIV uh)c := 1

V c

dV c

dt= 1

V c

∑n∈N (c)

(∂V c

∂xn

dxn

dt+ ∂V c

∂ yn

dyn

dt

)= 1

V c

∑n∈N (c)

(∂V c

∂xnun + ∂V c

∂ ynvn

). (6.17)

This discrete divergence operator applied to a constant vector in N 2h returns the zero vector in Ch since the volume of each

cell does not change under a rigid translation.

Remark 6.14. Definition (6.17) can be used in three dimensions once V c is well defined. Indeed, the faces of a three-dimensional cell cannot be flat, and their numerical treatment deserves special care. An effective approach is to triangulateeach face by connecting its center with its vertices.

Remark 6.15. The first equality in (6.17) is an example of a coordinate invariant definition that can be used in any coordinatesystem, e.g., Cartesian, cylindrical, or spherical.

Proposition 6.2. Formulas (6.15) and (6.17) are equivalent in the Cartesian coordinate system.

Proof. Let us first observe that the measure of cell c in a 2D Cartesian coordinate system is given by the shoelace formulas:

V c = 1

2

∑n∈N (c)

xn(yn+ − yn−) = −1

2

∑n∈N (c)

yn(xn+ − xn−),

and, therefore,



∂V c

∂xn= 1

2(yn+ − yn−),

∂V c

∂ yn= −1

2(xn+ − xn−), (6.18)

which are the coefficients in definition (6.17). Substituting (6.18) in (6.17), we obtain:

(DIVuh)c = 1

V c

∑n∈N (c)

(yn+ − yn−

2un − xn+ − xn−

2vn

). (6.19)

As shown in Fig. 12, it holds that

Sn−,n + Sn,n+ = Sn−,n+ =(

yn+ − yn−−(xn+ − xn−)

). (6.20)

Substituting (6.20) in (6.19), we obtain (6.16). This proves the assertion of the proposition. �6.3. The derived gradient operator

We construct the derived gradient operator through a discrete analog of the duality relation (6.14), which is equivalentto (2.4), to conserve the total energy as pointed out in the comments of (6.13). Note that we cannot use definition (2.20)because there the primary divergence operator acts from Fh to Ch , while now DIV acts from N 2

h to Ch . The mimetic innerproduct in N 2

h is given by

[uh,vh]N 2 :=∑n∈Th

Vnun · vn, (6.21)

where we recall that Vn is the volume associated with node n, i.e., the volume of a dual cell built around node n. We canrewrite this matrix product by introducing the inner product matrix MN 2 . This matrix is diagonal with volumes Vn on thediagonal. The mimetic inner product in Ch is given by (2.41) and (2.44). It is represented by the diagonal matrix MC whichhas volumes V c on the diagonal. Inserting the primary and adjoint operators DIV and DIV� into formula (2.8), using the

above matrix representations for the mimetic inner products and the identification GRAD ≡ −DIV� yield

[uh, GRAD ph]N 2 := −[uh,DIV� ph]N 2 = −[DIV uh,ph]C, (6.22)

which holds for every uh ∈N 2h and every ph ∈ Ch . As uh and ph are arbitrary vectors in N 2

h and Ch we readily obtain

GRAD ≡ −DIV� = −M−1N 2 DIV T MC, (6.23)

where DIV T is the transpose of the matrix, cf. Remark 2.2. From Eq. (6.23) we see that GRAD : Ch →N 2h . Since GRAD ph

contains two components for each mesh node n and we can write GRAD ph = (GRADxph, GRADyph)T .

6.4. Explicit formulas for the derived gradient operator

The discrete gradient operator GRAD given in (6.22) can be written in several equivalent forms. Let us note that

[DIV uh,ph]C =∑c∈Th

V c(DIV uh)c pc =∑c∈Th

pc

∑n∈N (c)

un · Sn−,n + Sn,n+2

.

Using the self-explanatory notation in Fig. 13, it holds that

1

2(Sn−,n + Sn,n+) = −(Sn−,n,c + Sn,n+,c).

Using this relation and taking into account the duality relation (6.22) that defines GRAD, we obtain:

[DIV uh,ph]C = −∑n∈N

Vnun · 1

Vn

∑c∈C(n)

pc(Sn−,n,c + Sn,n+,c) = −[uh, GRAD ph]N 2 ,

from which an explicit formula follows for the derived gradient at each node n:

(GRAD ph)n = 1

Vn

∑c∈C(n)

pc(Sn,n+,c + Sn,n−,c). (6.24)

From the right panel in Fig. 13, it holds that:



Fig. 13. Left panel: corner of the cell and corresponding notations. Right panel: dashed line shows the dual cell around node n.

Sn−,n,c + Sn,n+,c =(

yn+(c) − yn−(c)−xn+(c) + xn−(c)

).

Inserting this relation in (6.24) yields the following explicit expression for the derived gradient:

(GRADxph)n = − 1

Vn

∑c∈C(n)

pcyn+(c) − yn−(c)

2, (6.25)

(GRADyph)n = 1

Vn

∑c∈C(n)

pcxn+(c) − xn−(c)

2. (6.26)

Using the coefficients given in (6.18), we reformulate (6.25)–(6.26) as follows:

(GRADxph)n = − 1

Vn

∑c∈C(n)

∂V c

∂xnpc, (GRADyph)n = − 1

Vn

∑c∈C(n)

∂V c

∂ ynpc, (6.27)

which are the components of the discrete divergence operator DIV in (6.17). This result is expected in the Cartesiancoordinates due to Proposition 1. However, a stronger result can be proved.

Proposition 6.3. The derived gradient operator GRAD given by (6.27) is the negative adjoint of the divergence operator DIV givenby (6.17) in any coordinate system.

Proof. Using the definition of the discrete divergence operator, we obtain:

[DIV uh,ph]C =∑c∈Th

V c (DIV uh)c pc =∑c∈Th

pc

∑n∈N (c)

(∂V c

∂xnun + ∂V c

∂ ynvn

).

We rearrange the summation terms, use (6.27) and the definition of the mimetic inner product for N 2h given in (6.21) to

obtain:

[DIV uh,ph]C =∑n∈N

(un

( ∑c∈C(n)

pc∂V c

∂xn

)+ vn

( ∑c∈C(n)

pc∂V c

∂ yn

))= −

∑n∈N

Vn(un(GRADxph)n + vn(GRADyph)n

)= −[uh, GRAD ph]N 2

h,

which is the duality relation between DIV and GRAD. �The derived gradient operator GRAD is in the divergence form since formula (6.24) is an approximation of a contour

integral∮∂Vn

p n dS along the boundary of the dual cell built around node n. The derived gradient operator has a non-trivialnull space; therefore, it is different from the derived gradient defined in Section 2.4, which has a trivial null space, seeLemma 2.4. This leads to well-known pathological situations. For example, let us consider a square mesh with mesh size has shown in Fig. 14. Using the notation of such figure, we obtain the following expression for the discrete gradient at thecentral point:



Fig. 14. Illustration of a null space of the discrete gradient operator GRAD.

(GRAD ph)n =( (p1+p4)/2−(p2+p3)/2

h(p1+p2)/2−(p3+p4)/2

h

). (6.28)

If ph takes the checkerboard pattern presented in Fig. 14, its gradient is zero; thus, the non-constant discrete field ph belongs

to the kernel of GRAD.

Remark 6.16. The discrete gradient operator GRAD is not exact for linear function on general meshes. As proved in [237],this is a consequence of the fact that we have only one degree of freedom per cell.

Nonetheless, in spite of their shortcoming, the discrete operators DIV and GRAD are popular in hydrocodes. Moreover,they provide a discretization of the elliptic equation whose solution can be proved to converge at an optimal rate to theexact solution in a discrete energy-like norm, see [122].

6.5. Discrete operators for nodal velocities and subcell pressures

In this subsection, we discuss the notion of subcell pressures. From a practical viewpoint additional pressure unknownsare introduced to suppress artificial grid motion in the Lagrangian hydrodynamics, [71]. A new pair of a primary divergence

DIV and a derived gradient GRAD operator are defined following the strategy outlined in the previous subsections.Therefore, we will show only the final formulas for these operators.

A cell c is split into a few subcells by connecting its center with mid-points of its edges, see Fig. 15. The number ofsubcells is equal to the number of cell vertices xn and each subcell is labeled with two indices c and n. Thus, the subcell(c,n) is a quadrilateral with vertices

xn, xn+1/2 = xn + xn+2

, xc = 1

|N (c)|∑

n∈N (c)

xn, xn−1/2 = xn + xn−2

. (6.29)

Since the position of the vertices in subcell (c,n) depend on the position of the vertices in the parent cell c, the volumeV c,n of the subcell is a function of the primary mesh nodes. Thus, we can define the primary divergence operator DIVaccording to (6.17):

(DIV uh)c,n := 1

V c,n

dV c,n

dt= 1

V c,n

∑n′∈N (c)

(∂V c,n

∂xn′dxn′

dt+ ∂V c,n

∂ yn′dyn′

dt

). (6.30)

The derived gradient GRAD acting from subcells to nodes can be written in a form similar to (6.27) using the argument

of Proposition 6.3. The components of GRAD are given by

(GRADx ph)n =∑

c∈C(n)

∑n′∈N (c)

∂V c,n′

∂xnpc,n′ , (GRADyph)n =

∑c∈C(n)

∑n′∈N (c)

∂V c,n′

∂ ynpc,n′ .

Its stencil is shown in Fig. 15.

6.6. Compatible discretizations based on subcell forces

Let us now use the mimetic operators to build a semi-discrete form of the Lagrangian equations. The density is definedfor each cell c by the algebraic equation ρc = mc/V c . The mimetic analog of the momentum equation reads:

mndun = −Vn(GRADph)n, (6.31)
dt



Fig. 15. Stencil of the derived gradient operator GRAD for node n. Shaded ellipses correspond to subcell pressures.

where mn is the nodal mass that does not depend on time and is a sum of subcell masses in the associated dual controlvolume, see [237] for more detail. Note that other definitions of the nodal mass are also used in practice. We reformulatethe right-hand side of (6.31) using (6.24):

mndun

dt= −

∑c∈C(n)

fc,n, (6.32)

where fc,n is the subcell force associated with subcell (c,n),

fc,n = pc(Sn,n+,c + Sn,n−,c). (6.33)

Finally, the mimetic analog of the internal energy equation is:

mcdεc

dt= −pc V c(DIVuh)c. (6.34)

The right-hand side of (6.34) can be rewritten using (6.15) and (6.33):

mcdεc

dt=∑

n∈N (c)

un · pc(Sn,n+,c + Sn,n−,c) =∑

n∈N (c)

un · fc,n. (6.35)

The pair of Eqs. (6.32) and (6.35) form a compatible discretization that preserves the total discrete energy E = E + Kwhere

E =∑

c

mcεc and K =∑

n

mn|un|2

2. (6.36)

This property is a consequence of the mimetic discretization and is independent of the nature of the corner forces fc,n asstated in the following proposition.

Proposition 6.4. The total discrete energy E = E +K is preserved in time, i.e., dE/dt = 0.

Proof. The change of the total internal energy is readily obtained by summing up Eq. (6.35) over all the mesh cells:

dEdt

=∑

c

mcdεc

dt=∑

c

∑n∈N (c)

un · fc,n. (6.37)

To determine the change of the total kinetic energy K, we multiply the momentum equation (6.32) by un and we sum overall the mesh cells:

dKdt

=∑

n

mnd

dt

( |un|22

)= −

∑n

∑c∈C(n)

fc,n · un. (6.38)

The proposition follows from rearranging the summation in (6.38) and adding the result to (6.37). �The conservation of the total momentum,

M =∑n∈N

mnun, (6.39)

requires to impose restrictions on the subcell forces.



Fig. 16. Subcell pressures in a quadrilateral cell c.

Proposition 6.5. Let, for each cell c, the following holds true:∑n∈N (c)

fc,n = 0. (6.40)

Then, the total momentum M is conserved, i.e., dM/dt = 0.

Proof. The change of the total momentum is derived by summing up Eq. (6.32) over all the mesh nodes and rearrangingthe summation:

dMdt

= −∑

n

∑c∈C(n)

fc,n = −∑

c

∑n∈N (c)

fc,n = 0. (6.41)

This proves the assertion of the proposition. �Condition (6.40) is easy to verify for the subcell forces given by (6.33). Indeed, straightforward calculations lead to an

integral over a closed contour:∑n∈N (c)

fc,n =∑

n∈N (c)

pc(Sn,n+,c + Sn,n−,c) = − pc

2

∑n∈N (c)

(Sn,n+ + Sn−,n) = 0.

Another example of using Proposition 6.5 is given in [71] where fc,n are introduced to suppress the artificial grid motion.One additional example is given in the next subsection. Let us consider again subcell pressures. Using the notation in Fig. 16,we define subcell forces as follows:

fnc = pc,n + pc,n+

2Sn,n+,c + pc,n + pc,n−

2Sn,n−,c . (6.42)

Requirement (6.40) is clearly satisfied because of the geometrical interpretation of this force.

6.7. Artificial viscosity

Artificial viscosity is needed for hydrodynamic flows with shocks. The viscous forces, denoted as fn,c are added to themomentum equation (6.32). Similar to physical forces, fn,c must satisfy the conservation of momentum property (6.40) anddissipate kinetic energy into internal energy:∑

c∈C(n)

fn,c · un � 0. (6.43)

The artificial viscosity must be zero for self-similar motion and vanish smoothly away from shock discontinuities. Let uh ∈N 2

h and fh ∈ N 2h be the global vectors of all discrete velocities and viscous forces. The discrete calculus developed in

Section 2 allows us to satisfy (6.43) by taking

fh = DIV GRADuh ≡ Ahuh, (6.44)

where the primary gradient operator, GRAD : N 2h → E2

h , is given by (2.73) and the derived divergence operator, DIV :E2

h →N 2h , is given by (2.74).

This duality relationship leads immediately to the dissipation of the kinetic energy (6.38). In addition to the aboveproperty, the artificial viscosity is automatically zero for the linear velocity (uniform compression) and is sufficiently smallfor a smooth flow. To reduce the artificial viscosity even further for a smooth flow, the discrete divergence is derived with



Fig. 17. Example of reconstruction of curvilinear grids from logically rectangular grids.

respect to a weighted inner product. The weight is given by a tensor function μ(x) such that matrix Ah in the right-handside of (6.43) is an approximation of the continuous operator div(μgrad).

The function μ(x) is piecewise constant on mesh Th and is selected to satisfy other requirements for the artificialviscosity. In [186], we use the expression for the viscosity coefficient described in [273]:

μc = ψc ρc Lc

(cQ

γ − 1

4|�uh| +

√c2

Q

(γ − 1

4

)2

|�uh|2 + c2L s2

c

),

where sc is the local sound speed, cL and cQ are positive non-dimensional constants, Lc is the characteristic length of c,�uh is the measure of compressibility (for instance, the velocity jump across the shock), and ψc is a binary switch. Thisswitch ensures that the heating due to artificial viscosity occurs only for cells under compression.

As described in Section 3.3, an efficient method for calculating (6.44) builds matrix Ah directly rather than calculatingit as a product of two first-order operators. Recall that matrix Ah represents a mimetic semi-inner product. Numerical ex-periments presented in [186] show that discrete calculus approach reduces significantly the mesh imprint in the Lagrangianshock calculations.

6.8. Mimetic discretizations on curvilinear grids

In this subsection, we give one more example of a pair or a primary divergence and a derived gradient mimetic operators.These operators are constructed on curvilinear grids and are used to discretize the equations of Lagrangian hydrodynamicsin two-dimensional rz coordinates that preserve plane, cylindrical, and spherical symmetries. The presentation is based on[195].

The method developed in [195] is formulated on a curvilinear mesh that is reconstructed from a given logically rectangu-lar mesh, see Fig. 17. Thus, we relabeled the mesh nodes xn using two indices i and j. The edges in the reconstructed meshare arcs of circles and straight segments. The latter can be treated as arcs of circle of infinite radius. The major propertyof the reconstructed mesh is that the normal vector to a curvilinear edge varies along it. In the presented discretization,only the normal vectors at the edge end-points are used. The mesh reconstruction algorithm introduced in [195] is exactfor straight lines and circles, so that it does not change rectangular and polar grids. This is one of the main reasons whythe mimetic discretization described below preserves various symmetries.

We use the self-explanatory notation shown on the right panel in Fig. 18 that are more suitable for logically structuredmeshes. The nodes are labeled by i, j; the cells by i + 1/2, j + 1/2; the edges by i + 1/2, j and i, j + 1/2. The normalvectors at the edge end-points are labeled by the edge and node indices. The normal vectors are scaled by the length of thecorresponding half-edges. According to the generic mimetic approach, we first define a primary discrete divergence operator,

DIV :N 2h → Ch , and then use it to derive a discrete gradient operator, GRAD : Ch →N 2

h .The primary discrete divergence is based on (6.10) and reads as:

(DIV uh)i+1/2, j+1/2

= 1

V i+1/2, j+1/2

((ui+1, j · Si+1, j+1/2

i+1, j + ui+1, j+1 · Si+1, j+1/2i+1, j+1

)− (ui, j · Si, j+1/2i, j + ui, j+1 · Si, j+1/2

i, j+1

)+ (ui, j+1 · Si+1/2, j+1 + ui+1, j+1 · Si+1/2, j+1)− (ui, j · Si+1/2, j + ui+1, j · Si+1/2, j))

, (6.45)
i, j+1 i+1, j+1 i, j i+1, j



Fig. 18. Left panel: notation for a curvilinear mesh. Right panel: stencil for the discrete gradient and the dual cell.

where V i+1/2, j+1/2 is the volume of the cell with the curvilinear edges.There are several distinctive features of this definition. First, the normal vectors at the end points of an edge are different

and on a polar mesh they coincide with the normal vectors to the corresponding circle. Then, the measures of two half-edgesare different in the rz coordinate system. It is also important to note that the only difference in the definitions of theprimary divergence operators in the Cartesian and cylindrical coordinate systems lies in the definition of the geometricelements such as areas and volumes. This is another example of a coordinate invariant definition.

The derived gradient is obtained by the same duality relations used in Section 6.3, with a similar definition of the innerproduct matrices. Nonetheless, we point it out that the cell and nodal volumes that appear in the matrix definitions arevolumes of domains with curvilinear boundaries. The formula for the derived gradient operator is given by:

(GRAD ph)i, j

= 1

V i, j

((pi+1/2, j+1/2 − pi+1/2, j−1/2)Si+1/2, j

i, j + (pi−1/2, j+1/2 − pi−1/2, j−1/2)Si−1/2, ji, j

+ (pi+1/2, j+1/2 − pi−1/2, j+1/2)Si, j+1/2i+1/2, j+1/2 + (pi+1/2, j−1/2 − pi−1/2, j−1/2)Si, j−1/2

i, j

), (6.46)

where V i, j is the volume of the dual cell surrounding node i, j. This is also a coordinate invariant definition. The stenciland the corresponding dual cell for this operator are shown in Fig. 18.

In symmetry preserving calculations, we assume that the initial pressure, density and internal energy depend only onthe spherical radius; therefore, in the discrete setting they depend only on index j. The initial velocity is radial, i.e., it takesthe form ui, j = U j eR

i, j , where U j is the magnitude and eRi, j is the radial vector at node i, j. Then, we need to check that

the discrete divergence of such a vector on a polar mesh depends only on index j and that the discrete gradient has a formsimilar to the velocity, i.e., its magnitude depends only on index j and it is oriented in the radial direction. In [195] it isproved that the mimetic divergence and gradient operators defined above satisfy these conditions.

In Fig. 19 we show the results for the numerical resolution of the Noh implosion problem [213] on uniform and non-uniform polar meshes. The left panel in the figure shows the Lagrangian mesh obtained by a standard method when thenodes are connected by straight segments. The panel in the middle shows the same mesh in the mimetic method describedin this section. Note that the symmetry is perfectly preserved. Finally, on the right panel, we show the Lagrangian meshwhen the new mimetic discretization is applied to a mesh that is non-uniform azimuthal steps. Again, the symmetry isperfectly preserved.

7. Mimetic discretization of Lagrangian solid dynamics in 2D axisymmetric geometry

In this section, we describe how the mimetic approach can be used to discretize the equations of the Lagrangian soliddynamics in 2D axisymmetric geometry. Thus, all quantities depend only on the cylindrical coordinates r and z and do notdepend on the azimuthal coordinate θ .

In solid dynamics, instead of a cell-centered discretization of the pressure, we use a cell-centered discretization of thestress tensor σ . Thus, in the momentum equation, we need to discretize the divergence of a tensor field, and in the internalenergy equation, the symmetric part of the gradient of a vector field. In a staggered discretization approach, the discretedivergence acts from nodes to cells and the discrete gradient from cells to nodes; therefore, we cannot use the operatorsdefined in Section 2.



Fig. 19. The Lagrangian meshes in various methods for the Noh implosion problem.

We develop new mimetic operators that mimic important properties of continuum operators. For example, when thestress tensor is diagonal, σ = p I, we enforce a discrete analog of the identity divσ = grad p. We also guarantee thatthe trace of a discrete strain rate tensor equals the discrete divergence of the velocity. As usual, we will start with theadditional integral identities that are summarized in Section 7.1. The presentation in this section is an extension of themimetic discretizations developed in [196,228] for logically rectangular grids to general polygonal meshes.

7.1. Continuum equations and integral identities

The equations for the momentum and the internal energy can be written as follows:

ρdu

dt= divσ , ρ

dε

dt= σ : e, (7.1)

where σ is the symmetric stress tensor, e = (grad u + grad u)T /2 is the strain rate tensor, and the symbol “:” denotes thedouble inner product of two tensors. Compared with the Lagrangian hydrodynamics, now we have to consider two newdifferential operators: the divergence of a tensor field, divσ , and the gradient of a vector field, grad u.

In the axisymmetric geometry, the stress tensor has the following form:

σ =⎛⎝σrr 0 σrz

0 σθθ 0

σrz 0 σzz

⎞⎠ . (7.2)

Thus, the divergence of σ is the vector whose components are

(divσ )r = ∂σrr

∂r+ ∂σrz

∂z+ σrr − σθθ

r, (divσ )z = ∂σzz

∂z+ ∂σrz

∂r+ σrz

r. (7.3)

The strain rate tensor takes the form

e =⎛⎝ ∂u

∂r 0 12 ( ∂u

∂z + ∂v∂r )

0 ur 0

12 ( ∂u

∂z + ∂v∂r ) 0 ∂v

∂z

⎞⎠ , (7.4)

and its trace is equal to the divergence of the velocity,

tr e = div u = 1

r

(∂(r u)

∂r+ ∂(r v)

∂z

)= ∂u

∂r+ ∂v

∂z+ u

r. (7.5)

In order to develop mimetic discretizations of these operators, we make use of (6.14) as well as of the following identi-ties: ∫

V

φ ψ div A dV +∫V

φ A · gradψ dV +∫V

ψ A · gradφ dV =∮∂V

φ ψ A · n dS (7.6)

and ∫grad A : ψ σ T dV +

∫A · ψ divσ dV +

∫A · gradψ · σ dV =

∮ψ n · σ · A dS, (7.7)

V V V ∂V



which follow through a direct integration over the closed and bounded domain V of the differential identities:

div (φ A) = A grad φ + φ div A, (7.8)

div (ψ σ ) = ψ divσ + gradψ · σ , (7.9)

grad (φ ψ) = φ gradψ + ψ grad φ, (7.10)

div(ψ σ · A) = grad A : (ψ σ T )+ A · div(ψ σ ). (7.11)

In the next subsections, we will derive the mimetic operators starting from discrete analogs of the divergence of a vectorfield.

7.2. Primary mimetic operator

To discretize the divergence of a vector field in the axisymmetric case, we use the coordinate invariant definition (6.17).In the rz geometry, the volume of a polygon with vertices (rn, zn) is the volume of a figure of rotation,

V c = π

2

∑n∈N (c)

(zn+ − zn)r2

n+ + rn+ rn + r2n

3, (7.12)

and its derivatives with respect to the cylindrical coordinates rn and zn are

∂V c

∂rn= π

6

(rn+ (zn+ − zn) + 2 rn (zn+ − zn−) + rn− (zn − zn−)

),

∂V c

∂zn= −π

6

((rn+ − rn−) (rn+ + rn + rn−)

).

Using these derivatives we can write the discrete divergence operator DIV :N 2h → Ch as follows

(DIV uh)c = 1

V c

∑n∈N (c)

(∂V c

∂rnun + ∂V c

∂znvn

). (7.13)

It is worth noting that this expression is an approximation of the divergence of a vector field in the conservative form, cf.[19]:

div u = 1

r

(∂(ru)

∂r+ ∂(rv)

∂z

). (7.14)

7.3. Derived gradient operators

In accordance with (6.27) and Proposition 6.3, the components of the derived gradient operator GRAD : Ch → N 2h are

given by the following formulas:

(GRADr ph)n = − 1

Vn

∑c∈C(n)

dV c

drnpc, (GRADz ph)n = − 1

Vn

∑c∈C(n)

dV c

dznpc. (7.15)

These expressions are approximations of the gradient components of a scalar field in the conservative form, cf. [19]:

(grad p)r = 1

r

(∂(rp)

∂r− p

), (grad p)z = 1

r

∂(rp)

∂z. (7.16)

In order to derive the other discrete operators, we need the second derived discrete gradient operator acting from nodes

to cells, GRAD : Nh → C2h . To construct this operator, we use a discrete analog of the integral identity (7.6). As usual, we

assume that the surface integral is zero. Let us take φh ∈ Ch and ψh ∈ Nh . To approximate the first integral in (7.6) weintroduce an interpolation operator acting from cell unknowns to nodal unknowns, M : Ch → Nh , for consistency with theprevious publications, cf. [196,228,237], and its formal adjoint M∗ : Nh → Ch . The explicit form of these operators can befound in the cited publications. Their detailed description is beyond the scope of this paper; therefore, throughout thissubsection we will use only that both M and M∗ are exact on constant vectors.

We set either A = er or A = ez , where er and ez are the coordinate vectors in the cylindrical coordinate system, to obtainthe corresponding components of the gradient. Consider A = er and write the discrete analog of the integral identity (7.6)as follows∑

φc(M∗ψh

)c(div er)c V c +

∑φc(GRADrψh)c V c +

∑ψn(GRADrφh)n Vn = 0. (7.17)

c∈C c∈C n∈N



Here, (div er)c is a suitable approximation of div e = 1/r in cell c, which will be specified in a moment. Setting φc = 1 for

all mesh cell c, we obtain the definition of the r-component of the discrete gradient GRAD:

(GRADrψh)c = 1

V c

∑n∈N (c)

∂V c

∂rnψn − (div er)c

(M∗ψh

)c . (7.18)

This expression is a discrete analog of

gradrψ = 1

r

∂(rψ)

∂r− ψ

r. (7.19)

The r-component of the discrete gradient operator defined by (7.18) must be equal to zero when ψh is constant. SinceM∗ψh is a constant in Ch , we obtain:

(div er)c = 1

V c

∑n∈N (c)

∂V c

∂rn= Ac

V c= 1

rc, (7.20)

where (rc, zc) is the centroid of cell c.Consider now A = ez . Taking into account that div ez = 0, we obtain the following expression for the z-component of the

discrete gradient GRAD:

(GRADzψh)c = 1

V c

∑n∈N (c)

∂V c

∂znψn. (7.21)

We use the new derived gradient operator GRAD and a discrete analog of the integral identity (7.7) to derive themimetic divergence operator DIV acting from cells to nodes. Let us set A = er and note that only the θθ -component ofgrad er is not zero. The discrete analog of (7.7) can be written as follows∑

c∈C

((grad er)θθ

)c(σθθ )c

(M∗ψ

)c V c +

∑n∈N

ψn(DIVrσ h)n Vn

+∑c∈C

((GRADrψh)c(σrr)c + (GRADzψh)c(σzr)c

)= 0,

where ((grad er)θθ )c ∼ 1/rc will be defined later. This expression yields the following formula for the r-component of thediscrete divergence of a tensor:

(DIVrσ h)n = − 1

Vn

∑c∈C(n)

∂V c

∂rn(σrr)c − 1

Vn

∑c∈C(n)

∂V c

∂zn(σrz)c +M(σrr div er − σθθ gradθθ er)n. (7.22)

This expression corresponds to (7.3) in the continuum case. The condition that div pI= grad p, suggests us to define((grad er)θθ

)c = (div er)c = 1

rc,

which allows to write (7.22) in a more compact form:

(DIVr σ )n = − 1

Vn

∑c∈C(n)

∂V c

∂rn(σrr)c − 1

Vn

∑c∈C(n)

∂V c

∂zn(σrz)c +

(M(

σrr − σθθ

r

))n. (7.23)

Hereafter, notation (σrr − σθθ )/r denotes a vector from Ch with components ((σrr)c − (σθθ )c)/rc . We find that this abuse ofnotation improves the readability of long formulas.

Likewise, by setting A = ez , we obtain

(DIV zσ h)n = − 1

Vn

∑c∈C(n)

∂V c

∂zn(σzz)c − 1

Vn

∑c∈C(n)

∂V c

∂rn(σrz)c + (M(σrz div er)

)n, (7.24)

which corresponds to (7.3) in the continuum case. It is obvious that for a diagonal tensor, σ = p I, this discrete divergenceis equal to the gradient of p defined by Eqs. (7.18) and (7.21). Therefore, there holds the mimetic relation:

DIV σ h = GRADph.

Finally, we derive a discrete analog of the strain rate tensor, e = 12 (grad u + (grad u)T ). To derive a mimetic approxima-

tion, we make use of a discrete analog of the integral identity (7.7), where we set preliminarily ψ = 1 and A = u and weassume that σ is a symmetric tensor. With these assumptions, we obtain:


∫V
e : σ dV +∫V

u · divσ dV =∮∂V

n · σ · u dS. (7.25)

A discrete analog of this identity is given by:∑c∈C

((err)c(σrr)c + 2(erz)c(σrz)c + (ezz)c(σzz)c + (eθθ )c(σθθ )c V c

)−∑n∈N

un

( ∑c∈C(n)

∂V c

∂rn(σrr)c +

∑c∈C(n)

∂V c

∂zn(σrz)c − VnM

(σrr − σθθ

r

))n

−∑n∈N

vn

( ∑c∈C(n)

∂V c

∂zn(σzz)c +

∑c∈C(n)

∂V c

∂rn(σrz)c − VnM

(σrz

r

))n

= 0.

This identity leads to the following definition of the discrete strain rate tensor:

(err)c = 1

V c

∑n∈N (c)

un∂V c

∂rn− (M∗u)c

rc,

(err)c = 1

2

(1

V c

∑n∈N (c)

(un

∂V c

∂zn+ vn

∂V c

∂rn

)− (M∗v)c

rc

),

(ezz)c = 1

V c

∑n∈N (c)

vn∂V c

∂zn,

(eθθ )c = (M∗u)c

rc.

These expressions correspond to the components of the strain rate tensor defined by Eq. (7.4). It is also clear that

(tr eh)c = (DIV uh)c .

Therefore, the discrete operators eh and DIV are consistent with each other.

8. Conclusion

We described various mimetic finite difference (MFD) methods that have a common characteristics: they mimics impor-tant physical and mathematical properties of the underlying PDEs. These properties include conservation laws, symmetryand positivity of solutions, duality and self-adjointness of differential operators, and exact mathematical identities of vectorand tensor calculus. The MFD methods are designed to preserve these properties on the unstructured polyhedral meshesthat are often used in applications to represent complex static and moving models accurately.

We described two constructive frameworks based on distinct discrete vector and tensor calculus (DVTCs) that lead tomimetic and compatible discretizations of a wide class of PDEs. The first framework is focused on the discretization ofclassical second-order PDEs in primal and mixed forms. The numerical construction produces discrete analogs of gradient,curl and divergence operators that satisfy exact mathematical identities and discrete Helmholtz decomposition theorems.

We presented the MFD methods for the elliptic PDEs appearing in diffusion, electromagnetics, linear elasticity and fluidmechanics problems, and we summarized their convergence properties. The flexibility of this framework has been illustratedby a few examples that include the development of high-order MFD methods, the stability analysis of saddle-point systems,and the enforcement of a discrete maximum principle.

The second framework is focused on the equations of the Lagrangian hydrodynamics where conservation of mass,momentum, and total energy is paramount, especially in the case of multiple coupled physical processes such as magneto-hydrodynamics. The numerical construction is based on subcell-forces that may have different physical meaning. Usingartificial viscous forces, we illustrated how the discrete operators developed in the two frameworks can be used together.An extension of such framework to curvilinear meshes allowed us to build MFD methods that preserve the symmetry.

At the same time, we recognize that much work has to be done to complete the development and analysis of theseframeworks. In particular, we want to extend the mimetic approach to the discretization of nonlinear PDEs. We also want tocreate a new mimetic theory that will include the discretization in time. We would like to continue the development of themimetic discretizations for polyhedral meshes with strongly non-flat mesh faces [56,57], in particular, for meshes obtainedusing NURBS [79]. Much more has to be done on the discretization of tensor operators and the related extension of theDVTC (see, e.g., Section 7).

Last but not least, we want to extend the mimetic methodology further in non-traditional areas such as the asymptoticproperties of PDEs with a small parameter, the mesh data transfer, the reconstruction of vectors and tensors on generalmeshes, the multimaterial interface reconstruction, and the image recognition.



Acknowledgements

This work was performed under the auspices of the National Nuclear Security Administration of the US Departmentof Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The authors gratefully acknowledgethe partial support of the US Department of Energy Office of Science Advanced Scientific Computing Research (ASCR) Pro-gram in Applied Mathematics Research and the partial support of the US Department of Energy National Nuclear SecurityAdministration Advanced Simulation and Computing (ASC) Program.

The authors thank I. Aavatsmark, P. Antonietti, N. Ardelyan, M. Arioli, D. Arnold, A. Barlow, F. Bassi, L. Beirão de Veiga,L. Beghini, D. Benson, M. Berndt, E. Bertolazzi, N. Bigoni, J. Bishop, P. Bochev, D. Boffi, L. Bonaventura, F. Brezzi, A. Buffa,D. Burton, J. Campbell, A. Cangiani, E. Caramana, J. Castillo, Y. Coudière, L. Demkowicz, D. Di Pietro, J. Droniou, I. Duff, M.Edwards, A. Ern, R. Eymard, A. Favorskii, M. Floater, T. Gallouët, V. Ganzha, R. Garimella, F. Gardini, L. Gastaldi, E. Georgoulis,M. Gerritsma, V. Goloviznin, V. Gyrya, R. Herbin, A. Hirani, K. Hormann, F. Hubert, M. Hyman, A. Iollo, R. Klöfkorn, D. Knoll,T. Kolev, Y. Kuznetsov, R. Liska, R. Loubere, C. Lovadina, P.-H. Maire, L. Margolin, L.D. Marini, D. Mora, J. Morel, D. Moulton,R. Nicolaides, I. Perugia, C. Pierre, G. Paulino, J. Perot, A. Quarteroni, S. Rebay, B. Rider, R. Rieben, T. Russell, A. Russo,A. Samarskii, G. Sangalli, G. Scovazzi, V. Simoncini, P. Smolarkiewicz, A. Solov’ev, S. Steinberg, D. Svyatskiy, N. Sukumar,B. Swartz, C. Talischi, V. Tishkin, F. Teixeira, P. Vachal, M. Verani, M. Vohralík, E. Wachspress, P. Whalen, M. Wheeler, B.Wendroff, and I. Yotov for fruitful discussions and constructive comments over many years. We apologize in advance if weforgot to acknowledge somebody.

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