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The Mimetic Finite Difference Method
Gianmarco Manzini1
Istituto di Matematica Applicata e Tecnologie Informatiche(IMATI) C.N.R., Pavia, Italy
FVCA5 - June 08-13, 2008 Aussois, France
Manzini, G. The Mimetic Finite Difference Method
Outline
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Outline
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Outline
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
“Mimetic (mathematics)”From Wikipedia, the free encyclopedia
(i) The goal of numerical analysis is toapproximate the continuum, soinstead of solving a partialdifferential equation one aims insolve a discrete version of thecontinuum problem.
(ii) A numerical method is called mimetic when it mimics (orimitates) some properties of the continuum vector calculus.
An example: a mixed finite element method applied to Darcyflows strictly conserves the mass of the flowing fluid.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Some literature. . .
Mimetic schemes were first proposed in the early eighties:
Samarskii-Tishkin-Favorskii-Shashkov, OperationalFinite-Difference Schemes, Differential Equations, 1981;
many papers were published after this one. . .
Some recent joint work from Los Alamos-Pavia:
Brezzi-Lipnikov-Shashkov,SINUM,2005 (a priori estimates)Brezzi-Lipnikov-Simoncini, M3AS, 2005 (a family of MFDs). . .
Extensions:
Cangiani-M. CMAME, 2008 (post-processing)Beirao da Veiga-M., NME, 2008 (mesh adaptivity). . .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
. . . people and topics (in Pavia)
Main features:family of schemes based on mixed formulation;grids formed by elements of general shape (polygons,polyhedra);
people currently working in Pavia:Beirao da Veiga, Boffi, Brezzi, Buffa, Cangiani, M.,A. Russo, . . .
some topics under investigation:diffusion and convection-diffusion modelsa posteriori estimates and mesh adaptivityelectromagnetismStokes equations
2-D software implementation
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Linear diffusion in mixed form
Consider
div (−K∇p) = b, Ω ⊂ IRd , d = 2, 3
+boundary conditions
Let−→F be the flux vector variable:
−→F = −K∇p constitutive equation
div−→F = b conservation equation
(1)
Model problem:
solve (1) for p and−→F with suitable boundary conditions
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Formally
FIRST,
(let Th be a family of partitions of Ω formed by polygonalelements, h being the mesh size);
(i) degrees of freedom for− scalar fields −→ discrete scalars , Qh;− vector fields −→ discrete vectors , Xh;
Qh and Xh are not functions, but vectors of numbers!
(ii) “discrete” operators:− the discrete divergence DIVh : Xh → Qh;− the discrete flux (or gradient) Gh : Qh → Xh;
satisfying a duality relationship (discrete Gauss-Green formula).
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Formally
FIRST,
(let Th be a family of partitions of Ω formed by polygonalelements, h being the mesh size);
(i) degrees of freedom for− scalar fields −→ discrete scalars , Qh;− vector fields −→ discrete vectors , Xh;
Qh and Xh are not functions, but vectors of numbers!
(ii) “discrete” operators:− the discrete divergence DIVh : Xh → Qh;− the discrete flux (or gradient) Gh : Qh → Xh;
satisfying a duality relationship (discrete Gauss-Green formula).
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Formally
FIRST,
(let Th be a family of partitions of Ω formed by polygonalelements, h being the mesh size);
(i) degrees of freedom for− scalar fields −→ discrete scalars , Qh;− vector fields −→ discrete vectors , Xh;
Qh and Xh are not functions, but vectors of numbers!
(ii) “discrete” operators:− the discrete divergence DIVh : Xh → Qh;− the discrete flux (or gradient) Gh : Qh → Xh;
satisfying a duality relationship (discrete Gauss-Green formula).
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Formally
THEN,
mimic the continuous differential equations by using discreteoperators acting on the discrete scalar and flux unknownsph ∈ Qh and Fh ∈ Xh:
constitutive equation:
−→F = −K∇p −→ Fh = Ghph
conservation equation:
div−→F = b −→ DIVhFh = bI
(where bI is a suitable interpolation of b in Qh)Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Qh, degrees of freedom for scalar fields
Ω, Th
EqE
• q ∈ Qh means q =˘
qE¯
E∈Th
(equivalent to a piecewise
constant function)
• dim(Qh) = number of elements
of the mesh.
• ”interpolation” operator:
(pI)E =1|E |
Z
Ep dV .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Qh, degrees of freedom for scalar fields
Ω, Th
EqE
• q ∈ Qh means q =˘
qE¯
E∈Th
(equivalent to a piecewise
constant function)
• dim(Qh) = number of elements
of the mesh.
• ”interpolation” operator:
(pI)E =1|E |
Z
Ep dV .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Qh, degrees of freedom for scalar fields
Ω, Th
EqE
• q ∈ Qh means q =˘
qE¯
E∈Th
(equivalent to a piecewise
constant function)
• dim(Qh) = number of elements
of the mesh.
• ”interpolation” operator:
(pI)E =1|E |
Z
Ep dV .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Xh, degrees of freedom for vector fields
Ω, Th
E
E ′
e
• G ∈ Xh means G =˘
GeE
¯
e is an edge of E
• GeE + Ge
E′ = 0 ∀e ⊆ E ∩ E ′
dim(Xh) = number of edges
of the mesh.
• ”interpolation” operator:`−→
F I´e
E=
1|e|
Z
e
−→n eE ·
−→F dV
−→n eE
E
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Xh, degrees of freedom for vector fields
Ω, Th
E
E ′
e
• G ∈ Xh means G =˘
GeE
¯
e is an edge of E
• GeE + Ge
E′ = 0 ∀e ⊆ E ∩ E ′
dim(Xh) = number of edges
of the mesh.
• ”interpolation” operator:`−→
F I´e
E=
1|e|
Z
e
−→n eE ·
−→F dV
−→n eE
E
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Xh, degrees of freedom for vector fields
Ω, Th
E
E ′
e
• G ∈ Xh means G =˘
GeE
¯
e is an edge of E
• GeE + Ge
E′ = 0 ∀e ⊆ E ∩ E ′
dim(Xh) = number of edges
of the mesh.
• ”interpolation” operator:`−→
F I´e
E=
1|e|
Z
e
−→n eE ·
−→F dV
−→n eE
E
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Discrete divergence operator DIVh : Xh → Qh
Let−→G be a true vector field ; take the cell average of div
−→G
over E and use Gauss Theorem:
1|E |
∫
Ediv
−→G dV =
1|E |
∫
∂E
−→n E ·−→G dS =
1|E |
∑
e∈∂E
|e|(−→
GI)e
E
Let G ∈ Xh; the discrete divergence of G in Qh is definedelement by element as the constant value
(DIVhG)E =
1|E |
∑
e∈∂E
|e|GeE
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Discrete divergence operator DIVh : Xh → Qh
Let−→G be a true vector field ; take the cell average of div
−→G
over E and use Gauss Theorem:
1|E |
∫
Ediv
−→G dV =
1|E |
∫
∂E
−→n E ·−→G dS =
1|E |
∑
e∈∂E
|e|(−→
GI)e
E
Let G ∈ Xh; the discrete divergence of G in Qh is definedelement by element as the constant value
(DIVhG)E =
1|E |
∑
e∈∂E
|e|GeE
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
The mimetic conservation equation: DIVhFh = bI
Take the cell average of the conservation equation:
div−→F = b −→
1|E |
∫
Ediv
−→F dV =
1|E |
∫
Eb dV = bI |E
and define the flux Fh as the solution to
(DIVhFh)E = bI |E ≡
1|E |
∫
Ediv
−→F dV
using the inner product[p, q
]Qh
:=∑
E |E | pEqE =
∫
Ω
p q dV ,
variational form:[DIVhFh, q
]Qh
=[bI , q
]Qh
∀q ∈ Qh.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
The mimetic conservation equation: DIVhFh = bI
Take the cell average of the conservation equation:
div−→F = b −→
1|E |
∫
Ediv
−→F dV =
1|E |
∫
Eb dV = bI |E
and define the flux Fh as the solution to
(DIVhFh)E = bI |E ≡
1|E |
∫
Ediv
−→F dV
using the inner product[p, q
]Qh
:=∑
E |E | pEqE =
∫
Ω
p q dV ,
variational form:[DIVhFh, q
]Qh
=[bI , q
]Qh
∀q ∈ Qh.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
The mimetic constitutive equation: Fh = Ghph
Now, we discretize the constitutive equation :
−→F = −K∇p
Note that the conservation equation
1|E |
∑
e∈∂E
|e| (Fh)eE = (bI)E ∀E ∈ Th
holds for• many finite volume schemes;• the RT0−P0 mixed finite element method
. . . So let us first have a look at these approaches.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
The mimetic constitutive equation: Fh = Ghph
Now, we discretize the constitutive equation :
−→F = −K∇p
Note that the conservation equation
1|E |
∑
e∈∂E
|e| (Fh)eE = (bI)E ∀E ∈ Th
holds for• many finite volume schemes;• the RT0−P0 mixed finite element method
. . . So let us first have a look at these approaches.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Finite volume discretization
We use a direct formula for the flux:
−→F = −K∇p (with K = κI)
K
L
A
B
−→n AB
−−→n AB ·
−→F ≈κ
pK − pL∣∣−→x K −−→x L∣∣
K
L
A
B
−−→F ≈κ
(pK − pL∣∣−→x K −
−→x L∣∣−→n AB
γAB+
pB − pA∣∣−→x B −−→x A∣∣−→n KL
γKL
)
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
In mixed finite element the constitutive equation is discretized byusing an explicit representation of the flux field from the fluxdegrees of freedom inside each element.
Let E be a triangle and take the Raviart-Thomas space:
RT0(E) :=
−→v (x , y) =
(α
β
)+ γ
(xy
)(x , y) ∈ E
.
Reconstruct RE (G) inside E by using the degrees of freedomGe
E and the canonical basis functions of RT0(E).
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
In mixed finite element the constitutive equation is discretized byusing an explicit representation of the flux field from the fluxdegrees of freedom inside each element.
Let E be a triangle and take the Raviart-Thomas space:
RT0(E) :=
−→v (x , y) =
(α
β
)+ γ
(xy
)(x , y) ∈ E
.
Reconstruct RE (G) inside E by using the degrees of freedomGe
E and the canonical basis functions of RT0(E).
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
The Raviart-Thomas field RE (G) preserves:
1. the degrees of freedom: −→n eE · RE
(−→G)
= GeE
2. the elemental divergence: divRE(G)
= DIVh,EG
3. constant vector fields: P0-compatible;let G
−→c = (−→c )I with −→c constant inside E ; then,
RE(G
−→c ) =−→c .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
The Raviart-Thomas field RE (G) preserves:
1. the degrees of freedom: −→n eE · RE
(−→G)
= GeE
2. the elemental divergence: divRE(G)
= DIVh,EG
3. constant vector fields: P0-compatible;let G
−→c = (−→c )I with −→c constant inside E ; then,
RE(G
−→c ) =−→c .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
The Raviart-Thomas field RE (G) preserves:
1. the degrees of freedom: −→n eE · RE
(−→G)
= GeE
2. the elemental divergence: divRE(G)
= DIVh,EG
3. constant vector fields: P0-compatible;let G
−→c = (−→c )I with −→c constant inside E ; then,
RE(G
−→c ) =−→c .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
Given Fh, G in Xh (degrees of freedom):
1. reconstruct RE (Fh), RE (G) inside each element E ;
2. define the “RT0 inner product ”:
(Fh, G
)RT0
:=∑
E
∫
EK−1RE (Fh) · RE (G) dV
3. rewrite the RT0 − P0 variational form of K−1−→F = −∇pas:
(Fh, G
)RT0
=∑
E
∫
Eph divRE (G) dV ∀G ∈ Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
Given Fh, G in Xh (degrees of freedom):
1. reconstruct RE (Fh), RE (G) inside each element E ;
2. define the “RT0 inner product ”:
(Fh, G
)RT0
:=∑
E
∫
EK−1RE (Fh) · RE (G) dV
3. rewrite the RT0 − P0 variational form of K−1−→F = −∇pas:
(Fh, G
)RT0
=∑
E
∫
Eph divRE (G) dV ∀G ∈ Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mixed finite elements discretization
Given Fh, G in Xh (degrees of freedom):
1. reconstruct RE (Fh), RE (G) inside each element E ;
2. define the “RT0 inner product ”:
(Fh, G
)RT0
:=∑
E
∫
EK−1RE (Fh) · RE (G) dV
3. rewrite the RT0 − P0 variational form of K−1−→F = −∇pas:
(Fh, G
)RT0
=∑
E
∫
Eph divRE (G) dV ∀G ∈ Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
MFD: mimicking Mixed Finite Elements
FIRST, let RE(·)
be a reconstruction for E with the propertiesof mixed finite elements:
1. the degrees of freedom: −→n eE · RE
(−→G)
= GeE ;
2. the elemental divergence: divRE(G)
= DIVh,EG;
3. constant vector fields: P0-compatible;let G
−→c = (−→c )I with −→c constant inside E ; then,
RE(G
−→c ) =−→c .
We can build many operators: we do not have uniqueness!
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
MFD: mimicking Mixed Finite Elements
FIRST, let RE(·)
be a reconstruction for E with the propertiesof mixed finite elements:
1. the degrees of freedom: −→n eE · RE
(−→G)
= GeE ;
2. the elemental divergence: divRE(G)
= DIVh,EG;
3. constant vector fields: P0-compatible;let G
−→c = (−→c )I with −→c constant inside E ; then,
RE(G
−→c ) =−→c .
We can build many operators: we do not have uniqueness!
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
MFD: mimicking Mixed Finite Elements
FIRST, let RE(·)
be a reconstruction for E with the propertiesof mixed finite elements:
1. the degrees of freedom: −→n eE · RE
(−→G)
= GeE ;
2. the elemental divergence: divRE(G)
= DIVh,EG;
3. constant vector fields: P0-compatible;let G
−→c = (−→c )I with −→c constant inside E ; then,
RE(G
−→c ) =−→c .
We can build many operators: we do not have uniqueness!
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
MFD: mimicking Mixed Finite Elements
FIRST, let RE(·)
be a reconstruction for E with the propertiesof mixed finite elements:
1. the degrees of freedom: −→n eE · RE
(−→G)
= GeE ;
2. the elemental divergence: divRE(G)
= DIVh,EG;
3. constant vector fields: P0-compatible;let G
−→c = (−→c )I with −→c constant inside E ; then,
RE(G
−→c ) =−→c .
We can build many operators: we do not have uniqueness!
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
MFD: mimicking Mixed Finite Elements
THEN, given Fh, G in Xh (degrees of freedom):
1. reconstruct RE (Fh), RE (G) inside each element E ;
2. define the mimetic inner product:
[Fh, G
]Xh
:=∑
E
∫
EK−1RE (Fh) · RE (G) dV
3. mimetic discretization of K−1−→F = −∇p:
[Fh, G
]Xh
=∑
E
∫
EphdivRE (G)dV =
[ph,DIVhG
]Qh
∀G ∈ Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
MFD: mimicking Mixed Finite Elements
THEN, given Fh, G in Xh (degrees of freedom):
1. reconstruct RE (Fh), RE (G) inside each element E ;
2. define the mimetic inner product:
[Fh, G
]Xh
:=∑
E
∫
EK−1RE (Fh) · RE (G) dV
3. mimetic discretization of K−1−→F = −∇p:
[Fh, G
]Xh
=∑
E
∫
EphdivRE (G)dV =
[ph,DIVhG
]Qh
∀G ∈ Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
MFD: mimicking Mixed Finite Elements
THEN, given Fh, G in Xh (degrees of freedom):
1. reconstruct RE (Fh), RE (G) inside each element E ;
2. define the mimetic inner product:
[Fh, G
]Xh
:=∑
E
∫
EK−1RE (Fh) · RE (G) dV
3. mimetic discretization of K−1−→F = −∇p:
[Fh, G
]Xh
=∑
E
∫
EphdivRE (G)dV =
[ph,DIVhG
]Qh
∀G ∈ Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
The Mimetic Finite Difference Method
Scheme formulation:[Fh, G
]Xh
−[ph,DIVhG
]Qh
= 0 ∀G ∈ Xh[DIVhFh, q
]Qh
=[bI, q
]Qh
∀q ∈ Qh.
• Substitute Fh = Ghph in the first equation:[Ghph, G
]Xh
=[ph,DIVhG
]Qh
∀G ∈ Xh
to get the MFD discretization :
K−1−→F = −∇p −→ Fh = Ghph in Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
The Mimetic Finite Difference Method
Scheme formulation:[Fh, G
]Xh
−[ph,DIVhG
]Qh
= 0 ∀G ∈ Xh[DIVhFh, q
]Qh
=[bI, q
]Qh
∀q ∈ Qh.
• Substitute Fh = Ghph in the first equation:[Ghph, G
]Xh
=[ph,DIVhG
]Qh
∀G ∈ Xh
to get the MFD discretization :
K−1−→F = −∇p −→ Fh = Ghph in Xh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Features
Let R(·) = RE (·) be a lifting operator such that RE (·) isP0-compatible on E .
1. R(Xh) ⊂ H(div,Ω); hence R(Xh) − P0 is a conformingmixed discretization that generalizes RT0 − P0;
2. we can use elements of very general shapes, evennon-convex (but at least star-shaped) elements areadmissible;
3. the implementation and analysis of the MFD method aresimilar to those of the Mixed Finite Element method.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Formal constructionMimetic conservation equationMimetic constitutive equation
Mimetic scalar product for the flux
The scalar product for fluxes is given by assembling “elemental”scalar products, each one of which can be represented by asymmetric positive definite (SPD) matrix:
∫
EK−1RE (Fh) · RE (G) dV −→ ME = RE
K−1E
|E |RT
E + ME
RE(·)
is not unique −→ family of matrices
ME contains free positive parameters and ensures that ME
is an SPD matrix;
the formulas for RE , ME depend on the shape of E and canbe derived without the explicit knowledge of RE
().
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
A priori error estimates for Darcy’s equation[Brezzi-Lipnikov-Shashkov, SINUM 2005]
General assumptions:• Ω polygonal or polyhedral with Lipschitz continuous boundary;• Th is a partition satisfying some mesh regularity assumptions;• the scalar product
[·, ·]
Xhsatisfies local consistency and stability;
1. If p ∈ H2(Ω) then
∣∣∣∣∣∣−→F I − Fh∣∣∣∣∣∣
Xh≤ Ch
∣∣∣∣p∣∣∣∣
H2(Ω)
where∣∣∣∣∣∣ ·∣∣∣∣∣∣2
Xh=[·, ·]
Xh.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
A priori error estimates for Darcy’s equation[Brezzi-Lipnikov-Shashkov, SINUM 2005]
2. if p ∈ H2(Ω) then∣∣∣∣∣∣pI − ph
∣∣∣∣∣∣Qh
≤ Ch∣∣∣∣p∣∣∣∣
H2(Ω)
3. superconvergence: if p∈H2(Ω), b∈H1(Ω), and Ω is convex :
∣∣∣∣∣∣pI − ph
∣∣∣∣∣∣Qh
≤ Ch2(∣∣∣∣p
∣∣∣∣H2(Ω)
+∣∣∣∣b∣∣∣∣
H1(Ω)
)
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
A priori error estimates for Darcy’s equation[Brezzi-Lipnikov-Shashkov, SINUM 2005]
2. if p ∈ H2(Ω) then∣∣∣∣∣∣pI − ph
∣∣∣∣∣∣Qh
≤ Ch∣∣∣∣p∣∣∣∣
H2(Ω)
3. superconvergence: if p∈H2(Ω), b∈H1(Ω), and Ω is convex :
∣∣∣∣∣∣pI − ph
∣∣∣∣∣∣Qh
≤ Ch2(∣∣∣∣p
∣∣∣∣H2(Ω)
+∣∣∣∣b∣∣∣∣
H1(Ω)
)
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Post-processing of solution[Cangiani-M. CMAME 2008]
Let −→v be a vector field on E . Then,
−→vInterpolate−→
−→v I Reconstruct−→
−→v ∗ ∈(P0(E)
)d
where the elemental reconstructed vector is given by:∫
E
−→v ∗ · ∇q =[−→v I, (KE∇q)I
]E ∀q ∈ P1(E).
In effect,−→v ∗ = RT
E−→v I
where RE is the matrix used in the definition of the elementalscalar product; so, easy to implement and very cheap .
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Post-processing of solution[Cangiani-M. CMAME 2008, L. Beirao da Veiga Numer. Math. 2008]
1. Reconstruct the elemental gradient :
Fh|EReconstruct
−→−→F ∗
h|ECalculate−→ ∇Ep∗
h = −K−1E
−→F ∗
h;
2. reconstruct the piecewise linear pressure field as
ph|∗
E (−→x ) := pE + ∇E p∗
h · (−→x −
−→x E ), ∀−→x ∈ E
(−→x E center of gravity of E)
Under the hypothesis yielding scalar super-convergence:∣∣∣∣p − p∗
h
∣∣∣∣L2(Ω)
+ h∣∣∣∣∣∣p − p∗
h
∣∣∣∣∣∣1,h ≤ Ch2
(∣∣∣∣p∣∣∣∣
H2(Ω)+∣∣∣∣b∣∣∣∣
H1(Ω)
)
with∣∣∣∣∣∣q∣∣∣∣∣∣2
1,h=∑
E
∣∣∣∣∇q∣∣∣∣2
L2(E)+∑
e
h−1e
∣∣∣∣[[q]]∣∣∣∣2
e.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
An a posteriori error estimator for MFDs[L. Beirao da Veiga, Numer. Math. 2008]
p∗
h , post-processed pressure; [[q]]e jump of q across edge e.
error estimator: η2 :=∑
E ηE2 and
ηE2 :=
∣∣∣∣∣∣Fh + (KE∇p∗
h)I∣∣∣∣∣∣2
E+
12
∑
e∈∂E
h−1e
∣∣∣∣ [[p∗
h ]]e∣∣∣∣2
L2(e)
target error: err2 =∑
E errE2 with
errE2 :=
∣∣∣∣−→F −R(Fh)∣∣∣∣2
L2(E)+ h2
E
∣∣∣∣div (−→F −R(Fh))
∣∣∣∣2L2(E)
+∣∣∣∣∣∣p − p∗
h
∣∣∣∣∣∣1,E
and∣∣∣∣∣∣q∣∣∣∣∣∣2
1,E=∣∣∣∣∇q
∣∣∣∣2L2(E)
+∑
e∈∂E
h−1e
∣∣∣∣ [[q]]e∣∣∣∣2
L2(e)
efficiency: cηE ≤ errE ;
reliability: err ≤ Cη;.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
An a posteriori error estimator for MFDs[L. Beirao da Veiga, Numer. Math. 2008]
p∗
h , post-processed pressure; [[q]]e jump of q across edge e.
error estimator: η2 :=∑
E ηE2 and
ηE2 :=
∣∣∣∣∣∣Fh + (KE∇p∗
h)I∣∣∣∣∣∣2
E+
12
∑
e∈∂E
h−1e
∣∣∣∣ [[p∗
h ]]e∣∣∣∣2
L2(e)
target error: err2 =∑
E errE2 with
errE2 :=
∣∣∣∣−→F −R(Fh)∣∣∣∣2
L2(E)+ h2
E
∣∣∣∣div (−→F −R(Fh))
∣∣∣∣2L2(E)
+∣∣∣∣∣∣p − p∗
h
∣∣∣∣∣∣1,E
and∣∣∣∣∣∣q∣∣∣∣∣∣2
1,E=∣∣∣∣∇q
∣∣∣∣2L2(E)
+∑
e∈∂E
h−1e
∣∣∣∣ [[q]]e∣∣∣∣2
L2(e)
efficiency: cηE ≤ errE ;
reliability: err ≤ Cη;.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Some experiments on mesh adaptivity[L. Beirao da Veiga & M., IJNME, 2008]
The indicator ηE allows us to select elements needing refinement:
1. calculate local error estimates ηE ;
2. mark for refinement those elements such that ηE ≥ tol ηmax ;
3. subdivide the marked elements as follows:
No need for special treatment of hanging nodes!
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Mesh adaptivityL-shaped domain, K = II, p(r , θ) = r2/3 sin(2θ/3)
102
103
104
105
10610
-3
10-2
10-1
100
After 3 refinements η(), err(•) vs #elements
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Outline
1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation
2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
3 Summary
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Convection-Diffusion-Reaction Equation(In collaboration with A. Cangiani and A. Russo)
• Problem:−→F = −(K∇ p +
−→β p) in Ω
div−→F + c p = b in Ω
p = 0 on ∂Ω
with
• β, K and c smooth fields plus usual coercivity conditions,
• Ω ⊂ IR2 is a polygonal domain.
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Convection-Diffusion-Reaction EquationScheme formulation
discretization of convection-reaction terms
convection:∫
Ω
K−1−→β p · R(G)
dV −→∑
E pE[(−→β )I, G
]E
reaction:∫
Ω
c p q dV −→[cIph, q
]Qh
Scheme formulation[Fh, G
]Xh
−[p,DIVhG
]Qh
+∑
E pE[(−→β )I , G
]E = 0 ∀G ∈ Xh
[DIVhFh, q
]Qh
+[cIph, q
]Qh
=[b, q
]Qh
∀q ∈ Qh
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Convection-Diffusion-Reaction Equation
a priori estimate for the diffusive regime:
∣∣∣∣−→F −R(Fh)∣∣∣∣
H(div,Ω)+∣∣∣∣p − ph
∣∣∣∣L2(Ω)
≤ Ch(∣∣∣∣p
∣∣∣∣H1(Ω)
+ h∣∣∣∣p∣∣∣∣
H2(Ω)+∣∣∣∣b − bI
∣∣∣∣L2(Ω)
),
(with−→β ∈ W 2,∞(Ω), c ∈ W 1,∞(Ω), and coercivity assumptions)
superconvergence for pressure cell averages under sameassumptions of the pure elliptic problem and
−→β ∈ R(Xh):
∣∣∣∣∣∣pI − ph
∣∣∣∣∣∣Qh
≤ Ch2(∣∣∣∣p
∣∣∣∣H2(Ω)
+∣∣∣∣b∣∣∣∣
H1(Ω)
)
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Convection-Diffusion-Reaction EquationConvergence results
p(x , y) = sin(2π x) sin(2π y)
+x2 + y2 + 1,
−→β = (1, 3)T ,
c(x , y) = x y2
n h L2-error Rate Hdiv -error Rate1 9.135 10−2 9.134 10−2 −− 1.823 10−1 −−
2 4.654 10−2 4.630 10−2 1.007 8.572 10−2 1.1183 2.346 10−2 2.315 10−2 1.012 4.168 10−2 1.0524 1.175 10−2 1.164 10−2 0.995 2.039 10−2 1.0345 5.880 10−3 5.841 10−3 0.995 1.007 10−2 1.0186 2.940 10−3 2.927 10−3 0.996 5.027 10−3 1.002
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation
Convection-Diffusion-Reaction EquationSuperconvergence results
p(x , y) = sin(2π x) sin(2π y)
+x2 + y2 + 1,
−→β = (1, 3)T ,
c(x , y) = x y2
n h Qh-error Rate1 9.135 10−2 3.069 10−2 −−
2 4.654 10−2 1.078 10−2 1.5513 2.346 10−2 2.807 10−3 1.9644 1.175 10−2 7.483 10−4 1.9135 5.880 10−3 1.904 10−4 1.9756 2.940 10−3 4.796 10−5 1.989
Manzini, G. The Mimetic Finite Difference Method
Mimetic FormulationTheoretical results and applications
Summary
Summary
The MFD method for second-order elliptic problems
mimics properties of continuous operators; e.g. DIVh, Gh
satisfy discrete Green-like formulas ;
works on element of very general shape;
shows a strong connection with the lowest-order mixedfinite element method RT0 − P0, helpful in establishing thetheoretical foundation.
Manzini, G. The Mimetic Finite Difference Method
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