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Doctoral Dissertation
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Preconditioners for FETI-DP formulations with mortar methods
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Korea Advanced Institute of Science and Technology
2004
"!# $% &' (*),+.-/102 3465FETI-DP798:<;>=? @A preconditioner BDC EFG5IHKJL MONP QR
Preconditioners for FETI-DP formulations with
mortar methods
Preconditioners for FETI-DP formulations with
mortar methods
Advisor : Professor Lee, Chang-Ock
by
Kim, Hyea Hyun
Department of Mathematics, Division of Applied Mathematics
Korea Advanced Institute of Science and Technology
A dissertation submitted to the faculty of the Korea Ad-
vanced Institute of Science and Technology in partial fulfillment
of the requirements for the degree of Doctor of Philosophy in the
Department of Mathematics, Division of Applied Mathematics.
Daejeon, Korea
2003. 11. 22.
Approved by
Professor Lee, Chang-Ock
Advisor
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995113
àáâäãæåèçµéê . Kim, Hyea Hyun. Preconditioners for FETI-DP formulations
with mortar methods. ëìîíïñðÍò íïóÍôõ÷ö¥øùûúïîüýÿþ FETI-DP á pre-
conditioner "!#%$'&ê)(* . Department of Mathematics, Division of Applied
Mathematics. 2004. 115p. Advisor Prof. Lee, Chang-Ock. Text in English.
Abstract
In this dissertation, we consider FETI methods which are known as the most
efficient domain decomposition method especially for solving large scale problems.
In FETI methods, Lagrange multipliers are introduced to enforce the continuity of
solutions across subdomain interfaces. This gives a mixed problem with the conti-
nuity condition as constraints. After eliminating unknowns other than the Lagrange
multipliers, the resulting linear system is solved using the preconditioned conjugate
gradient method. There are three variants of FETI methods, FETI, two-level FETI
and dual-primal FETI(FETI-DP) method. Until now, FETI methods have been
developed for the problems discretized with conforming finite elements. Among
them, we extend FETI-DP methods to the problems with nonconforming discretiza-
tions, that arise from nonmatching triangulations across subdomain interfaces. The
nonmatching triangulations are important for problems with corner singularities,
contact problems as well as multi-physics problems. Moreover, the generation of
meshes can be done independently in each subdomain. To resolve the nonconfor-
mity of the approximation, we consider mortar methods, which gives the same order
of accuracy as conforming finite elements. In the mortar methods, the Lagrange
multiplier space is introduced to enforce the continuity of solutions across the sub-
domain interfaces. The saddle point formulation of mortar methods gives a similar
linear system to the mixed formulation of the FETI methods. The linear system is
ill-conditioned. Moreover, it is difficult to find a good preconditioner for this sys-
tem. We apply the FETI-DP method to solving this linear system efficiently and to
finding a good preconditioner easily.
This dissertation concerns elliptic problems both in 2D and 3D, and Stokes
problem in 2D. Especially, redundant continuity constraints are introduced for 3D
elliptic problems and Stokes problem. The Lagrange multipliers to the redundant
constraints are treated as the primal variables in the FETI-DP formulation. This
i
redundant constraints accelerate the convergence of FETI-DP methods. We propose
Neumann-Dirichlet preconditioners for the FETI-DP formulations of those problems
considered in this dissertation. The Neumann-Dirichlet preconditioner follows from
a dual norm on the Lagrange multiplier space. To define the dual norm, we consider
a duality pairing between the Lagrange multipliers and finite elements on nonmortar
sides. A norm for the finite elements on nonmortar sides are defined by using the
discrete harmonic extension or the Stokes extension. We show that the precondi-
tioner gives the condition number bound Cmaxi=1,··· ,N
(1 + log(Hi/hi))
2, where
C is a constant independent of meshes and the number of subdomains. Here, Hi and
hi are the subdomain size and mesh size associated with Ωi, and N is the number of
subdomains. For the elliptic problems with discontinuous coefficients, we can also
show that the constant C is not depending on the coefficients. In addition, numerical
results are provided.
ii
Contents
Abstract i
Contents iii
List of Tables v
List of Figures vi
1 Introduction 1
2 Sobolev spaces and finite elements 6
2.1 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Approximation by interpolation and inverse inequalities . . . . . . . 7
2.3 Vertex-edge-face lemmas for finite element functions . . . . . . . . . 11
3 Overview of FETI methods 13
3.1 A model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 FETI method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Dual-Primal FETI(FETI-DP) method . . . . . . . . . . . . . . . . . 20
3.4 Augmented FETI-DP method . . . . . . . . . . . . . . . . . . . . . . 26
4 Mortar methods 29
4.1 Nonconforming approximation . . . . . . . . . . . . . . . . . . . . . 30
4.2 Lagrange multiplier spaces . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 A priori error estimates . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Elliptic problems in 2D 44
5.1 A model problem and finite elements . . . . . . . . . . . . . . . . . . 44
5.2 FETI-DP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 47
iii
5.2.1 FETI-DP operator . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Condition number bound estimation . . . . . . . . . . . . . . . . . . 52
6 Elliptic problems in 3D 58
6.1 A model problem and finite elements . . . . . . . . . . . . . . . . . . 58
6.2 FETI-DP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2.1 FETI-DP operator . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3 Condition number bound estimation . . . . . . . . . . . . . . . . . . 64
7 Stokes problem in 2D 72
7.1 A model problem and finite elements . . . . . . . . . . . . . . . . . . 72
7.2 FETI-DP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.2.1 FETI-DP operator . . . . . . . . . . . . . . . . . . . . . . . . 76
7.2.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3 Condition number bound estimation . . . . . . . . . . . . . . . . . . 82
8 Numerical results 88
8.1 Elliptic problems in 2D . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.1.1 An elliptic problem with smooth coefficients . . . . . . . . . . 89
8.1.2 Elliptic problems with highly discontinuous coefficients . . . . 91
8.2 Stokes problem in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Appendix 104
Summary (in Korean) 108
References 110
iv
List of Tables
8.1 Comparison between FKL and FDW on matching(up) and nonmatch-
ing(down) grids when N = 4 × 4 . . . . . . . . . . . . . . . . . . . . 91
8.2 Comparison between FKL and FDW on matching(up) nonmatching(down)
grids when n− 1 = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.3 Comparison of preconditioners F−1KL, F−1
DW and F−1KW (γ = 2.0) for the
problem with highly discontinuous coefficients . . . . . . . . . . . . . 96
8.4 Condition numbers (number of iterations) of F−1KW (γ = 0.5, 1.0, 2.0, 10.0)
and F−1KL for the problems with highly discontinuous coefficients . . . 97
8.5 CG iterations(condition number) when N = 4 × 4 . . . . . . . . . . . 99
8.6 CG iterations(condition number) when n = 5 . . . . . . . . . . . . . 100
8.7 CG iterations(condition number) when n = 9 . . . . . . . . . . . . . 100
8.8 H1 and L2-errors(factor) on matching grids . . . . . . . . . . . . . . 101
8.9 H1 and L2-errors(factor) on nonmatching grids: N = 4 × 4 . . . . . 101
8.10 H1 and L2-errors(factor) on nonmatching grids: n = 5 . . . . . . . . 102
8.11 H1 and L2-errors(factor) on nonmatching grids: n = 9 . . . . . . . . 102
8.12 Inf-sup constant β0 when n = 5 and n = 9 . . . . . . . . . . . . . . . 103
8.13 Inf-sup constant β0 when N = 4 × 4 . . . . . . . . . . . . . . . . . . 103
v
List of Figures
3.1 Geometrically nonconforming(left) and conforming(right) partitions 14
3.2 FETI vs. FETI-DP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1 Basis functions for Wij and Mij(standard) . . . . . . . . . . . . . . . 34
4.2 Basis functions for Wij and Mij(dual) . . . . . . . . . . . . . . . . . 36
8.1 Partition of subdomains when N = 4 × 4 . . . . . . . . . . . . . . . . 89
8.2 Matching grids(left) and nonmatching grids(right) when n = 5 . . . 90
8.3 Triangulations for the case N = 2 × 2 and max(Hij/hij) = 16 . . . . 94
8.4 Triangulations Ωhii (left) and Ω2hi
i (right) when n = 5 . . . . . . . . . 99
vi
1. Introduction
Domain decomposition methods for solving partial differential equations have been
developed from the idea of Schwarz alternating method [45] which is an iterative
method for the solution of classical boundary value problems for harmonic functions.
In the method, moving alternately from one subdomain to the other subdomain,
similar problems are solved successively and the solutions formed by the iterative
process converge to the solution of the original single domain problem.
In fact, this iterative process can be regarded as a preconditioner for the bound-
ary value problem in the original domain. The preconditioner is essentially composed
of operators which solve local problems in each subdomain. Hence, the goal of do-
main decomposition methods is to develop a good preconditioner using the local
solvers from each subdomain. There are Schwarz methods for overlapping decom-
positions, substructuring methods and FETI (Finite Element Tearing and Intercon-
necting) methods for nonoverlapping decompositions. Among them, FETI methods
are most efficient and scalable especially when we solve large scale problems using
parallel machine. The scalability means that a method is robust to the increase of
subdomains with the fixed number of unknowns in each subdomain. Usually, we
need a coarse finite element space to obtain scalability of the method. In FETI
methods, coarse problem is naturally induced from the FETI formulation without
forming a special coarse finite element space.
The FETI method was first introduced by Farhat and Roux[29] for solving the
elastostatic problems. It is another variant of substructuring iterative methods. The
main idea is using Lagrange multipliers to match the solutions continuously across
subdomain boundaries. This gives a mixed problem. After eliminating unknowns
other than Lagrange multipliers, they obtained a linear system for the Lagrange
multipliers. In fact, this linear system, so called the FETI operator, is symmetric
and positive definite(s.p.d.) and solved using the preconditioned conjugate gradient
method(PCGM). We call these whole processes as the FETI method. Further, they
1
introduced a Dirichlet preconditioner and presented the numerical scalability of the
FETI method for second order elliptic problems.
Mandel and Tezaur [37] analyzed that the condition number of the FETI operator
with the Dirichlet preconditioner is bounded by C (1 + log (H/h))m with m ≤ 3 for
second order elliptic problems in 2D and 3D both, where H and h denote the sizes of
subdomains and meshes, respectively and C is a constant independent of mesh size
and subdomain size. For the same problem, Klawonn and Widlund[31] proposed a
new preconditioner using diagonal scaling matrix and showed that the bound of con-
dition number is C (1 + log (H/h))2. Moreover, they allowed jumps of coefficients of
elliptic problems across subdomain boundaries. However, for fourth order problems,
it was observed that the condition number grows faster than O (1 + log (H/h))3.
Farhat et al. [25, 27] developed the two level FETI method and they showed that
this method is numerically scalable for fourth order elliptic problems like as second
order elliptic problems.
The dual-primal FETI(FETI-DP) method was introduced in [26] with the similar
idea to the two level FETI method. The idea is to use primal variables at corner
points and Lagrange multipliers on edges to match solutions continuously across
subdomain boundaries. In the FETI-DP method, unknowns other than the primal
variables at corners and the Lagrange multipliers are eliminated first. Then, the
linear system for primal variables at corners and Lagrange multipliers follows. After
eliminating the primal variables at corners, we obtain the resulting linear system
of Lagrange multipliers, which is called a FETI-DP operator. This operator is
also s.p.d. and solved using PCGM as in the FETI method. However, we have
nonsingular local problems and a global corner problem in the FETI-DP operator.
These make the implementation of the FETI-DP method easier than the FETI
method. Moreover, the global corner problem fulfills the role of a coarse solver, which
globally transmits information between subdomains. They also showed numerically
that the FETI-DP method is scalable with respect to the mesh size, the subdomain
size and the number of elements per subdomain for second and fourth order elliptic
problems both. Mandel and Tezaur[38] analyzed that the condition number of the
FETI-DP method is bounded by C (1 + log (H/h))2 for both second and fourth order
2
elliptic problems in 2D. For 3D elliptic problems with heterogeneous coefficients,
Klawonn et el. [32] obtained the same bound of the condition number. In addition,
the FETI-DP method was applied to solving Stokes problem and Navier-Stokes
problem by Li [34, 35].
Recently, FETI(-DP) methods are applied to the problems discretized with non-
conforming finite elements [21, 22, 42, 48, 49]. Especially, the nonconforming fi-
nite elements arising from nonmatching triangulations across subdomain interfaces
are considered. Nonmatching discretizations are important for multiphysics simu-
lations, contact-impact problems, the generation of meshes and partitions aligned
with jumps in diffusion coefficients, hp-adaptive methods, and special discretizations
in the neighborhood of singularities (corners or joints). Of many methods for non-
matching methods, including [20] and [44], we consider mortar methods to resolve
the nonconformity of approximations. In mortar methods, orthogonality relations
between the jumps in the traces across subdomain interfaces are satisfied using a
discrete Lagrange multiplier space. Then, the mortar methods give the same ac-
curacy of approximations as conforming finite elements with the same polynomial
order. The sparse linear systems that arise in mortar methods are similar to the
systems solved by FETI methods on conforming discretizations [23, 29]. Hence,
FETI(-DP) methods can be applied to solving this linear system efficiently.
In [42, 48, 49], numerical study shows that FETI methods with mortar discretiza-
tions are efficient and the preconditioned FETI operator seems to have condition
number bound C(1 + log(H/h))2. After then, Dryja and Widlund [21] showed that
the Dirichlet preconditioner gives a condition number bound (1+log(H/h))2 with the
Neumann-Dirichlet ordering of substructures, where H and h denote the maximum
diameter of subdomains and minimum size of meshes of all subdomains, respectively.
In general cases, that is, without considering ordered substructures, they obtained
(1 + log(H/h))4 for the condition number bound. Moreover, in [22], they proposed
a different preconditioner which is similar to the one in [31], and proved the con-
dition number bound (1 + log(H/h))2. However, in their analysis, they imposed a
restriction that the sizes of meshes between neighboring subdomains are comparable.
This restriction is impractical when the coefficients of elliptic problems are highly
3
discontinuous between subdomains (see Wohlmuth[54]).
In this dissertation, we extend FETI-DP methods to the problems with mor-
tar discretizations. For the elliptic problems both in 2D and 3D, we obtain the
FETI-DP formulation differently with that of Dryja and Widlund [21, 22] and pro-
pose a Neumann-Dirichlet preconditioner which gives the condition number bound
C(1 + log(H/h))2 without the restriction on the mesh size between neighboring
subdomains. Moreover, for the elliptic problems with heterogeneous coefficients
the condition number bound is shown to be independent of the coefficients. The
Neumann-Dirichlet preconditioner follows from a dual norm on the Lagrange mul-
tiplier space. The dual norm is defined by using a duality pairing between the
Lagrange multiplier space and finite elements on nonmortar sides, and a norm for
the finite elements on nonmortar sides. The norm for the finite element function
on nonmortar sides is given by the discrete harmonic extension of that function. In
[32], it was shown that only considering corners as primal variables is inefficient for
3D problems. Hence, redundant mortar matching constraints are essentially needed
for 3D problems to obtain the same condition number bound as 2D problems. More-
over, the corresponding Lagrange multipliers are treated as primal variables in the
FETI-DP formulation.
For Stokes problem, we derive the FETI-DP operator with mortar matching
constraints and show that the Neumann-Dirichlet preconditioner gives the condition
number bound C(1 + log(H/h))2. In the FETI-DP formulation, we add redundant
continuity constraints to the coarse problems following the idea of Li [34]. These
constraints are introduced to solve the Stokes problem correctly and efficiently.
This dissertation is organized as follows: In Chapter 2, we introduce Sobolev
spaces and finite elements, and in Chapter 3, we overview FETI(-DP) methods.
Mortar methods are explained in Chapter 4. Chapter 5 and Chapter 6 are devoted to
FETI-DP formulations of the elliptic problems in 2D and 3D, and the analysis of the
condition number bound for the Neumann-Dirichlet preconditioner. In Chapter 7,
we extend the method to the Stokes problem. Numerical results are presented in
Chapter 8.
In the following, we make no distinction between a finite element function and
4
the corresponding vector of nodal values, i.e., we use the same symbol v both for the
finite element function and the vector of nodal values. Similarly, we use the same
notation for a finite element function space and a space of vectors of nodal values.
Moreover, the constant C is a generic constant which varies from place to place and
does not depend on the mesh size h and the subdomain size H.
5
2. Sobolev spaces and finite elements
2.1 Sobolev spaces
Let Ω ⊂ Rn(n = 2, 3) be a bounded polygonal(n = 2) or polyhedral(n = 3) domain
and L2(Ω) be the space of square integrable functions defined in Ω equipped with
the norm ‖ · ‖0,Ω:
‖v‖20,Ω :=
∫
Ωv2 dx.
The space L20(Ω) is a set of functions in L2(Ω) with zero average. The space H1(Ω)
is a set of functions in L2(Ω), which are square integrable up to the first weak
derivatives, and the norm is given by
‖v‖1,Ω :=
(∫
Ω∇v · ∇v dx+
1
d2Ω
∫
Ωv2 dx
)1/2
,
where dΩ denotes the diameter of Ω. For any set A, we denote dA as the diameter
of the set A.
Now, we introduce Sobolev spaces defined on the boundary ∂Ω of the domain Ω.
Let Σ ⊂ ∂Ω. For w ∈ L2(Σ), we define
|w|21/2,Σ :=
∫
Σ
∫
Σ
|w(x) − w(y)|2
|x− y|nds(x) ds(y).
Then H1/2(∂Ω) is the trace space of H1(Ω) normed by
‖w‖21/2,∂Ω := |w|21/2,∂Ω +
1
d∂Ω‖w‖2
0,∂Ω.
For any F ⊂ ∂Ω, H1/200 (F ) is the set of functions in L2(F ) such that the zero
extensions of the functions into ∂Ω are contained in H1/2(∂Ω). The norm for v ∈
H1/200 (F ) is given by
‖v‖H
1/200 (F )
:=
(|v|2
H1/200 (F )
+1
dF‖v‖2
0,F
)1/2
,
6
where
|v|2H
1/200 (F )
:= |v|2H1/2(F )
+
∫
F
v(x)2
dist(x, ∂F )ds.
The space H1/200 (F ) can be obtained by Hilbert scaling between the spaces L2(F )
and H10 (F ) or by the real method of interpolation between those spaces (see Lions
and Magenes [36]). From Section 4.1 in [56], we have the following relation:
C1‖v‖1/2,∂Ω ≤ ‖v‖H
1/200 (F )
≤ C2‖v‖1/2,∂Ω ∀v ∈ H1/200 (F ), (2.1)
where the constants C1 and C2 are independent of F and v is the zero extension of
v into ∂Ω. For the product spaces [H1/2(∂Ω)]2 and [H1/200 (F )]2, norms are defined
using the product norms and the inequalities (2.1) also hold.
In general, we use Wmp (Ω) to denote the Sobolev space with m-th weak deriva-
tives in Lp-norm. The norm is defined by
‖v‖Wmp (Ω) :=
∑
0≤k≤m
|v|pW kp (Ω)
1/p
,
and the semi-norm | · |W kp (Ω) is defined by
|v|W kp (Ω) :=
∑
|α|=k
∫
Ω|Dαv(x)|p dx
1/p
.
Here, α = (α1, · · · , αn) denotes a multi-index , |α| =∑n
i=1 αi and Dαv(x) is the
weak derivative of v(x) corresponding to the multi-index α. Note that we write
the Sobolev space with scaled H1-norm as H1(Ω) and the usual Sobolev space as
W 12 (Ω).
2.2 Approximation by interpolation and inverse inequal-
ities
In this section, we introduce several interpolation operators and review the approxi-
mation properties and the inverse inequalities for the finite element functions. These
results are used to analyze the approximation order of finite element methods.
7
Definition 2.1 A domain Ω is said to be star-shaped with respect to B if, for every
x ∈ Ω, the closed convex hull of x⋃B is contained in Ω.
For a star-shaped domain Ω, let
ρmax = supρ : Ω is star-shaped with respect to a ball of radius ρ.
Then, we state the following well-known result by Bramble and Hilbert [14, 15]:
Lemma 2.2 (Bramble-Hilbert) Let B be a ball in Ω such that Ω is star-shaped
with respect to B and such that its radius ρ > (1/2)ρmax. Then, for u ∈ Wm+1p (Ω)
with p ≥ 1, there exists Qmu of polynomial of degree m such that
|u−Qmu|W kp (Ω) ≤ Cdm+1−k
Ω |u|Wm+1p (Ω), 0 ≤ k ≤ m+ 1.
The polynomial Qmu is obtained from the Taylor polynomial of degree m of u
averaged over B, that is,
Qmu(x) =
∫
BTmy u(x)φ(y) dy,
where
Tmy u(x) =∑
|α|≤m
1
α!Dαu(y)(x− y)α,
and φ(x) ∈ C∞0 (Rn) is a cut-off function with supp(φ) = B and
∫Rnφ(x) dx = 1.
Now, we introduce finite elements in Ω. Let Ωh be a triangulation of Ω with
maximum diameter h. We assume that Ωh is regular, that is, there exists a constant
σ independent of h such that
hκ ≤ σρκ ∀κ ∈ Ωh,
where hκ is the diameter of κ and ρκ is the diameter of the circle inscribed in κ. For
each triangle κ, we consider Σκ as the principal lattice of order m in κ:
Σκ =
x =
n+1∑
j=1
λjaj :n+1∑
j=1
λj = 1 with λj ∈ 0, 1/m, · · · , (m− 1)/m, 1
,
8
where aj ∈ Rn denotes a vector corresponding to the j-th vertex of κ. Then, we
define
Xh(Ω) :=v ∈ C0(Ω) : v|κ ∈ Pm(κ) ∀κ ∈ Ωh
,
where Pm(κ) is the set of polynomials of degree up to m associated with Σκ.
Then, for v ∈ C0(Ω), we define the nodal value interpolation Ihv(x) ∈ Xh(Ω) by
Ihv(xl) = v(xl) ∀xl ∈ Σκ, ∀κ ∈ Ωh.
From the Sobolev imbedding theorem [3], we have
W kp (Ω) → Cj(Ω) with 0 ≤ j < k − n/p.
Applying the above inclusion with k = 1, 2 and j = 0, we obtain
W 2p (Ω) → C0(Ω) with p > n/2,
W 1p (Ω) → C0(Ω) with p > n.
Hence, we have the following approximation properties for the nodal value interpo-
lation Ihv(x):
Lemma 2.3 For all v ∈W s+1p (Ω) with p > n/2 and 1 ≤ s ≤ m, we have
|v − Ihv|W kp (Ω) ≤ Chs+1−k|v|W s+1
p (Ω), 0 ≤ k ≤ s+ 1.
Moreover, for all v ∈W 1p (Ω) with p > n, we have
|v − Ihv|W kp (Ω) ≤ Ch1−k|v|W 1
p (Ω), k = 0, 1.
For a non-smooth function v(x), interpolation operators with the same approxi-
mation order as the nodal value interpolation were developed by Clement [19] and
Scott and Zhang [46]. Both of them use the average values of v(x) near a nodal point
to obtain the interpolation. The interpolation by Clement does not regenerate the
functions in Xh, where as the interpolation by Scott and Zhang does regenerate the
functions in Xh. Both fit the zero boundary condition of v(x), where as the inter-
polation Qmv(x) by Bramble and Hilbert dose not fit the zero boundary condition.
The interpolation by Scott and Zhang can also fit more general boundary conditions.
9
Further, the idea of Scott and Zhang is generalized to construct Lagrange multipliers
with dual basis (see [53]).
We do not give the exact forms of those interpolations and only state the approx-
imation properties of those interpolations. Let Qv(x) and Iv(x) be the Clement,
and Scott and Zhang interpolations, respectively.
Lemma 2.4 For v ∈ W s+1p (Ω) with 0 ≤ s ≤ m, there exists Qv(x) ∈ Xh(Ω) such
that
‖v −Qv‖W kp (Ω) ≤ Chs+1−k‖u‖W s+1
p (Ω), 0 ≤ k ≤ s+ 1.
Lemma 2.5 For v ∈ W s+1p (Ω) with 0 ≤ s ≤ m, there exists Iv(x) ∈ Xh(Ω) such
that
|v − Iv|W kp (Ω) ≤ Chs+1−k|v|W s+1
p (Ω), 0 ≤ k ≤ s+ 1.
Now, we discuss relations among various norms on a finite element space Xh.
For a regular triangulation Ωh, we add an additional assumption that there exists a
constant γ independent of h such that
γh ≤ hκ ∀κ ∈ Ωh.
We call a regular triangulation Ωh with the above property as a quasi-uniform tri-
angulation. We have the following inverse inequalities for a finite element space Xh
associated with a quasi-uniform triangulation Ωh.
Lemma 2.6 For v ∈ Xh, let v|κ ∈W lp(κ)
⋂W kq (κ) with 1 ≤ p, q ≤ ∞ and 0 ≤ k ≤
l. Then there exists a constant C independent of h and κ such that
‖v‖W lp(κ)
≤ Chk−l+n/p−n/q‖v‖W kq (κ).
Using the above result when m = 1, that is, Xh is a piecewise linear finite element
space, and p = q = 2, l = 1, we obtain
‖v‖1,Ω ≤ Ch−1‖v‖0,Ω. (2.2)
10
Moreover, we have
‖v‖1/2,∂Ω ≤ Ch−1/2‖v‖0,∂Ω, (2.3)
‖v‖1,∂Ω ≤ Ch−1/2‖v‖1/2,∂Ω.
The above results are shown by Bramble et al. [17] and Xu [55].
2.3 Vertex-edge-face lemmas for finite element functions
In this section, we introduce several inequalities related to the interpolation of func-
tions on a part of ∂Ω, that is, a face, an edge, or a vertex. Those inequalities are
essentially used to analyze the condition number bound of substructuring methods,
Neumann-Neumann methods or FETI(-DP) methods.
Let Ωh be a quasi-uniform triangulation of Ω with maximum diameter h and
Xh(Ω) be a piecewise linear finite element space associated with Ωh. Then, we
consider subsets of ∂Ω, faces, edges and vertices. The faces and edges are open
subset of ∂Ω, that is, those sets do not include their boundaries. For Ω ⊂ R2, edges
are considered as faces. We use F , E and V to denote a face, an edge, and a vertex
of ∂Ω, respectively.
Recall the nodal value interpolation Ihv(x) in Section 2.2. Let Nh be the set of
nodes in Ωh and Xh(∂Ω) be the space of functions in Xh(Ω) restricted on ∂Ω. For
a set A ∈ ∂Ω, we define a nodal value interpolation IhAw(x) ∈ Xh(∂Ω) as follows:
IhAw(x) =
w(x) if x ∈ Nh
⋂A,
0 otherwise.
We have the following famous lemmas for the above interpolation [56]. In the fol-
lowing, H denotes the diameter of Ω and the constant C is a generic constant
independent of H and h.
Lemma 2.7 (Vertex lemma) Let V be a vertex of ∂Ω. Then for any w ∈
Xh(∂Ω), we have
‖IhV w‖1/2,∂Ω ≤ Ch(n−1)/2|w(V )| ≤ C
(1 + log
H
h
)1/2
‖w‖1/2,∂Ω.
11
Lemma 2.8 (Edge lemma) Assume that n = 3 and E is an edge of ∂Ω. Then
for any w ∈ Xh(∂Ω),
‖IhEw‖1/2,∂Ω ≤ C‖w‖0,E ≤ C
(1 + log
H
h
)1/2
‖w‖1/2,∂Ω.
Lemma 2.9 (Face lemma) Let F be a face (n = 3) or an edge (n = 2) of ∂Ω.
Then for any w ∈ Xh(∂Ω),
‖IhFw‖1/2,∂Ω ≤ C
(1 + log
H
h
)‖w‖1/2,∂Ω.
Lemma 2.10 Let F be a face (n = 3) or an edge (n = 2) of ∂Ω. Then we have
‖IhF 1‖1/2,∂Ω ≤ CH(n−2)/2
(1 + log
H
h
)1/2
.
Lemma 2.11 Let n = 2. For any edge E ⊂ ∂Ω and any w ∈ Xh(∂Ω),
‖w − I∂Ωw‖H1/200 (E)
≤ C
(1 + log
H
h
)|w|1/2,∂Ω,
where I∂Ωw = w on the corners of ∂Ω and is linear on each edge of ∂Ω.
12
3. Overview of FETI methods
FETI methods are iterative substructuring methods with Lagrange multipliers, which
are known as the most efficient parallel methods for large scale problems. There
are three variants of FETI methods, that is, FETI, two-level FETI and FETI-
DP(dual-primal FETI) method. FETI method has been developed into two-level
FETI method and FETI-DP method to solve more general problems efficiently and
easily.
3.1 A model problem
Let Ω be a bounded polygonal domain in R2. We consider the following elliptic
problem:
For f ∈ L2(Ω), find u ∈ H10 (Ω) such that
−∆u = f in Ω,
u = 0 on ∂Ω.(3.1)
Let Ωh be a regular triangulation of Ω. With the triangulation Ωh, we consider the
following P1-conforming finite elements
X :=v ∈ H1
0 (Ω) ∩ C0(Ω) : v|τ ∈ P1(τ) ∀τ ∈ Ωh. (3.2)
Then, the Galerkin approximation of (3.1) becomes:
Find u ∈ X such that
a(u, v) = f(v) ∀v ∈ X, (3.3)
where
a(u, v) :=
∫
Ω∇u · ∇v dx and f(v) :=
∫
Ωfv dx.
We decompose Ω into nonoverlapping subdomains Ω =⋃Ni=1 Ωi and assume that
the boundaries of each subdomain do not divide the triangles in Ωh. Hence, we obtain
13
Ω
ΩΩ
Ω
Ω Ω
ΩΩ1
2
34
1 2
3 4
Figure 3.1: Geometrically nonconforming(left) and conforming(right) partitions
the triangulation Ωhi from Ωh, that is, Ωh
i = Ωh ∩Ωi. Moreover, we assume that the
partition is geometrically conforming, which means that each subdomain intersects
with neighboring subdomains on the whole edge or at a vertex (see Figure 3.1). From
the triangulation Ωhi , we define the following finite element space in each subdomain
Ωi:
Xi :=v ∈ H1
D(Ωi) ∩ C0(Ωi) : v|τ ∈ P1(τ) ∀τ ∈ Ωh
i
,
where H1D(Ωi) is the set of Sobolev H1-functions in Ωi with zero trace value on
∂Ωi ∩ ∂Ω, that is, v|∂Ωi∩∂Ω = 0.
3.2 FETI method
Let
X :=
N∏
i=1
Xi. (3.4)
Then, for v ∈ X, the function values are not continuous across the subdomain
interfaces. FETI method was developed from the idea that the solution u of (3.3)
is obtained by solving a constraint minimization problem in X.
Let ∂Ωhi denote a set of nodes in ∂Ωi from the triangulation Ωh
i . Note that
X ⊂ X and X * H10 (Ω). Hence, we consider the following matching conditions for
14
v = (v1, · · · , vN ) ∈ X
vi(x) = vj(x) ∀x ∈ ∂Ωhi ∩ ∂Ωh
j , ∀i, j = 1, · · · , N, i 6= j. (3.5)
Since Ωhi is inherited from Ωh, we have
X =v ∈ X : v satisfies (3.5)
. (3.6)
The bilinear form a(·, ·) is s.p.d. on X. Hence, the problem (3.3) can be written
into the following minimization problem in X
J(u) = minv∈X
J(v), (3.7)
where J(v) = 12a(v, v) − f(v). For v ∈ X, we can rewrite a(v, v) and f(v) into
a(v, v) =N∑
i=1
ai(v, v), f(v) =N∑
i=1
fi(v), (3.8)
with
ai(u, v) =
∫
Ωi
∇u · ∇v dx, fi(v) =
∫
Ωi
fv dx.
Then, from (3.6), we write (3.7) into the following constraint minimization problem
in X:
J(u) = minv ∈ X,
v satisfies (3.5)
J(v). (3.9)
In the FETI method, we solve the problem (3.9) with a saddle point formulation.
Now, we define notations of matrices. Let Ki be the stiffness matrix from the
bilinear form ai(·, ·) and fi be the load vector from fi(·). Then we rewrite the
functional J(v) into the following matrix vector product form:
J(v) =1
2
N∑
i=1
vtiKivi −N∑
i=1
vtifi. (3.10)
We also rewrite the matching condition (3.5) into the following matrix vector product
form:
Bu = 0, (3.11)
15
where B =(B1 · · · BN
)and each Bi is a matrix which has -1,0 and 1 as entries,
with the number of columns equal to the number of nodes in the triangulation Ωhi
and the number of rows equal to the number of constraints in the matching condition
(3.5). Let M be the Lagrange multiplier space defined by
M = Range(B).
Then, the constraint minimization problem (3.9) can be written as the following
saddle point problem:
Find (u, λ) ∈ X ×M such that
L(u, λ) = maxµ∈M
minv∈X
L(v, µ), (3.12)
where
L(v, µ) =N∑
i=1
(1
2vtiKivi − vtifi
)+ (Bv)tµ.
Taking Euler-Lagrangian in the above saddle point problem, we get
Kiui + Btiλ = fi ∀i = 1, · · · , N, (3.13)
N∑
i=1
Biui = 0. (3.14)
For the floating subdomain Ωi, that is, ∂Ωi∩∂Ω = ∅, (3.13) becomes a full Neumann
boundary value problem and the matrix Ki has a null space. Hence, to solve (3.13)
for ui, we need the following admissible condition for λ:
fi − Btiλ ∈ Range(Ki) ∀i = 1, · · · , N.
From the fact that Ki’s are symmetric, the above condition is equivalent to
fi − Btiλ ⊥ Ker(Ki) ∀i = 1, · · · , N. (3.15)
Now, we define the admissible set
A :=µ ∈M : fi − Bt
iµ ⊥ Ker(Ki) ∀i = 1, · · · , N
16
and
V :=µ ∈M : Bt
iµ ⊥ Ker(Ki) ∀i = 1, · · · , N.
Then, taking some λ0 ∈ A, we have
A = λ0 + ν : ν ∈ V . (3.16)
The solution λ of the saddle point problem (3.12) should be in the admissible
set A, so that we consider the saddle point problem (3.12) in X × A. Let P be
a projection operator from M onto V. Then, from the relation (3.16) and taking
Euler-Lagrangian in the saddle point problem (3.12) on the set X ×A, the solution
(u, λ) ∈ X ×A of the problem (3.12) satisfies
Kiui + Btiλ = fi ∀i = 1, · · · , N, (3.17)
P tN∑
i=1
Biui = 0. (3.18)
Define K+i as a pseudo inverse of Ki and Ri as the matrix whose columns are basis
of Ker(Ki). Then the solution ui of (3.17) has the following form:
ui = K+i (fi − Bt
iλ) +Riαi, (3.19)
where αi is a vector, which will be chosen later and determine solution ui uniquely.
Substituting ui into (3.18) with P t(∑N
i=1 BiRi) = 0 and letting λ = λ0 + ν with
ν ∈ A, we obtain the following equation for ν:
Fν = d, (3.20)
where
F = P tN∑
i=1
BiK+i B
tiP, d = P t
N∑
i=1
Bi(K+i fi −K+
i Btiλ0).
We call F the FETI operator. In the FETI method, after solving for ν in (3.20) and
then substituting ν into (3.19), we obtain the solution ui’s.
Remark 3.1 The projection operator P and λ0 ∈ A can be chosen in the following
way. Let R = diagi=1,··· ,N (Ri) and G = BR. Then it can be shown that GtG is
invertible. Since λ0 ∈ A, we have
RtiBtiλ0 = Rtifi ∀i = 1, · · · , N.
17
From the above relation, λ0 satisfies
Gtλ0 = Rtf, (3.21)
with f =(f t1 · · · f tN
)t. Let λ0 = Gβ for some β and then substituting λ0 in
(3.21), we obtain
GtGβ = Rtf.
Hence, we can find λ0 ∈ A such that
λ0 = G(GtG)−1Rtf.
Moreover, we can compute the projection operator P = I − G(GtG)−1Gt. Then P
is the l2-orthogonal projection from M onto V.
Remark 3.2 After solving ν in (3.20), ui’s are computed from (3.19). In (3.19),
each αi is obtained from the condition that u = (ut1, · · · , utN )t satisfies Bu = 0.
Hence, we get
α = −(GtG)−1GtBK+(f − Btλ),
where α = (αt1, · · · , αN )t.
Since Btν ⊥ Ker(Ki) for ν ∈ V, it can be shown that F is a s.p.d. operator on
V. Hence we use the conjugate gradient method(CGM) to solve (3.20). In CGM,
the condition number of the operator F determines the reduction of relative errors
at each iteration. More precisely,
〈F (ν − νn), ν − νn〉12 ≤ C
(√κ(F ) − 1√κ(F ) + 1
)n〈F (ν − ν0), ν − ν0〉
12 ,
where 〈·, ·〉 denotes the l2-inner product, ν0 is the initial iterate, νn is the n-th iterate
of CGM and κ(F ) is the condition number of the operator F . The smaller κ(F ) is,
the faster CGM converges. Therefore, we consider a preconditioner F−1 for F to
reduce the condition number of the system
F−1/2FF−1/2ζ = F−1/2d, ν = F−1/2ζ.
The following is the preconditioned conjugate gradient method (PCGM) to solve
(3.20).
18
k = 0
ν0 is given
r0 = d− Fν0.
while (r0 6= 0)
Solve F zk = rk (zk = F−1rk)
k = k + 1
if k == 1
p1 = z0
else
βk = rtk−1zk−1/rtk−2zk−2
pk = zk−1 + βkpk−1
end
αk = rtk−1zk−1/ptkFpk
νk = νk−1 + αkpk
rk = rk−1 − αkFpk
Preconditioned Conjugate Gradient Method(PCGM)
The FETI method was first introduced by Farhat and Roux [29] for second order
elasticity problems in 2D. They observed that this method is numerically scalable
without considering a coarse space that is essentially needed for other domain de-
composition methods to achieve the scalability. It was realized that the projection
operator P plays the role of coarse solver in the FETI method.
After then, Farhat, Mandel and Roux [28] showed that the following condition
number bound for the FETI operator F for second order elasticity problems:
κ(F ) ≤ CH
h,
where H and h denote the size of subdomains and meshes, respectively and C is a
constant independent of H and h. From this bound, we can see that the condition
19
number of the FETI operator does not grow when the number of subdomains in-
creases maintaining the ratio of H and h, that is, the sizes of subdomain problems
are fixed. Hence, we can solve the problem (3.1) more accurately adding more sub-
domains with fixed bound of condition number for the operator F . This property
is called scalability. Furthermore, they introduced the Dirichlet preconditioner F−1D
such that
F−1D = PD−1P t with D−1 =
N∑
i=1
Bi
(0 0
0 Sibb
)Bti .
Here, Sibb is a Schur complement matrix which is obtained from Ki after eliminating
interior unknowns.
Mandel and Tezaur [37] analyzed that the condition number of FETI operator
with Dirichlet preconditioner is bounded by C (1 + log(H/h))m with m = 2 or 3, for
second order elliptic problems in 2D. With a different preconditioner, Klawonn and
Widlund [31] showed that the bound of the condition number is C (1 + log(H/h))2
and generalized the result for the elliptic problems with heterogeneous coefficients.
FETI method was extended to time dependent problems [24], advection-diffusion
problems [51] and plate-bending problems [39]. For plate-bending problems, the
condition number of the FETI operator with Dirichlet preconditioner grows faster
than O((1 + log(H/h))3). Since the plate-bending problems are of fourth order,
tearing the approximate solution at the cross point causes the drawback compared
with second order problems.
Farhat et al. [25, 27] introduced the two-level FETI method, a modification of
FETI method, for the fourth order problems. Adding additional Lagrange multi-
pliers, which makes the solution continuously at cross points(corners) in each CGM
iteration, to the original FETI formulation, they obtained the numerical scalability.
Tezaur [50] analyzed that the condition number bound of the two level FETI method
with Dirichlet preconditioner is C(1 + log(H/h))m with m = 2 or 3.
3.3 Dual-Primal FETI(FETI-DP) method
The dual-primal FETI(FETI-DP) method was first introduced by Farhat et al. [26]
with the similar idea to the two-level FETI method. However, the implementation
20
Ω Ω
Ω Ω
2
3 4
1
Ω Ω
ΩΩ
1 2
3 4
Figure 3.2: FETI vs. FETI-DP
is easier and the performance is better than existing FETI methods. The idea is
using primal variables at subdomain corners to match solutions directly across the
subdomain interfaces and using Lagrange multipliers to match solutions indirectly
across the remaining parts of the subdomain interfaces. Hence, the continuity of the
solutions at the subdomain corners holds for overall FETI-DP iterations.
In the FETI-DP method, we consider the following discrete space
Xc =v ∈ X : v is continuous at subdomain corners
,
where X is the space defined in (3.4). For v = (vt1, · · · , vtN )t ∈ X, we may write
vi =
(vir
vic
)for i = 1, · · · , N,
where vir and vic are vectors corresponding to the d.o.f. on the interior or edges, and
at the corners of the subdomain Ωi, respectively. For v ∈ Xc, since v is continuous
at subdomain corners and ΩiNi=1 is geometrically conforming, there exists a vector
vc such that Licvc = vic for i = 1, · · · , N where Lic is a matrix with entries 0 and
1, which restricts vc on the corners of subdomain Ωi. The vector vc has the d.o.f.
corresponding to the number of subdomain corners. To match v continuously on
the remaining parts of the subdomain interfaces, we need the following conditions:
vi(x) = vj(x) ∀x ∈ ∂Ωhi ∩ ∂Ωh
j ∩ Γ0ij , ∀i, j = 1, · · · , N, (3.22)
21
where Γ0ij is the interior part of Γij . Then we write (3.22) as
Brvr = 0, (3.23)
where Br =(B1r · · · BN
r
)and vr =
v1r...
vNr
. The matrix Bi
r has 0, 1 and -1 as
components, with the number of columns equal to the number of nodes on ∂Ωhi
excluding corners and the number of rows equal to the number of constraints in
(3.22). Then we have
X =v ∈ Xc : Brvr = 0
, (3.24)
where X is the finite element function space defined in (3.2). From (3.24), the
solution u of the problem (3.7) satisfies
J(u) = minv ∈ Xc
Brvr = 0
J(v). (3.25)
We introduce
M = Range(Br).
Then, M is equal to a space of vectors which have a d.o.f equal to the number of
constraints in (3.22).
In a saddle point formulation, (3.25) becomes: Find (u, λ) ∈ Xc ×M such that
L(u, λ) = maxµ∈M
minv∈Xc
L(v, µ), (3.26)
where
L(v, µ) =
N∑
i=1
1
2ai(v, v) − fi(v)
+ < Brvr, µ > .
Let Ki be the stiffness matrix from ai(·, ·) and fi be the load vector from fi(·). We
may assume that Ki and fi are ordered with
Ki =
(Kirr Ki
rc
Kicr Ki
cc
), fi =
(f ir
f ic
),
22
where the subscripts r and c denote the d.o.f. on interior or edges, and at corners,
respectively. Let
Krr =
Kirr
. . .
KNrr
,
Krc =
K1rcL
1c
...
KNrcL
Nc
,
Kcr = Ktrc,
Kcc =N∑
i=1
(Lic)tKi
ccLic,
ur =
u1r...
uNr
, fr =
f1r...
fNr
, fc =
N∑
i=1
(Lic)tf ic.
(3.27)
Using (3.27), L(v, µ) is written into
L(v, µ) =1
2
(vr
vc
)t(Krr Krc
Kcr Kcc
)(vr
vc
)−
(vr
vc
)t(fr
fc
)+ (Brvr)
tµ,
where vc is a vector that satisfies
Licvc = vic ∀i = 1, · · · , N
and vr = ((v1r )t, · · · , (vNr )t)t.
Taking Euler-Lagrangian in (3.26), (u, λ) satisfies
Krrur +Krcuc + Btrλ = fr, (3.28)
Kcrur +Kccuc = fc, (3.29)
Brur = 0. (3.30)
Since Krr is invertible, solving (3.28) for ur we have
ur = K−1rr (fr −Krcuc − Bt
rλ). (3.31)
23
Substituting ur into (3.30) and (3.29), we obtain
Frrλ+ Frcuc = dr, (3.32)
F trcλ− Fccuc = −dc, (3.33)
where
Frr = BrK−1rr B
tr,
Frc = BrK−1rr Krc,
Fcr = F trc,
Fcc = Kcc −KcrK−1rr Krc,
dr = BrK−1rr fr,
dc = fc −KcrK−1rr fr.
It can be shown that Fcc is invertible. Solving (3.33) for uc, we obtain
uc = F−1cc (Fcrλ+ dc). (3.34)
Then, substituting uc into (3.32), we obtain the following equation for λ:
(Frr + FrcF−1cc Fcr)λ = dr − FrcF
−1cc dc. (3.35)
We call
FDP = Frr + FrcF−1cc Fcr
a FETI-DP operator. It is shown that FDP is a s.p.d. operator. Hence, with a
suitable preconditioner, (3.35) is solved for λ using the PCGM. After solving for λ,
uc and ur are obtained from (3.34) and (3.31). As a preconditioner for the operator
FDP , we consider the following Dirichlet preconditioner:
F−1DP =
N∑
i=1
Bir
(0 0
0 Sirr
)(Bi
r)t,
where Sirr is a Schur complement operator obtained from K irr after eliminating
interior unknowns.
For second and fourth order elastostatic problems, FETI-DP method with Dirich-
let preconditioner is more robust and computationally efficient than existing FETI
24
methods, particularly when the number of subdomains is very large. In the FETI-DP
method, gluing the solution at corners, we do not have floating subdomain problems
as in the FETI methods. Hence we do not need a projection operator to eliminate
the null space of the floating subdomain problems. It was observed that F−1cc plays a
role of coarse solver in the FETI-DP method. That is, F−1cc globally transmits infor-
mation among the subdomains at each FETI-DP iteration. The bound of condition
number for the FETI-DP operator with Dirichlet preconditioner is
κ(F−1DPFDP ) ≤ C
(1 + log
H
h
)2
,
which was analyzed by Mandel and Tezaur [38] for both second and fourth order
elliptic problems.
For 3D problems, the FETI-DP method with the Dirichlet preconditioner needs
modifications to get the optimal condition number bound as in 2D problems. Farhat
et al. [26] extended the FETI-DP method to 3D problems by adding redundant
constraints to the coarse problem and obtained the numerical scalability as in 2D
problems. The Lagrange multipliers corresponding to the redundant constraints
are treated as the primal variables in the FETI-DP formulation. Hence, the coarse
problem is enlarged compared with the original FETI-DP method. They called
the FETI-DP method with redundant constraints, which are added to the coarse
problem, as the augmented FETI-DP method.
Klawonn, Widlund and Dryja [32] showed that with a different preconditioner
the condition number of the FETI-DP method is bounded by C(1 + log(H/h))2
for heterogenous coefficient elliptic problems in 3D. From the connection with the
existing substructuring iterative method for 3D problems, they showed that when
using only the d.o.f. at corners as primal variables, FETI-DP method is not effective.
They also, as Farhat et al. did in [26], added the redundant constraints to the coarse
problem and showed that FETI-DP method for 3D elliptic problems has the same
condition number bound as 2D problems. Moreover, they proposed an algorithm
choosing optimal primal variables with respect to the jumps of coefficients of elliptic
problems. The condition number bound was also shown to be C(1 + log(H/h))2 for
the case with the optimal primal variables.
25
Extensions of the FETI-DP method to the (Navier-)Stokes problem were done
by Li[34, 35] both in 2D and 3D cases. In the FETI-DP formulation, to solve
the Stokes problem more correctly and effectively at each FETI-DP iteration, the
redundant constraints are added to the coarse problem. Moreover, it is shown that
with a Dirichlet preconditioner, the bound of condition number of the FETI-DP
operator is C(1 + log(H/h))2. The Dirichlet preconditioner consists of local Stokes
problems on each subdomain.
3.4 Augmented FETI-DP method
In this section, we briefly review the augmented FETI-DP method, which was devel-
oped for solving 3D problems more efficiently. Further, we will use the augmented
FETI-DP formulation for solving the Stokes problem. In the FETI-DP formulation,
the continuity of the solution across the subdomain interfaces is enforced by the
Lagrange multipliers:
Brur = 0. (3.36)
Hence, the continuity of the solution holds when the FETI-DP iteration has con-
verged.
To accelerate the convergence of the FETI-DP method, we consider redundant
constraints
QtBrur = 0, (3.37)
where Q is some chosen matrix with a full column rank. Let N = Range(QtBr) and
M = Range(Br). Define
MQ =λ ∈M : Qtλ = 0
.
Then, introducing the Lagrange multipliers µ ∈ U and λ ∈ M for the constraints
(3.37) and (3.36), respectively, and then taking the Euler-Lagrangian to the saddle
point formulation
maxµ∈U,λ∈M
minv∈Xc
J(v)+ < QtBrvr, µ > + < Brvr, λ >
,
we obtain the followings:
26
Find (u, λ, µ) ∈ Xc ×M × U such that
Krrur +Krcuc + Btrλ+ Bt
rQµ = fr,
Kcrur +Kccuc = fc,
QtBrur = 0,
Brur = 0.
(3.38)
Let
uc =
(uc
µ
), fc =
(fc
0
),
Kcc =
(Kcc 0
0 0
),
Krc =(Krc Bt
rQ), Kcr = Kt
rc.
We consider uc as a primal variable like in the original FETI-DP formulation and
rewrite (3.38) into
Krrur + Krcuc = fr,
Kcrur + Kccuc = fc,
Brur = 0.
Then, eliminating unknowns ur and then uc, the FETI-DP operator follows:
(Frr + FrcF−1cc F
trc)λ = dr − FrcF
−1cc dc, (3.39)
where
Frc = BrK−1rr Krc,
Fcc = Kcc − KcrK−1rr Krc,
dc = fc − KcrK−1rr fr,
and the other terms are the same as those of the original FETI-DP formulation.
The invertibility of Fcc follows from the fact that Q has full column rank. Since Fcc
contains Fcc as a diagonal block, we call this method as the augmented FETI-DP
27
method. When Q = 0, the augmented FETI-DP formulation degenerates to that
of the basic FETI-DP method. Let FADP = Frr + FrcF−1cc F
trc. It can be shown that
FADP is s.p.d. on MQ. Therefore, the solution λ in (3.39) is uniquely determined in
MQ.
28
4. Mortar methods
Mortar methods were first introduced by Bernardi, Maday and Patera [11] for non-
conforming discretizations of the elliptic problems in 2D coupling finite element and
spectral methods. The methods were extended to coupling the finite elements with
nonmatching triangulations across subdomain interfaces and the spectral methods
with different orders between subdomains. In this dissertation, we consider the
mortar method for the finite elements with nonmatching triangulations across sub-
domain interfaces and call it the mortar finite element method. Nonmatching dis-
cretizations are important for multiphysics simulations, contact-impact problems,
the generation of meshes and partitions aligned with jumps in diffusion coefficients,
hp-adaptive methods, and special discretizations in the neighborhood of singularities
(corners or joints).
In the mortar methods, the orthogonality relations between jumps in the traces
across subdomain interfaces and Lagrange multipliers are imposed to obtain the
optimality of approximation like as conforming discretizations. Hence, the choice
of Lagrange multiplier space is crucial in the mortar methods. Until now, several
Lagrange multiplier spaces have been developed for the mortar finite element meth-
ods. Among them, the standard Lagrange multiplier space was naturally induced
from the finite elements on nonmortar sides. However, the basis of the mortar fi-
nite elements obtained from the standard Lagrange multiplier space are not locally
supported like as the finite element basis. In a mixed formulation of the mortar
methods, the Lagrange multipliers approximate the normal derivative of the solu-
tion. From this observation, the normal derivative of the solution can not be well
approximated by using the standard Lagrange multiplier space which consists of
continuous functions. Hence, a Lagrange multiplier space with dual basis was intro-
duced by Wohlmuth [53]. The locality of basis for mortar finite elements holds for
this type of Lagrange multiplier space. Hence, the implementation and the analysis
of the mortar methods are easier than those of the standard Lagrange multiplier
29
space.
4.1 Nonconforming approximation
In this section, we give two formulations of mortar method. They are the non-
conforming formulation and the saddle-point formulation. We consider a simple
elliptic problem (3.1). We assume that Ω is bounded polygonal(polyhedral) do-
main in Rn(n = 2, 3) and decomposed into nonoverlapping polygonal(polyhedral)
subdomains ΩiNi=1, which are geometrically conforming. Each subdomain Ωi is
associated with a quasi-uniform triangulation Ωhii with maximum diameter hi. On
the subdomain interfaces, these triangulations may not be aligned. Let
Xh :=N∏
i=1
Xi
with Xi defined in Section 3.1. Since the meshes are nonmatching across the sub-
domain boundaries, Xh is not contained in H10 (Ω). Hence, we need an appropriate
condition to find a good approximation uh in Xh for the solution u ∈ H10 (Ω) of the
problem (3.1). The mortar finite element method was developed for this purpose.
Before going into the mortar element method, we give a brief review where the idea
comes from.
In the following, we regard ‖ ·‖1,Ωi and ‖ ·‖1/2,Ωi as usual Sobolev norms without
scaling factor. Let us define
H :=N∏
i=1
H1D(Ωi)
equipped with the norm
‖v‖H
:=
(N∑
i=1
‖vi‖21,Ωi
)1/2
.
We introduce the following Sobolev space:
H0(div,Ω) =q ∈ [L2(Ω)]2 : ∇ · q ∈ L2(Ω), q · n|∂Ω = 0
normed by
‖q‖H(div,Ω) =(‖q‖2
0,Ω + ‖∇ · q‖20,Ω
)1/2.
30
Define
M =
ψ ∈ (ψi)
Ni=1 ∈
N∏
i=1
H−1/2(∂Ωi) : ∃q ∈ H0(div,Ω) such that ψi = q · ni, ∀i
normed by
‖ψ‖M = infq ∈ H0(div,Ω)
q · ni = ψ, ∀i
‖q‖H(div,Ω).
Note that H−1/2(∂Ωi) is the dual space for H1/2(∂Ωi) and H1/2(∂Ωi) is a function
space that is composed of traces of functions in H1(Ωi). We consider bilinear form
b(·, ·) : H ×M → R such that
b(v, ψ) :=N∑
i=1
∫
∂Ωi
viψi ds.
Then we can characterize H10 (Ω) as (see [43])
H10 (Ω) =
v ∈ H : b(v, ψ) = 0 ∀ψ ∈M
. (4.1)
Using the same idea as (4.1), we consider the following condition on X with suitable
Mij : ∫
Γij
(vi − vj)λij ds = 0 ∀λij ∈Mij , ∀i, j = 1, · · · , N, (4.2)
where (v1, · · · , vN ) ∈ X. The space Mij ’s will be defined later. On each interface
Γij(= Ωi ∩ Ωj), we determine one as a nonmortar side and the other as a mortar
side. Then, we define
mi := j : Ωi is the nonmortar side of Γij,
si := j : Ωi is the mortar side of Γij.(4.3)
Let
Mh =
N∏
i=1
∏
j∈mi
Mij
and a bilinear form b(·, ·) : Xh ×Mh → R
b(v, µ) :=
N∑
i=1
∑
j∈mi
∫
Γij
(vi − vj)µds. (4.4)
31
Then we define
Vh := v ∈ Xh : b(v, µ) = 0 ∀µ ∈Mh . (4.5)
Recall the definitions of a(v, v) and f(v) in (3.8). Then the nonconforming formu-
lation of the problem (3.1) becomes:
Find uh ∈ Vh satisfying
a(uh, v) = f(v) ∀v ∈ Vh. (4.6)
This formulation was first introduced by Bernardi et al. [11]. After then, considering
the mortar matching condition as constraints, the following saddle-point formulation
was introduced in [5]:
Find (uh, λh) ∈ (Xh,Mh) such that
a(uh, v) + b(v, λh) = f(v) ∀v ∈ Xh,
b(µ, uh) = 0 ∀µ ∈Mh.(4.7)
If the space Mh is suitably chosen, both of these two formulations have unique
solutions and those solutions are the same. The space Mh is associated with the tri-
angulations inherited from the nonmortar sides of interfaces. The inf-sup condition
of the space Xh×Mh is essential for the unique solvability of the formulation (4.7).
The solution λh in (4.7) approximates λ, the normal derivative of the solution u on
the subdomain interfaces. Further, if the inf-sup constant is not depending on the
mesh size and the space Mh has an approximation property like as the standard
finite elements, then the error λ − λh in (H1/200 )′-norm has the same order of ap-
proximation as the H1-norm of u − uh. The approximation order of u − uh is also
determined by the choice of Mh.
4.2 Lagrange multiplier spaces
In this section, we state the abstract multipliers conditions which will give the suit-
able Lagrange multiplier space ([30], [54]). In the following, C is a generic constant
which does not depend on the triangulations and Γij .
32
Let us define
W 0ij := w ∈ H1
0 (Γij) : w = v|Γij for v ∈ Xi. (4.8)
Then the abstract conditions for the Lagrange multiplier space Mij are
(A.1) 1 ∈Mij
(A.2) W 0ij and Mij have the same dimension.
(A.3) There is a constant C such that
‖φ‖0,Γij ≤ C supψ∈Mij
(φ, ψ)Γij‖ψ‖0,Γij
∀φ ∈W 0ij .
(A.4) For µ ∈ Hk−1/2(Γij), there exists µh ∈Mij such that
‖µ− µh‖20,Γij ≤ Ch2k−1
i |µ|2k−1/2,Γij,
where k is the order of finite elements in Xi.
Now, we define a mortar projection operator, which is essential in the analysis
of the mortar methods.
Definition 4.1 The mortar projection πij : L2(Γij) →W 0ij is defined by
∫
Γij
(w − πijw)µds = 0 ∀µ ∈Mij .
The condition (A.1) gives the coercivity of the bilinear form a(·, ·) in Vh, which
is independent of number of subdomains and mesh size. From (A.2) and (A.3), we
can see that the mortar projection is well-defined. Furthermore, from (A.2) and
(A.3), the continuity of the mortar projection in H1/200 -norm can be shown. Then,
we can see that the inf-sup constant of Xh ×Mh is independent of mesh size from
the continuity of the projection operator. Hence, both problems (4.6) and (4.7) have
unique solutions. The approximation order of the space Vh is calculated by using
the Lagrange interpolation and the continuity of the mortar projection and it is the
same as the conforming finite elements. For the error u − uh, we can obtain the
optimal order of approximation using the approximation property of the space Vh
and (A.4). For the error λ − λh, the order of approximation can be shown by the
inf-sup condition and (A.4).
33
Γ
Γij
ij
Wij
Mij
Figure 4.1: Basis functions for Wij and Mij(standard)
Now, we illustrate several Lagrange multiplier spaces that satisfy above condi-
tions (A.1)-(A.4). The standard Lagrange multiplier space was first introduced in
[11] for the elliptic problems in 2D. After then, Belgacem and Maday [9] extended
the result to 3D problems.
First, we consider a two-dimensional case. On Γij with j ∈ mi, we define
Wij := w : w = v|Γij for v ∈ Xi. (4.9)
and let
φ0, φ1, · · · , φL, φL+1
be the nodal basis functions for Wij . Moreover, we assume that the basis functions
are sequentially ordered according to the location of the nodes on Γij . From the
basis functions in Wij , Mij is defined as
Mij := spanφ0 + φ1, φ2, · · · , φL−1, φL + φL+1.
The basis for Wij and Mij are illustrated in Figure 4.1. The standard Lagrange
multiplier space is similarly defined for the higher order finite elements or three-
dimensional cases. For the case of higher order finite elements, let us assume that
each subdomain Ωi is associated with Pk-conforming finite elements. Let Tij be the
triangulation of Γij inherited from the nonmortar side of Γij . Then, Mij is defined
by
Mij := µ : µ|τ ∈ Pl(τ), if τ ∩ ∂Γij = ∅, l = k
otherwise, l = k − 1, ∀τ ∈ Tij.
34
For the three-dimensional cases, common faces are considered as the interfaces
of subdomains. Let us assume that Ωi is equipped with the P1-conforming finite
elements. The interface Γij(= ∂Ωi ∩ ∂Ωj) consists of triangulations induced from
the nonmortar side. We distinguish nodes on the interior of Γij and the boundary
of Γij . Let I and B be the sets of nodes on the interior and the boundary of Γij ,
respectively. For each a ∈ B, we assume that there exist Na(≥ 1) interior nodes
which are vertices of triangles with a as a vertex and denote them by aqNaq=1. For
each a ∈ B, we choose positive real numbers caqNaq=1 such that
∑Naq=1 c
aq = 1. Then
the Lagrange multiplier space is defined as
Mij :=
µ ∈Wij : µ =
∑
a∈I
µ(a)φa +∑
a∈B
(
Na∑
q=1
caqµ(aq))φa
,
where φa is the nodal basis function at the node a.
The standard Lagrange multiplier space Mij is contained in the finite element
space on the nonmortar side of Γij . Hence, it consist of continuous functions. From
the observation that λh approximates the normal derivative of the solution on the
interfaces, the standard space Mij is not correct one to approximate the normal
derivative because the normal flux of u may not be continuous on the interfaces
even though u ∈ H2(Ω). To overcome the discrepancy, the Lagrange multiplier
space with dual basis was developed by Wohlmuth [53]. The concept of dual basis
was first introduced in [46]. In [53], it was also shown that the Lagrange multiplier
space with the dual basis gives the same approximation property as the standard
one. Further, Kim et al. [30] generalized the result to three-dimensional problems.
First, we consider 2D case. The interface Γij is equipped with the triangulation
Tij from the nonmortar side. Let φlnl=1 be the nodal basis for W 0
ij . These basis
functions are sequentially ordered according to the location of nodes. Then, the dual
basis ψlnl=1 is defined by
∫
Γij
φlψk ds = δlk
∫
Γij
φl ds ∀l, k = 1, · · · , n,
and 1 ∈ span ψ1, · · · , ψn.
We follow [53] to give an example of dual basis. For τ ∈ Tij , let φl and φl+1
be the nodal basis functions at the end points of τ . On τ whose end points do not
35
Mij
ij WΓij
Γij
Figure 4.2: Basis functions for Wij and Mij(dual)
intersect with ∂Γij , we find ψl,τ = a1φl|τ + a2φl+1|τ and ψl+1,τ = b1φl|τ + b2φl+1|τ
such that ∫
τφmψs,τ ds = δms
∫
τφm ds for m, s = l, l + 1. (4.10)
Then, we obtain (a1, a2) = (2,−1) and (b1, b2) = (−1, 2). On τ whose one end point
intersects with ∂Γij , we let, say l, be the index of the point which does not intersect
with ∂Γij . Then, ψl,τ is given by
ψl,τ = 1,
which satisfy the condition (4.10). For φl ∈W 0ij , whose support consists of triangles
τl−1 and τl in Tij with m = 1 or 2, ψl is defined as
ψl|τi = ψl,τi for i = l − 1, l,
and zero on the remaining part of Γij . From the construction, it can be seen easily
that ψini=1 is a dual basis of φi
ni=1. The dual basis of Mij and nodal basis of Wij
are illustrated in Figure 4.2.
The dual basis can be extended to 3D problems similarly. In 3D case, the
interface Γij(= ∂Ωi ∩ ∂Ωj) consists of two-dimensional triangulations. For τ ∈ Tij ,
we label the vertices of τ by 1, 2, 3. There are four possible cases: First case is
that all of three nodes are on the interior of Γij . Second, one of them is on the
boundary of Γij . Third, two of them are on the boundary of Γij . Fourth, all vertices
are on the boundary of Γij .
36
For the first case, let φτ,l3l=1 be a nodal basis for three vertices. We want to
find ψτ,l =∑3
k=1 alkφτ,k, l = 1, 2, 3 such that
∫
τφτ,kψτ,l ds = δkl
∫
τφτ,k ds, k, l = 1, 2, 3.
Then, we obtain
(alk) =
3 −1 −1
−1 3 −1
−1 −1 3
.
For the second case, we may assume that the node with the label 3 is on the
boundary of Γij . Let ψτ,l =∑3
k=1 alkφτ,k, l = 1, 2. Then we find alk’s
(alk) =
(5/2 −3/2 1/2
−3/2 5/2 1/2
),
which satisfy
∫
τφτ,kψτ,l ds = δkl
∫
τφτ,k ds, k, l = 1, 2,
ψτ,1 + ψτ,2 = 1.
For the third case, we assume that the node with the label 1 is on the interior
of Γij . Then, we let ψτ,1 = 1.
For the fourth case, there is no interior node. Hence, we do not have an extra
ψτ,l. Instead, we find an interior node xτ , which is a vertex of a triangle τ that
shares a common edge with τ . Then, for µ ∈Mij , we let µ|τ = µ|τ .
We consider nodal basis functions φini=1 for W 0
ij and obtain ψini=1 using the
local dual basis ψτ,l similarly as is 2D case. If there is a triangle τ of the fourth
case, we modify the dual basis function ψi corresponding to the interior node xτ by
extending ψi = 1 on the triangle τ . Then, Mij is given by
M := span ψ1, · · · , ψn .
From the construction of the local dual basis, we can also see that ψlnl=1 is a dual
basis for φlnl=1.
37
There have been various extensions of the mortar element methods. To prob-
lems other than elliptic problems, such as advection-diffusion problems, Stokes prob-
lem, Maxwell equations and plate problems, the mortar element methods are also
applicable[1, 6, 7, 40]. Due to its generality and optimality, the mortar methods
have been widely used for problems with realistic importances[2, 4, 8, 18].
4.3 A priori error estimates
In this section, we provide proofs for the approximation properties of the mortar
methods. As mentioned before, using the Lagrange multipliers satisfying (A.1)-
(A.4), we can obtain the same order of approximations as conforming finite elements.
The space Vh is not a subspace of H10 (Ω). Hence, we are in a nonconforming
setting. The uniform ellipticity of the bilinear form a(·, ·) on Vh×Vh is well known for
the standard mortar space Vh; see [11]. In [10], it was shown that a(·, ·) is uniformly
elliptic on Y × Y , where
Y := v ∈N∏
i=1
H1D(Ωi) :
∫
Γij
(vi − vj) ds = 0, ∀i = 1, · · · , N, j ∈ mi.
Further, the ellipticity constant on the space Y was shown to be independent of the
number of subdomains in [47]. The approximation property of V h to H10 (Ω) was also
shown in [11]. In addition to the uniform ellipticity and the approximation property
of V h, we need to consider the consistency error to obtain a stable and convergent
finite element discretization. In the following, C is a generic constant independent
of the number of subdomains and mesh size.
Now, let us show the following stabilities of the mortar projection:
Lemma 4.2 We have
‖πijw‖0,Γij ≤ C‖w‖0,Γij ∀w ∈ L2(Γij),
and
‖πijw‖1,Γij ≤ C‖w‖1,Γij ∀w ∈ H10 (Γij).
38
Proof. First, we show the L2-stability. From (A.3), the definition of πij and Holder
inequality, we obtain
‖πijw‖0,Γij ≤ C supµ∈Mij
(πijw, µ)Γij‖µ‖0,Γij
≤ C supv∈W 0
ij
(w, µ)Γij‖µ‖0,Γij
≤ C‖w‖0,Γij .
For w ∈ H10 (Γij), there exists Qw ∈W 0
ij such that
‖w −Qw‖0,Γij ≤ Chi‖w‖1,Γij , ‖Qw‖1,Γij ≤ C‖w‖1,Γij (see Lemma 2.4).
Using the fact that πij(Qw) = Qw, the inverse inequality in (2.2) and the L2-stability
of the mortar projection, we have
‖πij(w −Qw)‖1,Γij ≤ Ch−1i ‖w −Qw‖0,Γij
≤ C‖w‖1,Γij .
Then, from the triangle inequality, the above inequality and the approximation
property of Q, we obtain
‖πijw − w‖1,Γij ≤ C‖w‖1,Γij ∀w ∈ H10 (Γij).
This completes the proof.
Remark 4.3 Using an interpolation between L2(Γij) and H10 (Γij), we have
‖πijw‖H1/200 (Γij)
≤ C‖w‖H
1/200 (Γij)
∀w ∈ H1/200 (Γij). (4.11)
This result also holds for 3D case.
For v ∈∏Ni=1H1(Ωi), let us define a broken H1-norm as
‖v‖2∗ =
N∑
i=1
‖v‖21,Ωi .
From the stability of mortar projection, we have the following approximation prop-
erty of the space Vh.
39
Lemma 4.4 Assume that v|Ωi ∈ H2(Ωi) for i = 1, · · · , N . Then we have
infvh∈Vh
‖v − vh‖2∗ ≤ C
N∑
i=1
h2i ‖v‖
22,Ωi .
Proof. Let Ihv ∈ Xh be the Lagrange interpolation of v. Take
χ = Ihv +N∑
i=1
∑
j∈mi
Eijπij [Ihv] ∈ Vh,
where [Ihv] = (Ihv)i − (Ihv)j and Eij is an extension operator from W 0ij to Xi,
which is continuous
‖Eijw‖1,Ωi ≤ C‖w‖H
1/200 (Γij)
and Eijw = 0 on ∂Ωi\Γij . The discrete harmonic extension can be such an extension
operator.
Then, we have
‖N∑
i=1
∑
j∈mi
Eijπij [Ihv]‖2
∗ ≤ CN∑
i=1
∑
j∈mi
‖πij [Ihv]‖2
H1/200 (Γij)
.
We observe that Ihv ∈ H1/200 (Γij) for 2D case, but not for 3D case. So that we
analyze each case differently.
For 2D case, using the stability of mortar projection in H1/200 -norm and coloring
argument, we obtain
‖N∑
i=1
∑
j∈mi
Eijπij [Ihv]‖2
∗ ≤ CN∑
i=1
h2i ‖v‖
22,Ωi .
For 3D case, using the inverse inequality, the stability of mortar projection in
L2-norm and coloring argument, we get
‖N∑
i=1
∑
j∈mi
Eijπij [Ihv]‖2
∗ ≤ C maxi=1,··· ,N,j∈mi
(1 + hj/hi)N∑
i=1
h2i ‖v‖
22,Ωi .
Then, using the above inequalities, approximation property of Ihv and triangle
inequality, we obtain
‖v − χ‖2∗ ≤ C
N∑
i=1
h2i ‖v‖
22,Ωi .
This completes the proof.
40
Remark 4.5 For 3D case, the constant C in the approximation property depends
on the ratio of meshes between mortar and nonmortar sides.
From the second Lemma of Strang [13], we have the following well-known result:
Lemma 4.6
‖u− uh‖∗ ≤ C
infvh∈Vh
‖u− vh‖∗ + supvh∈Vh
∑Ni=1
∑j∈mi
∫Γij
∂u∂n [vh] ds
‖vh‖∗
.
The first term is called an approximation error and the second term is called a
consistency error.
For the consistency error, we have
Lemma 4.7
supvh∈Vh
∑Ni=1
∑j∈mi
∫Γij
∂u∂n [vh] ds
‖vh‖∗≤ C
(N∑
i=1
h2i ‖u‖
22,Ωi)
)1/2
.
Proof. Since vh ∈ Vh, we have∫
Γij
∂u
∂n[vh] ds =
∫
Γij
(∂u
∂n− µh)[vh] ds ∀µh ∈Mij .
From (A.4) with k = 1 and the definition of dual norm (H1/2(Γij))′, we get
‖∂u
∂n− µh‖(H1/2(Γij))′
≤ Chi|∂u
∂n|1/2,Γij ,
where µh is chosen as the L2-projection of ∂u∂n onto Mij . It follows that
∫
Γij
∂u
∂n[vh] ds ≤ Chi|
∂u
∂n|1/2,Γij (|vh|1,Ωi + |vh|1,Ωj ).
Using the above inequality, a coloring argument and a trace theorem, we obtain
|N∑
i=1
∑
j∈mi
∫
Γij
∂u
∂n[vh] ds| ≤ C(
N∑
i=1
h2i ‖u‖2,Ωi)
1/2‖vh‖∗
and complete the proof.
From Lemma 4.6, Lemma 4.4 and Lemma 4.7, we obtain the following a priori
estimate for u− uh:
41
Theorem 4.8 Assume that u|Ωi ∈ H2(Ωi) for i = 1, · · · , N . Then, we have
‖u− uh‖∗ ≤ C
(N∑
i=1
h2i ‖u‖
22,Ωi
)1/2
.
Now, we derive a priori estimate of λ−λh with a suitable norm, where λ = ∂u∂n on
the interface of subdomains and λh is a solution of the saddle-point formulation (4.7).
Let us define a norm for µh ∈M by
‖µh‖2
(H1/200 (Γ))′
=N∑
i=1
∑
j∈mi
‖µh‖2
(H1/200 (Γij))′
The following inf-sup condition is essential in the a priori estimate of λ− λh. From
the continuity of mortar projection in H1/200 norm, we can easily obtain the following
result.
Lemma 4.9 There exists a constant β independent of mesh sizes and the number
of subdomains such that
inf06=µh∈M
sup06=vh∈Xh
b(vh, µh)
‖µh‖(H1/200 (Γ))′
a(vh, vh)1/2≥ β.
From the above result and Lemma 4.7, we have
Theorem 4.10
‖λ− λh‖2
(H1/200 (Γ))′
≤ CN∑
i=1
h2i ‖u‖
22,Ωi ,
where C depends on the inf-sup constant β.
Until now, we review the a priori error estimates of mortar methods for the
elliptic problem (3.1). For an elliptic problem with heterogeneous coefficients, we
also obtain the similar a priori error bounds by following [53]. For that case, we
obtain
‖u− uh‖2∗ ≤ C
N∑
i=1
Cih2i ‖u‖
22,Ωi ,
‖λ− λh‖2
(H1/200 (Γ))′
≤ CN∑
i=1
Cih2i ‖u‖
22,Ωi ,
42
where
Ci = max
supk∈mi
min
(1 +
aiak, 1 +
(hihk
)2), sup
1 ≤ j ≤ N
i ∈ mj
min
(1 +
ajai, 1 +
(hjhi
)2) .
Here, the constant ai is the positive coefficient of the elliptic problem in Ωi. On
Γij , if we choose nonmortar side with smaller ai, then we always have Ci ≤ 2 for all
i. For 3D case, the ratio of meshes between mortar and nonmortar sides occurs in
the constant C of the a priori estimates. We can also see that the term dose not
give significant effect when it is multiplied by Ci’s even though we choose smaller
mesh size on nonmortar side. The elliptic problems with discontinuous coefficients
can be approximated by the elliptic problems with heterogeneous coefficients. For
the problems with continuous coefficients, the assumption of comparable sizes of
meshes is reasonable in practice. Hence, the ratio of meshes can be bounded by
some number in this case.
43
5. Elliptic problems in 2D
5.1 A model problem and finite elements
Let Ω be a bounded polygonal domain in R2. We consider a FETI-DP method on
nonmatching grids for the following elliptic problem:
For f ∈ L2(Ω), find u ∈ H1(Ω) such that
−∇ · (A(x)∇u(x)) + β(x)u(x) = f(x) in Ω,
u(x) = 0 on ΓD, (5.1)
n · (A(x)∇u(x)) = 0 on ΓN .
Here, A(x) = (αij(x)) ∈ R2×2 and n is the outward unit vector normal to ΓN . We
assume that αij(x), β(x) ∈ L∞ (Ω), A(x) is uniformly elliptic, β(x) ≥ 0 for all x ∈ Ω
and |ΓD| 6= 0, where |ΓD| denotes the measure of ΓD.
Let Ω be partitioned into nonoverlapping polygonal subdomains ΩiNi=1. We
assume that the partition is geometrically conforming, which means that the sub-
domains intersect with neighboring subdomains on the whole edge or at a ver-
tex(corner). Let Ωhii be a quasi-uniform triangulation of the subdomain Ωi with
the maximum diameter hi. The meshes may not be aligned across the subdomain
interfaces. For each subdomain Ωi, we introduce a P1-conforming finite element
space
Xi := v ∈ H1D(Ωi) : v|τ ∈ P1(τ), τ ∈ Ωh
i ,
where H1D(Ωi) := v ∈ H1(Ωi) : v = 0 on ΓD ∩ ∂Ωi and P1(τ) is a set of
polynomials of degree ≤ 1 in τ . For (ui, vi) ∈ Xi ×Xi, define a bilinear form
ai(ui, vi) :=
∫
Ωi
A(x)∇ui · ∇vi dx+
∫
Ωi
β(x)uividx.
To get the FETI-DP formulation, we need a finite element space in Ω as follows:
X :=
v ∈
N∏
i=1
Xi : v is continuous at subdomain vertices
.
44
By restricting the space Xi’s on the boundaries of each subdomains, we define
Wi := Xi|∂Ωi ∀i = 1, · · · , N.
Then we let
W :=
w ∈
N∏
i=1
Wi : w is continuous at subdomain corners
. (5.2)
Let Si be the Schur complement matrix of the bilinear form ai(·, ·) over the finite
elements Xi. That is,
Si = AiBB −AiBI(AiII)
−1AiIB,
where Ai is a stiffness matrix associated with the bilinear form a(·, ·) and ordered
with
Ai =
(AiII AiIB
AiBI AiBB
).
Here, the subscripts I and B represent the d.o.f. on interior and boundary of Ωi,
respectively. Then, a semi-norm is defined for wi ∈Wi
|wi|2Si :=< Siwi, wi >,
where < ·, · > is the l2-inner product of vectors. For w ∈ W , since w is continuous
at subdomain vertices, by summing up these semi-norms, we define a norm
‖w‖2W :=
N∑
i=1
|wi|2Si , wi = w|∂Ωi . (5.3)
Moreover, we define a subspace of W
Wr := w ∈W : w vanishes at subdomain vertices . (5.4)
We note that the spaceX is not contained inH1(Ω). To approximate the solution
of the problem (5.1) in X, we impose the mortar matching condition (4.2) on v ∈ X
with a suitable Lagrange multiplier space satisfying the assumptions (A.1)-(A.4) in
Section 4.2. On each Γij , we determine mortar and nonmortar sides and define
the index sets mi and si as (4.3). We define the spaces Wij , W0ij and Mij as in
45
Section 4.2. Especially, we consider the standard Lagrange multiplier space Mij .
However, our theory can be extended to a general Lagrange multiplier space which
satisfies the assumptions (A.1)-(A.4). Then the global Lagrange multiplier space is
defined by
M :=N∏
i=1
∏
j∈mi
Mij .
Similarly, we let
W 0 :=N∏
i=1
∏
j∈mi
W 0ij .
Now, we define norms for the spaces W 0 and M . For wij ∈W 0ij , wij ∈Wi is the
zero extension of wij into ∂Ωi. Let wi =∑
j∈miwij and w = (w1, · · · , wN ). Since w
is continuous at subdomain vertices, w ∈ W . Hence, we define a norm for w ∈ W 0
as
‖w‖W 0 := ‖w‖W . (5.5)
Let < ·, · >m be a duality pairing between M and W 0 such that
< λ,w >m:=N∑
i=1
∑
j∈mi
∫
Γij
λijwij ds for (λ,w) ∈M ×W 0. (5.6)
Using this, we define a dual norm on M by
‖λ‖M := maxw∈W 0\0
< λ,w >m‖w‖W 0
. (5.7)
Recall the following mortar matching condition for (v1, · · · , vN ) ∈ X:
∫
Γij
(vi − vj)λij ds = 0 ∀λij ∈Mij , ∀i = 1, · · · , N, j ∈ mi. (5.8)
Now, we rewrite the mortar matching condition (5.8) into a matrix form. Let
φmk Km+1k=0 be the basis function for Wm|Γij with m = i, j and ψl
Ll=1 be the basis
function for Mij . Then we define matrices Biji and Bij
j with entries
(Bijm
)lk
= ±
∫
Γij
ψlφmk ds, l = 1, · · · , L, k = 0, · · · ,Km + 1 for m = i, j, (5.9)
46
where the + sign is chosen if m is the nonmortar side of Γij , otherwise, the − sign
is chosen. Then we rewrite (5.8) into
Biji vi|Γij +Bij
j vj |Γij = 0, ∀i = 1, · · · , N, j ∈ mi.
Let Eij : Mij →M be an extension operator from Mij to M by zero and Rlij : Wl →
Wl|Γij for l = i, j be a restriction operator. Using these operators, we define
Bi =∑
j∈mi
EijBiji R
iij +
∑
j∈si
EjiBjii R
iji.
Then the mortar matching condition (5.8) becomes
Bw = 0,
where B =(B1 · · · BN
)and w =
(wt1 · · · wtN
)twith wi = vi|∂Ωi .
5.2 FETI-DP formulation
5.2.1 FETI-DP operator
In this section, we construct the FETI-DP operator for the problem (5.1) with the
mortar matching condition as constraints. The derivation of FETI-DP equation for
the Lagrange multipliers follows [38]. However, the FETI-DP operator with mortar
matching condition is new. Dryja and Widlund[21, 22] eliminate unknowns both
on interior and vertex nodal points, and impose a mortar matching condition over
Wr in (5.4). Hence, the resulting solution u does not satisfy the mortar matching
condition (5.8). We only eliminate interior nodal points, and impose the mortar
matching condition on the function over W in (5.2).
For wi ∈Wi we write
wi =
(wir
wic
),
where r and c stand for the nodal values on the edges and vertices. From now on,
we use the subscript symbol r and c to represent the degrees of freedom(d.o.f.) on
edges and at vertices, respectively. Define Wc as the set of vectors which have d.o.f.
47
corresponding to the union of subdomain vertices, that is, global corner points. For
w = (w1, · · · , wN ) ∈ W , since w is continuous at subdomain vertices, there exists
wc ∈ Wc such that Licwc = wic for i = 1, · · · , N , where the matrix Lic consists of
0 and 1 and restricts the value of wc on the vertices of subdomain Ωi. Hence, for
w = (w1, · · · , wN ) ∈W , we write
wi =
(wir
Licwc
)∀i, for some wc ∈Wc.
Recall that Si is the Schur complement matrix obtained from the bilinear form
ai(·, ·) and let gi be the Schur complement forcing vector obtained from∫Ωifvi dx.
The matrix Si and vector gi are ordered into
Si =
(Sirr Sirc
Sicr Sicc
), gi =
(gir
gic
).
Let Bi,r and Bi,c be matrices that consist of the columns of Bi corresponding to the
nodal points on edges and at vertices, respectively.
Then, the saddle-point formulation of the problem (5.1) with the mortar con-
straints gives:
Find (wr, wc, λ) ∈Wr ×Wc ×M such that
Srrwr + Srcwc +Btrλ = gr, (5.10)
Scrwr + Sccwc +Btcλ = gc, (5.11)
Brwr +Bcwc = 0, (5.12)
where
Srr = diagi=1,··· ,N
(Sirr),
Src =
S1rcL
1c
...
SNrcLNc
,
Scr = Strc,
48
Scc =
N∑
i=1
(Lic)tSiccL
ic,
Br = (B1,r, · · · , BN,r) , Bc =N∑
i=1
Bi,cLic,
gr =
g1r...
gNr
, gc =
N∑
i=1
(Lic)t gic, wr =
w1r...
wNr
.
Since Srr is invertible, we solve (5.10) for wr to get
wr = S−1rr
(gr − Srcwc −Bt
rλ).
After substituting wr into (5.12) and (5.11), we obtain
BrS−1rr B
trλ+
(BrS
−1rr Src −Bc
)wc = BrS
−1rr gr,
(ScrS
−1rr B
tr −Bt
c
)λ−
(Scc − ScrS
−1rr Src
)wc = −
(gc − ScrS
−1rr gr
).
Let
FIrr = BrS−1rr B
tr,
FIrc = BrS−1rr Src −Bc,
FIcr = ScrS−1rr B
tr −Bt
c
(= F tIrc
),
FIcc = Scc − ScrS−1rr Src,
dr = BrS−1rr gr,
dc = gc − ScrS−1rr gr.
(5.13)
Then (λ,wc) satisfies(FIrr FIrc
FIcr −FIcc
)(λ
wc
)=
(dr
−dc
).
Eliminating wc in the above equation, we obtain
(FIrr + FIrcF
−1IccFIcr
)λ = dr − FIrcF
−1Iccdc. (5.14)
Here, FDP = FIrr + FIrcF−1IccFIcr is called the FETI-DP operator for the prob-
lem (5.1). In Section 5.3, it will be shown that FDP is a s.p.d. operator. Hence, the
equation (5.14) will be solved by the PCGM.
49
Remark 5.1 In the formulation by Dryja and Widlund [21, 22], the mortar con-
straints are
Brwr = 0.
Hence, letting Bc = 0 in (5.13), (5.14) gives the FETI-DP operator developed by
Dryja and Widlund.
5.2.2 Preconditioner
From now on, we find an operator FDP that gives
< FDPλ, λ >= ‖λ‖2(W 0)′ . (5.15)
Then, the operator F−1DP will be proposed as a preconditioner for FDP .
Let Eiij : W 0
ij → Wi be an extension operator by 0 and Rij : W 0 → W 0ij be a
restriction operator. We have
wij = Eiijwij for wij ∈W 0
ij ,
where wij ∈ Wi is the zero extension of wij into ∂Ωi. Then, by (5.5) and (5.3), we
get
‖w‖2W 0 =
N∑
i=1
⟨Si
∑
j∈mi
EiijRijw
,
∑
j∈mi
EiijRijw
⟩.
Let Ei =∑
j∈miEiijRij , then the above relation is written into
‖w‖2W 0 =< Sw,w > with S =
N∑
i=1
(Ei)tSiEi. (5.16)
Assume that Ωi is the nonmortar side of Γij . We recall that
(Biji )lk =
∫
Γij
ξijl φijk ds, l = 1, · · · , L, k = 0, 1, · · · ,Ki + 1.
Since Ωi is the nonmortar side of Γij , we have Ki = L. We take (Biji,r)lk = (Bij
i )lk
for l, k = 1, · · · , L and it gives
λtijBiji,rwij =
∫
Γij
λijwij ds for wij ∈W 0ij .
50
Let
B = diagi=1,··· ,N
(diagj∈mi
(Biji,r
)).
Then, the following holds for (w, λ) ∈W 0 ×M :
λtBw =
N∑
i=1
∑
j∈mi
∫
Γij
λijwij ds, (5.17)
where λij = λ|Γij and wij = w|Γij .
From the definition of the dual norm (5.7), (5.6), (5.17) and (5.16), we have
‖λ‖2M = max
w∈W 0\0
< λ, Bw >2
< Sw,w >.
Since S is symmetric and positive definite on W 0, the maximum in the above equa-
tion occurs when Btλ = Sw. This gives that
‖λ‖2M =< BS−1Btλ, λ > .
Therefore, we have
FDP = BS−1Bt.
Then we take F−1DP =
(BS−1Bt
)−1as a preconditioner for FDP and we call it a
Neumann-Dirichlet preconditioner. Since B consists of diagonal blocks Biji,r’s, which
are invertible and symmetric, we get
F−1DP =
N∑
i=1
∑
j∈mi
Rtij(Biji,r)
−1(Eiij)
t
Si
∑
j∈mi
Eiij(Biji,r)
−1Rij
.
Hence, the work for multiplying F−1DP by a vector can be done parallely in each
subdomain. Let
Bi =∑
j∈mi
Rtij(Biji,r)
−1(Eiij)
t.
Moreover, from the operator Bi, we can see that the preconditioner F−1DP is different
from the preconditioners in [21, 22, 29, 31, 38]. Only on the slave sides of interfaces,
the function values are transferred between the spaces Wi and M . Hence, the cost
needed to compute Biwi and Btiλ is reduced by half compared with other FETI(-DP)
preconditioners.
51
5.3 Condition number bound estimation
The following well-known result is given when ai(u, v) =∫Ωi
∇u·∇v dx (see Theorem
4.1.3 in [41]). With slight modification, we can obtain the similar result for a general
case.
Lemma 5.2 For wi ∈Wi, we have
C1|wi|21/2,∂Ωi
≤< Siwi, wi >≤ C2‖wi‖21/2,∂Ωi
,
where C1 and C2 are constants depending on A(x) and β(x), but not depending on
Hi and hi.
In the following, we obtain a formula that is useful to analyze the condition
number bound of the FETI-DP operator and the result is the same as Lemma 4.3
of Mandel and Tezaur [38]. However, in our formulation, the continuity constraints
are imposed on w ∈W , that is, the d.o.f. on edges and global corners; see (5.12).
Lemma 5.3 For λ ∈M , we have
maxw∈W\0
< Bw, λ >2
‖w‖2W
=< FDPλ, λ > .
Proof. We rewrite the equations (5.10)-(5.12) into
Sbwb +Btbλ = gb,
Bbwb = 0,
where
Sb =
(Srr Src
Scr Scc
), Bb =
(Br Bc
),
wb =
(wr
wc
), gb =
(gr
gc
).
Since Sb is invertible, elimination wb in the above equations, we obtain
BbS−1b Bt
bλ = BbS−1b gb,
52
which is the same as (5.14). Therefore, we have
FDP = BbS−1b Bt
b. (5.18)
For w ∈W , using the notations in Section 5.2, we write
< Bw, λ > =< Brwr +Bcwc, λ >,
‖w‖2W =
(wr
wc
)t(Srr Src
Scr Scc
)(wr
wc
).
Then, we have
maxw∈W\0
b(w, λ)2
‖w‖2W
= maxwb∈Wr×Wc\0
< Bbwb, λ >2
wtbSbwb. (5.19)
Since Sb is s.p.d. on Wr ×Wc, in the R.H.S. of (5.19) the maximum occurs when
Sbwb = Bbtλ. Hence we have
maxw∈W\0
< Bw, λ >2
‖w‖2W
=< BbS−1b Bt
bwb, λ > . (5.20)
Combining (5.20) and (5.18), we complete the proof.
Remark 5.4 Since Sb is s.p.d. on Wr ×Wc, from (5.18), we can see that FDP is
s.p.d. on M.
Now, we estimate the lower bound of the condition number for the operator
F−1DPFDP .
Lemma 5.5 For any λ ∈M , we have
maxw∈W\0
< Bw, λ >2
‖w‖2W
≥ ‖λ‖M .
Proof. For w ∈W 0, let w = (w1, · · · , wN ) be the zero extension into W . Then, it
follows that
maxw∈W\0
< Bw, λ >2
‖w‖2W
≥ maxw∈W 0\0
< Bw, λ >2
‖w‖2W
. (5.21)
53
Since wj = 0 on Γij , for j ∈ mi, we have
< Bw, λ >=N∑
i=1
∑
j∈mi
∫
Γij
wijλij ds =< λ,w >m, (5.22)
where wij = w|Γij . Combining (5.22), (5.5) and (5.7), we obtain
maxw∈W 0\0
< Bw, λ >2
‖w‖2W
= maxw∈W 0\0
< λ,w >2m
‖w‖2W 0
= ‖λ‖2M . (5.23)
From (5.21) and (5.23), we complete the proof.
To estimate the upper bound of < FDPλ, λ >, we need the following estimate
for ‖wi − wj‖2
H1/200 (Γij)
.
Lemma 5.6 For w ∈W , let wi = w|∂Ωi and wj = w|∂Ωj . Then we have
‖wi − wj‖2
H1/200 (Γij)
≤ C maxl∈i,j
(1 + log
Hl
hl
)2(
|wi|21/2,∂Ωi
+ |wj |21/2,∂Ωj
),
where C is a constant independent of hi’s and Hi’s and may depend on A(x) and
β(x).
Proof. Let IHw be a linear function on Γij that has the same value with w at the
end points of Γij . From Lemma 2.11, we have
‖wl − IHwl‖H1/200 (Γij)
≤ C
(1 + log
Hl
hl
)|wl|1/2,∂Ωl for l = i, j.
Using this, we prove the lemma.
Recall the definition of the mortar projection πij in Section 4.2 and the stability
of πij :
‖πijv‖H1/200 (Γij)
≤ C‖v‖H
1/200 (Γij)
∀v ∈ H1/200 (Γij), (5.24)
where C is a constant independent of Hi’s and hi’s. Now, we estimate the upper
bound of the operator F−1DPFDP .
Lemma 5.7 For λ ∈M , we have
maxw∈W\0
< Bw, λ >2
‖w‖2W
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2‖λ‖2
M ,
where C is a constant depending on A(x) and β(x), but independent of hi’s and
Hi’s.
54
Proof. From the definitions of the matrix B and πij in (4.1), we have
< Bw, λ >2=
N∑
i=1
∑
j∈mi
∫
Γij
πij(wi − wj)λij ds
2
.
We let z ∈ W 0 be such that z|Γij = πij(wi − wj). Then the above equation is the
duality pairing between λ and z. Hence, using the definition of dual norm on λ, we
get
< Bw, λ >2≤ ‖λ‖2M‖z‖2
W 0 . (5.25)
Let z = (z1, · · · , zN ) ∈ W be the zero extension of z. Then, from (5.5), (5.3),
Lemma 5.2, (2.1), (5.24) and Lemma 5.6,
‖z‖2W 0 =
N∑
i=1
< Sizi, zi >
≤ CN∑
i=1
‖zi‖21/2,∂Ωi
≤ CN∑
i=1
∑
j∈mi
‖πij(wi − wj)‖2
H1/200 (Γij)
≤ CN∑
i=1
∑
j∈mi
‖wi − wj‖2
H1/200 (Γij)
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2
N∑
i=1
|wi|21/2,∂Ωi
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2‖w‖2
W .
(5.26)
Here, C denotes a generic constant independent of hi’s and Hi’s, which may vary
from occurrence and occurrence. Combining (5.25) and (5.26), we complete the
proof.
Since the preconditioner F−1DP follows from the dual norm of λ ∈M (see (5.15)),
combining Lemma 5.3, Lemma 5.5 and Lemma 5.7, we obtain the following estimate.
55
Theorem 5.8 For λ ∈M , we have
< FDPλ, λ >≤< FDPλ, λ >≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2< FDPλ, λ >,
where C is a constant depending on A(x) and β(x), but independent of Hi’s and
hi’s.
Corollary 5.9 We have the condition number estimate
κ(F−1DPFDP
)≤ C max
i=1,··· ,N
(1 + log
Hi
hi
)2,
where C is a constant depending on A(x) and β(x), but independent of Hi’s and
hi’s.
Remark 5.10 On each Γij, the choice of master and slave side is arbitrary.
Remark 5.11 In Corollary 5.9, the condition number depends on A(x) and β(x).
Now we consider a problem:
−∇ · (α(x)∇u(x)) = f(x) in Ω,
u = 0 on ∂Ω,
where α(x) is a piecewise constant and has jumps across the subdomain boundaries,
i.e., α(x) = ρi for all x ∈ Ωi for some constant ρi > 0. On Γij, we choose Ωhi |Γij as
the slave side if ρi ≤ ρj. Otherwise, we choose Ωhi |Γij as the master side. Then we
have
C1ρi|wi|21/2,∂Ωi
≤< Siwi, wi >≤ C2ρi‖wi‖21/2,∂Ωi
,
where C1 and C2 are constants independent of ρi’s, hi’s and Hi’s. Following the proof
56
of Lemma 5.7 and using the above inequalities instead of Lemma 5.2, we obtain
‖z‖2W 0 ≤ C
N∑
i=1
∑
j∈mi
ρi‖wi − wj‖2
H1/200 (Γij)
≤ CN∑
i=1
∑
j∈mi
maxl∈i,j
(1 + log
Hl
hl
)2
×(ρi|wi|
21/2,∂Ωi
+ ρi|wj |21/2,∂Ωj
)
≤ C
N∑
i=1
∑
j∈mi
maxl∈i,j
(1 + log
Hl
hl
)2
×
(< Siwi, wi > +
ρiρj
< Sjwj , wj >
),
where C is a generic constant independent of ρi’s, Hi’s and hi’s. Since ρi ≤ ρj, we
can see that the constant C in Lemma 5.7 is bounded independently of the coeffi-
cients. Hence, the condition number bound is independent of ρi’s.
57
6. Elliptic problems in 3D
6.1 A model problem and finite elements
Let Ω be a bounded polyhedral domain in R3. We consider the same elliptic prob-
lem (5.1) with A(x) ∈ R3×3. The domain Ω is partitioned into nonoverlapping
polyhedral subdomains ΩiNi=1, which are geometrically conforming. This means
that each subdomain intersects with neighboring subdomains on the whole face,
whole edge or at a vertex. Among them, we call faces the interfaces of subdomains
and use Γij to denote the interface of subdomain Ωi and Ωj . Let Ωhii be a quasi-
uniform triangulation of Ωi with the maximum diameter hi. These meshes may not
be aligned across the subdomain interfaces.
For each subdomain Ωi, we introduce a finite element space Xi, Wi, X and W
as in 2D case in Chapter 5. We define a bilinear form
ai(ui, vi) :=
∫
Ωi
A(x)∇ui · ∇vi dx+
∫
Ωi
β(x)uividx.
and let Si be the Schur complement matrix obtained from the bilinear form ai(·, ·)
over the finite elements Xi. Using this operator, a semi-norm is defined for wi ∈Wi:
|wi|2Si :=< Siwi, wi >,
where < ·, · > is the l2-inner product of vectors. For w ∈ W , since w is continuous
at subdomain vertices, by summing up these semi-norms, we define a norm
‖w‖2W :=
N∑
i=1
|wi|2Si , wi = w|∂Ωi . (6.1)
On each Γij , we determine a nonmortar side and a mortar side and define the
index sets mi and si; see (4.3). The spaces Wij , W0ij and Mij are defined as in
Section 4.2. For 3D case, examples of Mij , which satisfy the assumptions (A.1)-
58
(A.4), are given in Section 4.2. Then, we define
W 0 :=
N∏
i=1
∏
j∈mi
W 0ij , (6.2)
M :=N∏
i=1
∏
j∈mi
M0ij . (6.3)
The space W 0 is equipped with the norm ‖ · ‖W 0
‖w‖W 0 := ‖w‖W , (6.4)
where w ∈W is the zero extension of w.
Using the Lagrange multiplier space M , we impose the mortar matching condi-
tion (5.8) on the space X. It is already known from the numerical results in [26]
that using the primal variables at corners is not enough to get the same condition
number bound as 2D problems. Hence, we add redundant continuity constraints to
the coarse problem and follow the augmented FETI-DP formulation. The redundant
constraints are∫
Γij
vi ds =
∫
Γij
vj ds ∀i = 1, · · · , N, j ∈ mi. (6.5)
That is, the averages of functions are the same across the common face Γij . Since
1 ∈Mij , the above constraints are redundant to the mortar constraints (5.8). Then,
those constraints are written into the following algebraic equations:
Bw = 0
RtBw = 0,
where the matrix B is defined similarly as 2D case and R is a matrix that gives the
redundant constraints. More precisely, Rtλ = 0 means that sum of λ|Γij is zero for
each Γij and R has 0 or 1 as entries.
For the 3D elliptic problems with conforming discretizations, Klawonn et al. [32]
developed FETI-DP methods with various redundant constraints. They showed
that the method is not competitive when only using the primal variables at corners.
Additional continuity constraints on edges or on faces are needed to obtain the same
59
condition number bound as 2D elliptic problems. The continuity constraints on an
edge is that the averages of functions across the common edge are the same. The
same is applied to a face also. From their results, it seems that the continuity
constraints on edges are essential. Further, in [33], they extended the results to
the case with face constraints only. In mortar context, the constraints on edges
are not redundant to the mortar matching condition. We will only impose the face
constraints as the redundant constraints. This is a different feature of our method
from that of Klawonn et al. [32].
6.2 FETI-DP formulation
6.2.1 FETI-DP operator
In 3D, we have a face, an edge or a vertex as an intersection of subdomains. Hence,
we use the symbol r to represent the d.o.f. on faces and edges and c to represent the
d.o.f. at corners(vertices). Then, we write
wi =
(wir
wic
)for wi ∈Wi
and define wc and wr for w ∈ W as in Section 5.2.1. The spaces Wr and Wc
consist of the vectors wr and wc, respectively. Let U be a Lagrange multiplier space
corresponding to the redundant constraints (6.5). We use that same notations of
matrices and vectors as in Section 5.2.1 except that the symbol r represents the
d.o.f. on faces and edges. Then, we have the following saddle point formulation of
the problem (5.1):
Find (wr, wc, µ, λ) ∈Wr ×Wc × U ×M satisfying
Srrwr + Srcwc +BtrRµ+Bt
rλ = gr,
Scrwr + Sccwc +BtcRµ+Bt
cλ = gc,
RtBrwr +RtBcwc = 0,
Brwr +Bcwc = 0.
(6.6)
60
In the above equations, we regard wc =
(wc
µ
)as the primal variables in the FETI-
DP formulation and follow the augmented FETI-DP formulation introduced in Sec-
tion 3.4. Let
Krr = Srr,
Krc =(Src Bt
rR), Kcr = Kt
rc,
Kcc =
(Scc Bt
cR
RtBc 0
),
Bc =(Bc 0
), gc =
(gc
0
).
Then we have
Krr Krc Btr
Kcr Kcc Btc
Br Bc 0
wr
wc
λ
=
gr
gc
0
. (6.7)
Since Krr is invertible, after eliminating wr in (6.7), we obtain
(−Fcc Fcl
F tcl Fll
)(wc
λ
)=
(−dc
dl
),
where
Fcc = Kcc −KcrK−1rr Kcr,
Flc = BrK−1rr Krc − Bt
c, Fcl = F tlc,
Fll = BrK−1rr B
tr
dl = BrS−1rr gr, dc = gc −KcrK
−1rr gr.
From the fact that BtcR has a full column rank, we can show that Fcc is invertible.
Hence, eliminating wc in the above equation, the FETI-DP equation of (6.6) follows
FDPλ = dl − F tclF−1cc dc, (6.8)
with FDP = Fll + F tclF−1cc Fcl. we call FDP the FETI-DP operator. Since, we added
61
the redundant mortar matching constraints to the FETI-DP formulation, the solu-
tion of FETI-DP equation is not uniquely determined in M . Let us define
MR :=λ ∈M : Rtλ = 0
. (6.9)
In Section 6.3, we will show that FDP is s.p.d. on MR. Hence, the solution λ ∈MR
is uniquely determined.
6.2.2 Preconditioner
Since FDP is s.p.d. on MR, we will solve (6.8) by preconditioned conjugate gradient
method using a suitable preconditioner. We derive a preconditioner from the similar
idea with 2D case.
Let us define the following subspaces equipped with the norms induced from W
and W 0:
WR :=w ∈W : RtBw = 0
, (6.10)
W 0R :=
w ∈W 0 : RtBw = 0
, (6.11)
where w is the zero extension of w into the space W . Recall the definition of the
space MR in (6.9). A duality pairing between the spaces MR and W 0R is defined as
< λ,w >m=N∑
i=1
∑
j∈mi
∫
Γij
λijwij ds. (6.12)
Then, a dual norm on λ ∈MR is given by
‖λ‖MR:= max
w∈W 0R
< λ,w >m‖w‖W 0
. (6.13)
Similarly to the 2D problems, we will find an operator FDP which gives
< FDPλ, λ >= ‖λ‖2MR
(6.14)
and propose F−1DP as a preconditioner for the operator FDP .
Now, we derive a matrix form of the operator F−1DP . Since the dual norm is defined
on the subspaces MR and W 0R, we need the following l2-orthogonal projections:
P ijW 0R
: W 0|Γij →W 0R|Γij ,
P ijMR: M |Γij →MR|Γij .
62
From the above projection operators, the l2-orthogonal projections PW 0R
: W →WR
and PMR: M →MR are obtained
PW 0R
= diagNi=1diagj∈mi(PijW 0R),
PMR= diagNi=1diagj∈mi(P
ijMR
).
We recall the following restriction and extension
Rij : W 0 →W 0ij ,
Eiij : W 0ij →Wi.
and the matrices Biji and Bij
j in (5.9). We obtain the matrices Biji,r from Bij
i after
deleting columns corresponding to the d.o.f. on the boundary of Γij . Let
S =N∑
i=1
(∑
j∈mi
EiijRij)Si(∑
j∈mi
EiijRij)t,
B = diagNi=1diagj∈mi(Biji ).
Then we have
‖w‖2W 0 =< Spw,w > for w ∈W 0
R,
< λ,w >m = λtBpw for λ ∈MR, w ∈W 0R,
where
Sp = P tW 0RSPW 0 ,
Bp = P tMRBPW 0
R.
It can be shown that Sp and Bp are invertible on W 0R and Bt
p is invertible on MR.
Hence, the maximum in (6.13) occurs when Spw = Btpλ and this gives
< BpS−1p Bt
pλ, λ >= ‖λ‖2MR
for λ ∈MR.
63
As a result, we have FDP = BpS−1p Bt
p. From the observation that Bp consists of
invertible block matrices Bijp = (P ijMR
)tBiji,rP
ijW 0R, we get
F−1DP =
N∑
i=1
(∑
j∈mi
Eiij(Bijp )−1Rij)
tSi(∑
j∈mi
Eiij(Bijp )−1Rij). (6.15)
Hence, the computation of F−1DPλ can be done parallely in each subdomain.
6.3 Condition number bound estimation
We have the following result as in 2D problems.
Lemma 6.1 For wi ∈Wi, we have
C1|wi|21/2,∂Ωi
≤< Siwi, wi >≤ C2‖wi‖21/2,∂Ωi
,
where C1 and C2 are constants depending on A(x) and β(x), but independent of Hi
and hi.
Lemma 6.2 For λ ∈MR, we have
maxw∈WR\0
< Bw, λ >2
‖w‖2W
=< FDPλ, λ > .
Proof. The saddle-point problem (6.7) is equivalent to solving the following prob-
lem
maxλ∈B(WR)
minw∈WR
(1
2wtSw + wtg + λtBw
),
where g is a vector composed of the vectors gr and gc in (6.6). It can be shown
easily that B(WR) = MR. We recall the l2-orthogonal projections PMR: M → MR
and PWR: W →WR. Then, taking Euler-Lagrangian in the above problem, we get
Spw +Btpλ = P tWR
g,
Bpw = 0,
where
Sp = P tWRSPWR
,
Bp = P tMRBPWR
.
64
We can see that Sp is s.p.d. on WR. Hence, eliminating w in the above equations,
we obtain
BpS−1p Bt
pλ = d,
where d = BpS−1p P tWR
g. Since this equation is obtained from the same problem with
(6.7), we have
FDP = BpS−1p Bt
p. (6.16)
Using the identity
‖w‖2W =< Sw,w >
and the projections PWRand PMR
, we can see that
maxw∈WR\0
< Bw, λ >2
‖w‖2W
=< BpS−1p Bt
pλ, λ > for λ ∈MR. (6.17)
From (6.16) and (6.17), we prove the lemma.
Remark 6.3 For λ ∈ MR, Btpλ = 0 gives λ = 0 and Sp is s.p.d. on WR. Hence,
from (6.16), we can see that FDP is s.p.d. on MR.
Now, we estimate the lower bound of the operator FDP .
Lemma 6.4 For any λ ∈MR, we have
maxw∈WR\0
< Bw, λ >2
‖w‖2W
≥ ‖λ‖2MR
.
Proof. Let w ∈ W be the zero extension of w ∈ W 0R. Then, we can see that
w ∈WR. Using the definitions of ‖λ‖MR, ‖w‖W 0 and < λ,w >m, we get
‖λ‖2MR
= maxw∈W 0
R\0
< λ,w >2m
‖w‖2W 0
= maxw∈W 0
R\0
< Bw, λ >2
‖w‖2W
≤ maxw∈WR\0
< Bw, λ >2
‖w‖2W
.
This completes the proof.
65
To estimate the upper bound of the operator FDP , we define an interpolation
Ii0wi ∈Wi by
(I i0wi)(x) =
wi(x), x ∈ ∂F ∩ ∂Ωh
i ,
CF , x ∈ F ∩ ∂Ωhi ,
where ∂Ωhi is the set of nodes on the boundary of Ωi and CF is an average of wi on
the face F ⊂ ∂Ωi, that is,
CF =
∫F wi ds∫F ds
.
Note that faces and edges are open sets which do not include their boundaries. In
the following, C is a generic constant which does not depend on the mesh size or the
number of subdomains and may depend on A(x) or β(x). Recall the definition of
norms ‖ · ‖H
1/200 (F )
and ‖ · ‖1/2,∂Ωi in Section 2.1. Using the definition of CF , Holder
inequality and the definition of ‖ · ‖1/2,∂Ωi , we obtain
|CF | ≤ CH−1/2i ‖wi‖1/2,∂Ωi . (6.18)
For a set A ⊂ ∂Ωi, IhAwi denotes a nodal value interpolation of wi on the set A. The
interpolation I i0wi has the following approximation properties.
Lemma 6.5 For wi ∈Wi, we have
‖wi − I i0wi‖H1/200 (F )
≤ C
(1 + log
Hi
hi
)|wi|1/2,∂Ωi , (6.19)
‖I i0wi − CF ‖0,F ≤ Ch1/2i
(1 + log
Hi
hi
)1/2
|wi|1/2,∂Ωi . (6.20)
Proof. First, we consider
‖wi − I i0wi‖H1/200 (F )
= ‖IhFwi − IhFCF ‖H1/200 (F )
≤ ‖IhFwi‖H1/200 (F )
+ |CF |‖IhF 1‖
H1/200 (F )
.
Then from the Lemma 2.9, Lemma 2.10 and (6.18), we get
‖wi − I i0wi‖H1/200 (F )
≤ C
(1 + log
Hi
hi
)‖wi‖1/2,∂Ωi .
Since wi−Ii0wi is invariant to a constant addition, we can replace the norm ‖·‖1/2,∂Ωi
by the semi-norm | · |1/2,∂Ωi .
66
Now, we consider the second estimate. From the definition of I i0wi and the
quasi-uniform assumption on the triangulation, we get
‖I i0wi − CF ‖0,F = ‖Ih∂F (wi − CF )‖0,F
≤ Ch1/2i ‖Ih∂F (wi − CF )‖0,∂F
≤ Ch1/2i
∑
E⊂∂F
‖IhE(wi − CF )‖0,E
≤ Ch1/2i
(∑
E⊂∂F
‖wi‖0,E +∑
E⊂∂F
‖CF ‖0,E
),
where E is a closed edge on ∂F . Using the Lemma 2.8, we have
‖wi‖0,E ≤ C
(1 + log
Hi
hi
)1/2
‖wi‖1/2,∂Ωi , (6.21)
and
‖CF ‖0,E ≤ |E|1/2|CF |.
From (6.18) and |E| ≤ CHi, it follows that
‖CF ‖0,E ≤ C‖wi‖1/2,∂Ωi . (6.22)
From (6.21), (6.22) and the invariance of I i0wi − CF to the constant addition, we
complete the proof of (6.20).
Using the above estimates, we have the following result similarly to the 2D case.
Lemma 6.6 For w ∈WR, we have
‖πij(wi−wj)‖H1/200 (Γij)
≤ Cmaxi,j
(1 + log
Hl
hl
)(|wi|1/2,∂Ωi +
(hjhi
)1/2
|wj |1/2,∂Ωj
).
Proof. Using the interpolations I i0wi and Ij0wj , the inverse inequality (2.3) and
the continuity of πij , we get
‖πij(wi − wj)‖H1/200 (Γij)
≤ ‖πij(wi − I i0wi)‖H1/200 (Γij)
+ ‖πij(wj − Ij0wj)‖H1/200 (Γij)
+ ‖πij(Ii0wi − Ij0wj)‖H1/2
00 (Γij)
≤ ‖wi − I i0wi‖H1/200 (Γij)
+ ‖wj − Ij0wj‖H1/200 (Γij)
+ Ch−1/2i ‖I i0wi − Ij0wj‖0,Γij .
67
Since w ∈WR, we have the same CF for wi and wj on F (= Γij). Then, we have
‖I i0wi − Ij0wj‖0,Γij ≤ ‖I i0wi − CF ‖0,Γij + ‖Ij0wj − CF ‖0,Γij .
From the above equation and the approximation properties of I i0wi in Lemma 6.5,
we obtain
‖πij(wi−wj)‖H1/200 (Γij)
≤ C
((1 + log
Hi
hi)|wi|1/2,∂Ωi + (
hjhi
)1/2(1 + logHj
hj)|wj |1/2,∂Ωj
)
and complete the proof.
Now, we estimate the upper bound of the operator FDP . Let us define
ri = maxj∈mi
1 +
hjhi
for i = 1, · · · , N.
Lemma 6.7 For λ ∈MR, we have
maxw∈WR\0
< Bw, λ >2
‖w‖2W
≤ C maxi=1,··· ,N
ri
(1 + log
Hi
hi
)2‖λ‖2
MR,
where C is a constant depending on A(x) and β(x), but independent of hi’s and
Hi’s.
Proof. From the definitions of B and πij , we have
< Bw, λ >2=
N∑
i=1
∑
j∈mi
∫
Γij
πij(wi − wj)λij ds
2
.
We consider z ∈ W 0 such that z|Γij = πij(wi − wj). Since w ∈ WR, we can see
that z ∈ W 0R. Then the above equation is the duality pairing between λ ∈ MR and
z ∈W 0R. Hence, using the definition of dual norm on λ, we get
< Bw, λ >2≤ ‖λ‖2MR
‖z‖2W 0 . (6.23)
68
Let z = (z1, · · · , zN ) ∈ W be the zero extension of z. Then from (6.4), (6.1),
Lemma 6.1, (2.1), (4.11) and Lemma 6.6,
‖z‖2W 0 =
N∑
i=1
< Sizi, zi >
≤ CN∑
i=1
‖zi‖21/2,∂Ωi
≤ CN∑
i=1
∑
j∈mi
‖πij(wi − wj)‖2
H1/200 (Γij)
≤ CN∑
i=1
∑
j∈mi
((1 + log
Hi
hi)2|wi|
21/2,∂Ωi
+hjhi
(1 + logHj
hj)2|wj |
21/2,∂Ωj
)
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2
ri
‖w‖2
W .
(6.24)
Here, C denotes a generic constant independent of hi’s and Hi’s, which may vary
from occurrence and occurrence. Combining (6.23) and (6.24), we complete the
proof.
Remark 6.8 When the coefficients A(x) and β(x) do not change rapidly across
subdomain interfaces, it is appropriate to use triangulations which have similar mesh
sizes between neighboring subdomains. Hence, in this case, we may assume that ri
is bounded independent of the mesh sizes.
Now, we consider the following elliptic problem with discontinuous constant co-
efficients:
−∇ · (α(x)∇u) = f in Ω,
u = 0 on ∂Ω,(6.25)
with α(x)|Ωi = ρi(> 0) for all i = 1, · · · , N . Then, we have the similar estimates to
Lemma 6.1
C1ρi|wi|1/2,∂Ωi ≤< Siwi, wi >≤ C2ρi‖wi‖1/2,∂Ωi ,
69
where C1 and C2 are constants not depending on ρi, hi and Hi. Using the above
bound, we follow the proofs of Lemma 6.7 and obtain
‖z‖2W 0 ≤ C
N∑
i=1
∑
j∈mi
((1 + log
Hi
hi
)2
|wi|21/2,∂Ωi
+hjhi
ρiρj
(1 + log
Hj
hj
)2
|wj |21/2,∂Ωj
),
(6.26)
where C is a constant independent of ρi’s, hi’s and Hi’s. For the same elliptic
problem in 2D, Wohlmuth [52] observed that the ratio hihj
tends to become(ρiρj
)1/4
as an adaptivity strategy is applied successively. In this stage, we make a reasonable
assumption on the ratio of meshes for 3D problems.
Assumption on meshes: For each Γij , we assume that
hjhi
≤ C
(ρjρi
)γ, with 0 ≤ γ ≤ 1, (6.27)
where C is a constant independent of hi’s, ρi’s and Hi’s.
On Γij , if we choose Ωi with smaller ρi as a slave side, then from the above
assumption and (6.26) we get
‖z‖2W 0 ≤C
N∑
i=1
∑
j∈mi
((1 + log
Hi
hi
)2
|wi|21/2,∂Ωi
+
(ρiρj
)1−γ (1 + log
Hj
hj
)2
|wj |21/2,∂Ωj
),
where C is a constant independent of ρi’s, hi’s and Hi’s. Since the slave side has
smaller ρi’s, in the above equation(ρiρj
)1−γ≤ 1. Therefore, we obtain the following
result.
Lemma 6.9 With the assumption (6.27) on meshes, for the elliptic problem (6.25)
we have
maxw∈WR\0
< Bw, λ >2
‖w‖2W
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2‖λ‖2
MR,
where C is a constant independent of ρi’s, hi’s and Hi’s.
70
Remark 6.10 The result is the same as 2D case. However, we need an additional
assumption on the ratio of meshes for 3D problems.
Now, we restrict ourselves to the elliptic problems with coefficients A(x) and
β(x) that do not change rapidly across subdomain interfaces or with discontinu-
ous coefficients ρi’s. From Remark 6.8 and Lemma 6.9, we can see that the term
ri disappears on the condition number bound for those cases. From Lemma 6.2,
Lemma 6.4, Lemma 6.7 and Lemma 6.9, we have the following result.
Theorem 6.11 Assume that the elliptic problem has coefficients A(x) and β(x)
which do not change rapidly across subdomain interfaces or the elliptic problem has
discontinuous coefficients ρi’s. Then, for λ ∈MR,
< FDPλ, λ >≤< FDPλ, λ >≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2< FDPλ, λ >,
where C is a constant depending on A(x) and β(x), but independent of Hi’s and
hi’s. For the elliptic problems with discontinuous coefficients ρi’s, the constant C is
independent of the coefficients.
From (6.14) and the above theorem, we obtain the condition number bound:
Corollary 6.12 Under the assumption of Theorem 6.11, we have
κ(F−1DPFDP ) ≤ C max
i=1,··· ,N
(1 + log
Hi
hi
)2,
where the constant C is the same as one in the above theorem.
71
7. Stokes problem in 2D
In this chapter, we consider a FETI-DP formulation of the Stokes problem with
mortar methods. Under the conforming discretizations, Li [34, 35] extended the
FETI-DP methods to the Stokes problem and linearized Navier-Stokes problem both
in 2D and 3D. The analysis of the mortar methods for the Stokes problem was done
by Belgacem [6].
7.1 A model problem and finite elements
Let Ω be a bounded polygonal domain in R2. In the following, we consider the
Stokes problem: For f ∈ [L2(Ω)]2, find (u, p) ∈ [H10 (Ω)]2 × L2
0(Ω) satisfying
−4u + ∇p = f in Ω,
−∇ · u = 0 in Ω,
u = 0 on ∂Ω.
(7.1)
We assume that Ω is partitioned into nonoverlapping bounded polygonal subdomains
ΩiNi=1 and the partition is geometrically conforming. For each subdomain, we
introduce the following Sobolev spaces:
H1D(Ωi) :=
v ∈ H1(Ωi) : v = 0 on ∂Ωi ∩ ∂Ω
,
L20(Ωi) :=
q ∈ L2(Ωi) :
∫
Ωi
q dx = 0
,
Π0 :=q0 ∈ L2
0(Ω) : q0|Ωi is a constant for each i.
Then, the variational form of the Stokes problem (7.1) is:
72
Find(u, pI , p
0)∈∏Ni=1
[H1D(Ωi)
]2×∏Ni=1 L
20(Ωi) × Π0 such that
N∑
i=1
(∇u,∇v)Ωi −N∑
i=1
(pI + p0,∇ · v)Ωi =
N∑
i=1
(f ,v)Ωi ∀ v ∈N∏
i=1
[H1D(Ωi)
]2,
−N∑
i=1
(∇ · u, qI)Ωi = 0 ∀ qI ∈N∏
i=1
L20(Ωi),
−N∑
i=1
(∇ · u, q0)Ωi = 0 ∀ q0 ∈ Π0,
(7.2)
and the velocity u is continuous across the subdomain interfaces Γ =⋃Ni,j=1(∂Ωi ∩
∂Ωj). Here, (·, ·)Ωi denotes the inner product in [L2(Ωi)]n for n = 1 or 2.
For each subdomain Ωi, we consider a quasi-uniform triangulation Ω2hii with the
maximum diameter 2hi. After bisecting each edge of triangles in Ω2hii , we obtain
a finer triangulation Ωhii from Ω2hi
i . Note that these triangulations need not match
across the subdomain interfaces. From these triangulations, we consider the inf-sup
stable P1(hi) − P0(2hi) finite elements in each subdomain Ωi and let
Xi :=vi ∈
[H1D(Ωi) ∩ C
0(Ωi)]2
: vi|τ ∈ [P1(τ)]2 ∀ τ ∈ Ωhi
i
,
Qi :=qi ∈ L2(Ωi) : qi|τ ∈ P0(τ) ∀ τ ∈ Ω2hi
i
,
Q0i := Qi ∩ L
20(Ωi),
where Pl(τ) is a set of polynomials of degree ≤ l in τ .
To get a FETI-DP formulation, we define the following spaces:
X :=
v ∈
N∏
i=1
Xi : v is continuous at subdomain corners
,
Q :=
N∏
i=1
Q0i ,
Wi := Xi|∂Ωi for i = 1, · · · , N,
W =
w ∈
N∏
i=1
Wi : w is continuous at subdomain corners
.
73
For v = (vt1, · · · ,vtN )t ∈ X, we write
vi =
viI
vir
vic
,
where the symbol I, r and c represent the d.o.f. on interior, on edges and at cor-
ners(vertices), respectively. Since v is continuous at subdomain corners, there exists
a vector vc satisfying vic = Licvc for all i = 1, · · · , N , with a restriction map Lic. The
vector vc has the d.o.f. corresponding to the union of subdomain corners. Let
vtI =((v1I)t · · · (vNI )t
),vtr =
((v1r)t · · · (vNr )t
).
We define the spaces XI ,Wr and Wc which consist of vectors vI , vr and vc, respec-
tively. Similarly, for w ∈W , we define wr ∈Wr and wc ∈Wc.
Note that the space X is not contained in [H10 (Ω)]2. To approximate the solution
of the problem (7.1) in the space X, we impose the mortar matching condition on
the velocity functions. Let Γij = ∂Ωi ∩ ∂Ωj . Since the triangulations are different
across Γij , we distinguish them by choosing one as a mortar side and the other as a
nonmortar side. Then the index sets mi and si are defined as (4.3). We may write
∂Ωi \ ∂Ω = (⋃
j∈mi
Γij)⋃
(⋃
j∈si
Γij).
Now, we define the following spaces from the finite elements on the nonmortar sides
of interfaces:
Wij := Wi|Γij for j ∈ mi, i = 1, · · · , N,
W 0ij := wij ∈Wij : wij vanishes at the end points of Γij ,
W 0 :=N∏
i=1
∏
j∈mi
W 0ij
and consider the Lagrange multiplier space Mij introduced in Section 4.2. More
precisely, the standard Lagrange multiplier space Mij is defined as
Mij := ψ ∈[C0(Γij)
]2: ψ|τ ∈ [Pl(τ)]
2, if τ ∩ ∂Γij = ∅, l = 1,
otherwise l = 0, ∀τ ∈ Tij,
74
where Tij is a triangulation on Γij inherited from the nonmortar side of Γij . Then
we take the Lagrange multiplier space
M :=N∏
i=1
∏
j∈mi
Mij
and impose the following mortar matching condition on the velocity functions:
For v = (v1, · · · ,vN ) ∈ X, v satisfies that∫
Γij
(vi − vj) · λij ds = 0 ∀λij ∈Mij , ∀ i = 1, · · · , N, ∀j ∈ mi. (7.3)
Let us define the spaces
V := v ∈ X : v satisfies (7.3) ,
P :=q ∈ L2
0(Ω) : q|Ωi ∈ Qi ∀ i = 1, · · · , N
for the velocity and pressure, respectively. The space P is written into a direct sum
of the L2-orthogonal spaces Q and Π0, that is,
P = Q⊕ Π0.
When Hood-Taylor finite elements P2(h) − P1(h) are used for each subdomain, the
spaces M , V and P are defined similarly to the P1(h) − P0(2h) finite elements. It
was shown in [6] that the best approximation property holds for the approximation
space V × P with Hood-Taylor finite elements. The inf-sup constant of the space
V ×P is crucial in the analysis of the approximation order. If the inf-sup constant is
independent of mesh size and subdomain size then the best approximation property
holds. In [6], it was shown that the inf-sup constant is independent of the mesh
size. However, it was not proved for the subdomain size. Following the similar
idea to Belgacem [6], we can see that the inf-sup constant of the space V × P with
P1(h)− P0(2h) finite elements is independent of the mesh size . For the subdomain
size H, we compute the inf-sup constant numerically and observe that the constant
seems to be independent of H (see Section 8.2).
Now, we rewrite (7.3) into a matrix form. Let Biji be a matrix with entries
(Biji )lk = ±
∫
Γij
ψl · φk ds ∀l = 1, · · · , L, ∀k = 1, · · · ,K, (7.4)
75
where ψlLl=1 is a basis for Mij and φk
Kk=1 is a nodal basis for Wi|Γij . Here,
Wi|Γij means the restriction of functions in Wi on Γij . In (7.4), the +sign is chosen
if Ωi|Γij is a nonmortar side, otherwise the −sign is chosen. Then we rewrite (7.3)
as
Biji vi|Γij +Bij
j vj |Γij = 0 ∀i = 1, · · · , N, ∀j ∈ mi. (7.5)
Define Eij : Mij →M to be an extension operator by zero and Rlij : Wl →Wl|Γij
for l = i, j to be a restriction operator and let Bi =∑
j∈mi∪siEijB
iji R
iij . Then (7.5)
becomes
Bw = 0, (7.6)
where
B =(B1 · · · BN
),
w =(wt
1 · · · wtN
)twith wi = vi|∂Ωi , ∀i = 1, · · · , N.
Let Bi,r and Bi,c be matrices that consist of the columns of Bi corresponding to the
d.o.f. on edges and corners, respectively. Then, using the notations introduced in
Section 7.1, (7.6) is written into
Brwr +Bcwc = 0, (7.7)
where Br =(B1,r · · · BN,r
)and Bc =
∑Ni=1Bi,cL
ic.
7.2 FETI-DP formulation
7.2.1 FETI-DP operator
In this section, we formulate a FETI-DP operator with the continuity constraints (7.7)
which are obtained from the mortar matching condition (7.3). To solve the Stokes
problem efficiently and correctly, we will add the redundant continuity constraints
to the coarse problem:
∫
Γij
(vi − vj) ds = 0 ∀i = 1, · · · , N, ∀j ∈ mi. (7.8)
76
In the FETI-DP method, the mortar matching condition holds when the solution
has converged. Hence, the convergence of the FETI-DP method is enhanced by
adding the redundant constraints to the coarse problem. When preconditioning the
FETI-DP operator, we solve a Dirichlet problem, i.e. a local Stokes problem, in each
subdomain. Furthermore, the compatibility condition of the local Stokes problem
follows from the redundant constraints.
We rewrite (7.8) as
Rt(Brwr +Bcwc) = 0, (7.9)
where the matrix R has the number of columns corresponding to two times of the
number of Γij ’s(interfaces) and rows corresponding to the d.o.f. on the space M and
has entries 1 and 0. For λ ∈ M , at each interior nodal point of Γij , λ|Γij has two
components corresponding to horizonal and vertical parts of velocity function. For
λ ∈ M , Rtλ = 0 means that for all Γij , the sums of λ|Γij corresponding to each
horizonal and vertical parts of velocity function are zero.
Let U be the Lagrange multiplier space corresponding to the constraints (7.9)
and for µ ∈ U , µ|Γij has two components that correspond to the constraints for
horizontal velocity and vertical velocity. Introducing Lagrange multipliers λ and
µ to enforce the constraints (7.7) and (7.9), the followings are induced from the
Galerkin approximation to (7.2):
Find (uI , pI ,ur,uc, p0,µ,λ) ∈ XI ×Q×Wr ×Wc × Π0 × U ×M such that
AII GII AIr AIc GI0 0 0
GtII 0 GtrI GtcI 0 0 0
ArI GrI Arr Arc Gr0 BtrR Bt
r
AcI GcI Acr Acc Gc0 BtcR Bt
c
GtI0 0 Gtr0 Gtc0 0 0 0
0 0 RtBr RtBc 0 0 0
0 0 Br Bc 0 0 0
uI
pI
ur
uc
p0
µ
λ
=
f I
0
f r
f c
0
0
0
, (7.10)
77
where
AII AIr AIc
ArI Arr Arc
AcI Acr Acc
is a stiffness matrix induced from
N∑
i=1
(∇u,∇v)Ωi ,
GII
GrI
GcI
is a matrix induced from
N∑
i=1
(−∇ · v, pI)Ωi ,
GI0
Gr0
Gc0
is a matrix induced from
N∑
i=1
(−∇ · v, p0)Ωi
and the subscripts I, r and c denote the interior, edges and corners, respectively.
Since p0|Ωi is constant, we have GI0 = 0. Let
zr =
uI
pI
ur
, zc =
uc
p0
µ
.
We regard zc as a primal variable. Then (7.10) can be written as
Krr Krc Btr
Ktrc Kcc Bt
c
Br Bc 0
zr
zc
λ
=
f r
f c
0
.
After eliminating zr, we obtain the following equation for zc and λ:(−Fcc Fcl
F tcl Fll
)(zc
λ
)=
(−dc
dl
)
where
Fll = BrK−1rr B
tr,
Fcl = KtrcK
−1rr B
tr − Bt
c,
Fcc = Kcc −KtrcK
−1rr Krc,
dl = BrK−1rr f r,
dc = f c −KtrcK
−1rr fr.
78
Note that
(Gr0 Bt
rR
Gc0 BtcR
)(p0
µ
)= 0 implies that
(p0
µ
)= 0. Using this it can be
shown easily that Fcc is invertible. Hence eliminating zc, we obtain the following
equation for λ:
(Fll + F tclF−1cc Fcl)λ = dl − F tclF
−1cc dc. (7.11)
Let FDP = Fll + F tclF−1cc Fcl and call it the FETI-DP operator. Since we add the
redundant constraints to the coarse problem, λ is not uniquely determined in M .
Let us define
MR =λ ∈M : Rtλ = 0
. (7.12)
In Section 7.3, we will show that FDP is s.p.d. on MR and λ ∈ MR is uniquely
determined. In the following section, we define several norms on the finite element
function spaces and propose a preconditioner for the operator FDP .
7.2.2 Preconditioner
For wi ∈Wi, we define Siwi by
AiII GiII AiIr AiIc
GiIIt
0 GirItGicI
t
AirI GirI Airr Airc
AicI GicI Aicr Aicc
uiI
piI
wir
wic
=
0
0
Si
(wir
wic
)
,
where the superscript i denotes submatrices corresponding to the subdomain Ωi.
Let us define
S := diag(S1, · · · , SN )
and it can be seen easily that S is s.p.d. on W . Hence, we define
‖w‖W :=
(N∑
i=1
< Siwi,wi >
)1/2
(7.13)
as a norm for w ∈ W . For a function wij ∈ W 0ij with j ∈ mi, let wij be the zero
extension of wij into Wi. Using this, for w ∈W 0 we define an extension w ∈W by
w = (w1, · · · , wN ) with wi =∑
j∈mi
wij ∀i = 1, · · · , N,
79
and define a norm on W 0 as
‖w‖W 0 := ‖w‖W . (7.14)
We introduce the following subspaces with the norms induced from the spaces W
and W 0:
WR :=w ∈W : Rt(Brwr +Bcwc) = 0
,
WR,G :=w ∈WR : Gtr0wr +Gtc0wc = 0
W 0R :=
w ∈W 0 : w ∈WR
.
Recall the definition of MR in (7.12) and let < ·, · >m be a duality pairing between
MR and W 0R defined as
< λ,w >m=N∑
i=1
∑
j∈mi
∫
Γij
λij ·wij ds.
Then we define a dual norm for λ ∈MR by
‖λ‖2MR
:= maxw∈W 0
R\0
< λ,w >2m
‖w‖2W 0
. (7.15)
Now, we will find an operator FDP which gives
< FDPλ,λ >= ‖λ‖2MR
(7.16)
and propose F−1DP as a preconditioner for the FETI-DP operator in (7.11). Define
Rij : W 0 → W 0ij as a restriction operator and Ei
ij : W 0ij → Wi as an extension
operator by zero. Then for w ∈W 0R,
‖w‖2W 0 = ‖w‖2
W
=N∑
i=1
< Siwi, wi >
=N∑
i=1
< Si(∑
j∈mi
EiijRijw),∑
j∈mi
EiijRijw > .
80
Let S =∑N
i=1(∑
j∈miEiijRij)
tSi(∑
j∈miEiijRij). Moreover, we have
< λ,w >m=< Bw,λ > (7.17)
where B = diagi=1,··· ,N
(diagj∈miB
iji
)and Bij
i is a matrix obtained from Biji after
deleting the columns corresponding to the d.o.f. at the end points of Γij . Note that
Biji is invertible. Since, we restrict λ ∈ MR and w ∈ W 0
R, to find FDP in a matrix
form we need the following l2-orthogonal projections:
PW 0R
: W 0 →W 0R,
PMR: M →MR.
For λ ∈MR and w ∈W 0R, we may write
< λ,w >m=< Bpw,λ >, ‖w‖2W 0 =< Spw,w >, (7.18)
where
Sp = P tW 0RSPW 0
R, Bp = P tMR
BPW 0R.
Then it can be shown that the operators
Sp : W 0 →W 0R,
Bp : W 0 →MR
are invertible on W 0R and Sp is s.p.d. on W 0
R. Hence, using (7.18), the maximum in
(7.15) occurs when w ∈W 0R satisfies Spw = Bt
pλ. Therefore, we have
‖λ‖2MR
=< BpS−1p Bt
pλ,λ > .
Let
F−1DP = (BpS
−1p Bt
p)−1 = (Bt
p)−1SpB
−1p .
and we call it a Neumann-Dirichlet preconditioner for the operator FDP . Define
l2-orthogonal projections
P ijW 0R
: W 0|Γij →W 0R|Γij ,
P ijMR: M |Γij →MR|Γij .
81
Then the projection operators PW 0R
and PMRare composed of diagonal blocks of
P ijW 0R’s and P ijMR
’s, respectively. Moreover, it can be shown easily that
(P ijMR)tBij
i PijW 0R
: W 0R|Γij →MR|Γij
is invertible. Hence, it follows that
B−1p = diagi=1,··· ,Ndiagj∈mi
(B−1ij
),
where Bij = (P ijMR)tBij
i PijW 0R
and
F−1DP =
N∑
i=1
∑
j∈mi
EiijB−1ij Rij
t
Si
∑
j∈mi
EiijB−1ij Rij
.
Therefore, the computation of F−1DPλ can be done parallely in each subdomain.
7.3 Condition number bound estimation
Lemma 7.1 We have
B(WR,G) = B(WR) = MR.
Proof. Since WR,G ⊂WR, B(WR,G) ⊂ B(WR).
Now, we will show that B(WR) ⊂ B(WR,G). Let w = (w1, · · · , wN ) ∈W be the
zero extension of w ∈W 0. Since wj |Γij = 0 for j ∈ mi and w is zero at subdomain
corners, we have
Bw = Bw, (7.19)
with B as defined in (7.17). From the fact that B is a 1− 1 mapping from W 0 onto
M and the definitions of W 0R and MR, we get
B(W 0R) = MR. (7.20)
For w ∈W 0R, the zero extension w = (w1, · · · , wN ) satisfies
∫
∂Ωi
wi ds = 0 ∀ i = 1, · · · , N
82
and then applying the divergence theorem
Gtr0wr +Gtc0wc = 0
holds for w. Hence, for w ∈W 0R, we have w ∈WR,G and from (7.19) we obtain
B(W 0R) ⊂ B(WR,G). (7.21)
From the definitions of WR and MR,
B(WR) = MR. (7.22)
Combining (7.22), (7.20) and (7.21), we have B(WR) ⊂ B(WR,G).
Remark 7.2 For w ∈W 0R, we have w ∈WR,G.
Lemma 7.3 For λ ∈MR, we have
< FDPλ,λ >= maxw∈WR,G\0
< Bw,λ >2
‖w‖2W
.
Proof. The problem (7.10) is equivalent to solving the following min-max problem:
maxλ∈B(WR,G)
minw∈WR,G
N∑
i=1
(1
2< Siwi,wi > − < di,wi >
)+ < Bw,λ >
, (7.23)
where di is the Schur complement forcing vector obtained from(f tI 0t f tr f tc
)t
after solving Stokes problem in each subdomain Ωi.
Let PWR,Gbe the l2-orthogonal projection from W onto WR,G. Recall that
B(WR,G) = MR from Lemma 7.1 and PMRis the projection operator from M onto
MR introduced in Section 7.2.2. Then taking Euler-Lagrangian in (7.23), we obtain
(Sp Bt
p
Bp 0
)(w
λ
)=
(P tWR,G
d
0
), (7.24)
where
Sp = P tWR,GSPWR,G
, Bp = P tMRBPWR,G
,
d =(dt1 · · · dtN
)t.
83
Since Sp is s.p.d. on WR,G, the equation for λ follows by eliminating w in (7.24):
BpS−1p Bt
pλ = BpS−1p d, (7.25)
which is the same as (7.11). Therefore we have
BpS−1p Bt
p = FDP . (7.26)
For λ ∈MR, we consider
maxw∈WR,G\0
< Bw,λ >2
‖w‖2W
. (7.27)
From (7.13), the definition of ‖ · ‖W , we may write
‖w‖2W =< Spw,w > for w ∈WR,G.
Since Sp is s.p.d. on WR,G, the maximum in (7.27) occurs when w ∈WR,G satisfies
Spw = Btpλ. Hence, we have
maxw∈WR,G\0
< Bw,λ >2
‖w‖2W
=< BpS−1p Bt
pλ,λ > . (7.28)
Combining (7.26) and (7.28), we complete the proof.
Remark 7.4 For λ ∈MR, Btpλ = 0 gives λ = 0 and Sp is s.p.d. on WR,G. Hence,
from (7.26), we can see that FDP is s.p.d. on MR.
Lemma 7.5 For λ ∈MR, we have
maxw∈WR,G\0
< Bw,λ >2
‖w‖2W
≥ ‖λ‖2MR
.
Proof. By definition, we have
‖λ‖2MR
= maxw∈W 0
R\0
< λ,w >2m
‖w‖2W 0
. (7.29)
Let w ∈W be the zero extension of w ∈W 0R. Then, w ∈WR,G. Moreover, we get
< λ,w >m=< Bw,λ > . (7.30)
From (7.29) and (7.30), we prove the lemma.
Let us define a notation | · |Si :=< Si ·, · >1/2. Then the following lemma can be
found in Bramble and Pasciak [16].
84
Lemma 7.6 For wi ∈Wi, we have
C1β|wi|Si ≤ |wi|1/2,∂Ωi ≤ C2|wi|Si ,
where β is the inf-sup constant for the finite elements of subdomain Ωi and the
constants C1 and C2 are independent of hi and Hi.
Since we have chosen inf-sup stable P1(h)− P0(2h) finite elements for each sub-
domain, the constant β is independent of hi and Hi. Therefore, we have
C1|wi|Si ≤ |wi|1/2,∂Ωi ≤ C2|wi|Si , (7.31)
where C1 and C2 are constants independent of hi and Hi.
From Lemma 2.11, we have the following result for the space W .
Lemma 7.7 For w ∈W , we have
‖wi −wj‖2
H1/200 (Γij)
≤ C maxl∈i,j
(1 + log
Hl
hl
)2(
|wi|21/2,∂Ωi
+ |wj |21/2,∂Ωj
),
where wi is the restriction of w onto ∂Ωi for i = 1, · · · , N and C is a constant
independent of hi’s and Hi’s.
Definition 7.8 We define a projection πij : [H1/200 (Γij)]
2 →W 0ij for v ∈ [H
1/200 (Γij)]
2
by ∫
Γij
(v − πijv) · λij ds = 0 ∀λij ∈Mij .
From (4.11), πij is a continuous operator on H1/200 (Γij). By extending the result to
the product space [H1/200 (Γij)]
2, we obtain
‖πijv‖H1/200 (Γij)
≤ C‖v‖H
1/200 (Γij)
∀v ∈ [H1/200 (Γij)]
2, (7.32)
with the constant C independent of Hi’s and hi’s.
Lemma 7.9 For λ ∈MR, we have
maxw∈WR,G\0
< Bw,λ >2
‖w‖2W
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2‖λ‖2
MR,
where C is a constant independent of hi’s and Hi’s.
85
Proof. Note that
< Bw,λ >=
N∑
i=1
∑
j∈mi
∫
Γij
(wi −wj) · λij ds.
Since wi −wj ∈ [H1/200 (Γij)]
2, from the definition of πij , we have
< Bw,λ >=N∑
i=1
∑
j∈mi
∫
Γij
πij(wi −wj) · λij ds. (7.33)
Let zij = πij(wi −wj) and z ∈ W 0 with z|Γij = zij . Since
(1
0
),
(0
1
)∈ Mij and
w ∈WR,G, ∫
Γij
zij ds =
∫
Γij
(wi −wj) ds = 0. (7.34)
From (7.34), we can see that RtBz = 0 with z ∈ W as the zero extension of z.
Hence, z ∈ W 0R and (7.33) is the duality pairing between z ∈ W 0
R and λ ∈ MR.
From (7.15), we get
< Bw,λ >2=< λ, z >2m≤ ‖λ‖2
MR‖z‖2
W 0 . (7.35)
From (7.14), (7.13), (7.31), (2.1), (7.32) and Lemma 7.7, we obtain
‖z‖2W 0 = ‖z‖2
W
=N∑
i=1
|zi|2Si
≤ CN∑
i=1
|zi|21/2,∂Ωi
≤ CN∑
i=1
∑
j∈mi
‖wi −wj‖2
H1/200 (Γij)
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2
N∑
i=1
|wi|21/2,∂Ωi
≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2‖w‖2
W .
(7.36)
86
Here, C is a generic constant which is independent of hi’s and Hi’s. Combining
(7.35) and (7.36), we complete the proof.
From Lemma 7.3, Lemma 7.5 and Lemma 7.9, we have
Theorem 7.10 For λ ∈MR,
‖λ‖2MR
≤ < FDPλ,λ > ≤ C maxi=1,··· ,N
(1 + log
Hi
hi
)2‖λ‖2
MR,
where C is a constant independent of hi’s and Hi’s.
Consequently, from (7.16) we obtain the following condition number estimate:
Corollary 7.11
κ(F−1DPFDP ) ≤ C max
i=1,··· ,N
(1 + log
Hi
hi
)2.
87
8. Numerical results
In this chapter, we provide numerical tests for the FETI-DP formulation developed
in this dissertation. The numerical tests are done for elliptic problems in 2D and
Stokes problem in 2D. Especially for the elliptic problems, we compare our results
with the previously developed FETI-DP formulation and FETI-DP preconditioners.
We present the approximate errors as well as the number of iterations in CGM.
8.1 Elliptic problems in 2D
Let Ω = [0, 1] × [0, 1] ∈ R2. We consider the following model problem:
−∇ · (α(x, y)∇u) = f in Ω,
u = 0 on ∂Ω,(8.1)
where α(x, y) > 0 and f ∈ L2(Ω). As mentioned in Section 5.2.1, our formulation is
different from that of Dryja and Widlund [21, 22]. We compare these two formula-
tions for the same problem on matching and nonmatching discretizations both. To
compare them, we consider the elliptic problem with continuous coefficients. Fur-
ther, we show the efficiency of the Neumann-Dirichlet preconditioner compared with
the existing FETI preconditioners for the elliptic problems with highly discontinuous
coefficients.
To distinguish our FETI-DP formulation from that of Dryja and Widlund, we
denote them by FKL and FDW , respectively. Also, for the preconditioners, we use the
notation F−1KL for our preconditioner, that is, the Neumann-Dirichlet preconditioner,
and F−1DW for Dryja and Widlund’s. The preconditioner F−1
DW has the form
F−1DW = (BrB
tr)
−1BrSrrBtr(BrB
tr)
−1,
where Br is the scaled matrix of Br divided by the mesh parameters of each subdo-
mains (see (3.13) in [22]). We also consider a preconditioner F−1KW by Klawonn and
88
Ω00
Ω01
Ω10
Ω
Ω
ij
33
Figure 8.1: Partition of subdomains when N = 4 × 4
Widlund [31], which was developed for solving the heterogeneous coefficient elliptic
problems with FETI formulation. Stefanica [48] observed that this preconditioner
is the most efficient one for the FETI formulation with mortar constraints. We
adapt the preconditioner to the FETI-DP formulation with mortar methods and
compare it with the preconditioners F−1KL and F−1
KW for elliptic problems with highly
discontinuous coefficients. The preconditioner F−1KW is given by
F−1KW = (BrD
−1r Bt
r)−1BrD
−1r SrrD
−1r Bt
r(BrD−1r Bt
r)−1, (8.2)
where Dr is a diagonal matrix whose entries are determined by the coefficients of
the elliptic problem. The matrix Dr will be described later.
8.1.1 An elliptic problem with smooth coefficients
We consider an elliptic problem with smooth coefficients. Simply, we take α(x, y) = 1
and the exact solution u(x, y) = y(1−y) sinπx in (8.1). In CG(Conjugate Gradient)
iteration, the stopping criterion is when the relative residual is less than 10−6. We
use n to denote the number of nodes on edges including end points and N to denote
the number of subdomains. In this problem, we use the same n for all subdomains,
divide Ω into rectangular subdomains as Figure 8.1 and denote each subdomain by
Ωij .
To make nonmatching grids across subdomain interfaces, we generate triangula-
tions in each subdomain in the following way: For each subdomain, we have chosen
89
Figure 8.2: Matching grids(left) and nonmatching grids(right) when n = 5
n random quasi-uniform nodes on each horizontal and vertical edges. Using these
nodes, we generate nonuniform structured grids in each subdomain. Since we use the
same n for all subdomains, the sizes of meshes between neighboring subdomains are
comparable. For matching grids, we use uniform meshes. Figure 8.2 shows examples
of matching and nonmatching grids.
First, we divide Ω intoN = 4×4 subdomains and increase the number of nodes n.
Table 8.1 shows L2 andH1-errors and the number of CG iterations between those two
formulations on both matching and nonmatching discretizations. For the H1-error,
we compute the broken H1-norm of errors over all subdomains. Table 8.2 shows the
numerical results when we fix n − 1 = 4 and increase the number of subdomains
N . For the cases N = 8 × 8, 16 × 16 and 32 × 32, we divide Ω into subdomains
as the same manner with N = 4 × 4. In the case of matching grids, Bc = 0 in the
FETI-DP formulation. Hence, two formulations are the same. However, they are
different on nonmatching grids. From the Tables 8.1 and 8.2, we can see that in
FDW -formulation the approximated solution does not converge to the exact solution
under nonmatching grids as n and N increase. Since the mortar matching condition
is imposed incorrectly, FDW -formulation dose not give the correct approximation. In
FKL-formulation, O(h2) and O(h) convergences are observed for L2 and H1-errors,
respectively. Furthermore, we can see that both preconditioners seem to give log2-
90
FKL, FDW -formulationn− 1
L2-error H1-error F−1KL F−1
DW
4 4.1293e-4 5.7497e-2 10 5
8 1.0399e-4 2.8798e-2 12 6
16 2.6046e-5 1.4405e-2 14 6
32 6.5127e-6 7.2036e-3 15 7
FKL-formulation FDW -formulationn− 1
L2-error H1-error F−1KL F−1
DW L2-error H1-error F−1DW
4 5.0850e-4 6.0126e-2 10 7 8.2409e-3 1.4987e-1 7
8 1.2865e-4 3.0128e-2 13 8 9.4588e-3 1.5738e-1 8
16 3.2235e-5 1.5072e-2 15 10 9.6715e-3 1.5766e-1 9
32 8.0627e-6 7.5374e-3 16 10 9.5528e-3 1.5599e-1 10
Table 8.1: Comparison between FKL and FDW on matching(up) and nonmatch-
ing(down) grids when N = 4 × 4
growth of the condition number bound and the CG iteration of F−1DW is smaller than
F−1KL.
8.1.2 Elliptic problems with highly discontinuous coefficients
We consider the problem (8.1) when α(x, y) is highly discontinuous across subdomain
interfaces and the mesh sizes are not comparable between subdomains. Under this
situation, we will compare preconditioners F−1KL, F−1
DW and F−1KW in FKL-formulation.
Recall the preconditioner F−1KW in (8.2). The diagonal matrix Dr consists of diagonal
matrices Dir’s:
Dr = diagi=1,··· ,N (Dir).
Here, we describe the matrix Dir precisely. For each subdomain Ωi, let Ni be the set
of nodes on the boundary of Ωi except ∂Ω. Let us define
µi(x) =∑
∂Ωj 3x
ργj for x ∈ Ni,
91
FKL, FDW -formulationN
L2-error H1-error F−1KL F−1
DW
4 × 4 4.1293e-4 5.7497e-2 10 5
8 × 8 1.0399e-4 2.8798e-2 11 6
16 × 16 2.6045e-5 1.4405e-2 11 6
32 × 32 6.5144e-6 7.2036e-3 11 6
FKL-formulation FDW -formulationN
L2-error H1-error F−1KL F−1
DW L2-error H1-error F−1DW
4 × 4 5.0850e-4 6.0126e-2 10 7 8.2409e-3 1.4987e-1 7
8 × 8 1.1744e-4 2.9900e-2 11 8 2.5171e-2 2.5418e-1 8
16 × 16 2.9743e-5 1.4980e-2 12 8 6.8789e-2 4.2452e-1 9
32 × 32 7.4318e-6 7.4917e-3 12 8 1.0531e-1 5.2951e-1 12
Table 8.2: Comparison between FKL and FDW on matching(up) nonmatching(down)
grids when n− 1 = 4
92
where α(x) = ρj(> 0) for x ∈ Ωj and γ ∈ [1/2,∞). Then the matrix Dir is given by
Dir = diagx∈Ni
(ργiµi(x)
).
We consider the cases of N = 2×2, 4×4, 8×8 subdomains. For each subdomain
Ωij , we choose the coefficient α(x, y) in the following way:
α(x, y) =
1 if both i and j are even,
250 if i is odd and j is even,
5000 if i is even and j is odd,
10 if both i and j are odd,
and denote them by ρij . In addition, we consider the exact solution u(x, y), which
belongs to H1(Ω), according to the partition of the domain:
u(x, y) =
p1(x, y) sin(πx) sin(πy)/α(x, y) when N = 2 × 2,
p2(x, y) sin(2πx) sin(2πy)/α(x, y) when N = 4 × 4,
sin(8πx) sin(8πy)/α(x, y) when N = 8 × 8,
where
p1(x, y) = (x− 1/2)(y − 1/2),
p2(x, y) = (x− 1/4)(x− 3/4)(y − 1/4)(y − 3/4).
Following [54] (see Section 1.5.3), we have chosen different mesh size in each
subdomain according to the ratio of coefficients between neighboring subdomains,
that is,hijhkl
' 4
√ρijρkl
,
where hij is the mesh size of the subdomain Ωij . Using the mesh sizes of these ratios,
we divide each subdomain into uniform meshes. LetHij be the size of the subdomain
Ωij . When N = 2×2 and max(Hij/hij) = 16, we obtain triangulations as Figure 8.3
and the triangulations are not comparable between neighboring subdomains.
In Section 1.5.3 of [54], it was shown that a good approximation of the solution is
obtained when the slave side is chosen to give a Lagrange multiplier space of higher
93
Ω
Ωρ
Ωρ =10
Ωρ=1
00
00
01
01ρ =5000
11
11
10
10=250
Figure 8.3: Triangulations for the case N = 2 × 2 and max(Hij/hij) = 16
dimension. Hence, choosing the subdomain with smaller hij (smaller ρij) as the
slave side, we can approximate the exact solution more accurately. This observation
coincides with the choice of master and slave sides in Remark 5.11.
Table 8.3 shows L2 and H1-errors and CG iterations with F−1KL, F−1
DW and F−1KW
as preconditioners under the FKL-formulation. In CG iteration, we use the same
stopping criterion 10−6 as before. Increasing max(Hij/hij), we observe the O(h2)
and O(h) convergences of L2 and H1-errors, respectively, for all cases of N . Further-
more, we see that the CG iterations of F−1KL and F−1
KW are much smaller than F−1DW .
The number of iterations of F−1KL and F−1
KW show similar behaviors in Table 8.3.
In Table 8.4, we compare the number of iterations and condition numbers of F−1KL
and F−1KW with various γ. From the results, we can observe that as γ goes to the
infinity, the number of iterations and condition numbers of F−1KW converge to those
of F−1KL. Moreover, we can show that
F−1KW → F−1
KL as γ → ∞.
Since the nonmortar sides have smaller ρi’s on the interfaces, the followings hold:
(Dir)
−1|Γij → ∞ as γ → ∞, if j ∈ mi,
(Dir)
−1|Γij → 0 as γ → ∞, otherwise.(8.3)
94
We rewrite
BrD−1r Bt
r =(Br,n Br,m
)(D−1r,n 0
0 D−1r,m
)(Br,n Br,m
)t,
where the subscripts n and m represent submatrices on nonmortar and mortar sides,
respectively. From (8.3), it holds
BrD−1r Bt
r → Br,nD−1r,nB
tr,n as γ → ∞.
Hence, we have
(BrD−1r Bt
r)−1 → (B−1
r,n)tD−1r,nB
−1r,n as γ → ∞. (8.4)
Similarly, we obtain
BrD−1r =
(Br,n Br,m
)(D−1r,n 0
0 D−1r,m
)→(Br,nD
−1r,n 0
)as γ → ∞. (8.5)
Therefore, from (8.4) and (8.5), it follows that
F−1KW →
(B−1r,n 0
)Srr
((B−1
r,n)t
0
)(= FKL) as γ → ∞.
From our numerical results, we conclude that our formulation gives the cor-
rect approximation of the model problem with nonmatching grids. For the case of
continuous coefficients and comparable meshes between subdomain interfaces, the
preconditioner F−1DW by Dryja and Widlund gives smaller number of iterations than
our preconditioner F−1KL. However, the preconditioner F−1
KL turns out to be much
more efficient than F−1DW for the problem with highly discontinuous coefficients on
noncomparable meshes. Furthermore, the preconditioner F−1KL is the limit of F−1
KW
as γ goes to the infinity.
8.2 Stokes problem in 2D
In this section, we present numerical results for the FETI-DP formulation of the
Stokes problem. Let Ω = [0, 1] × [0, 1] ⊂ R2 and consider the following Stokes
95
N max(Hij/hij) L2-error H1-error F−1DW F−1
KL F−1KW
16 3.0571e-5 7.6362e-3 17 3 3
32 7.8276e-6 3.8249e-3 26 3 3
2 × 2 64 1.9747e-6 1.9133e-3 39 4 3
128 4.9571e-7 9.5675e-4 50 4 4
256 1.2421e-7 4.7839e-4 60 4 4
16 2.1574e-6 1.0939e-3 75 4 3
4 × 4 32 5.4460e-7 5.4805e-4 81 4 4
64 1.3799e-7 2.7415e-4 111 4 4
128 3.4810e-8 1.3709e-4 130 4 4
16 1.0262e-3 8.8753e-1 113 3 3
8 × 8 32 2.4870e-4 4.4462e-1 136 4 4
64 6.4579e-5 2.2240e-1 168 4 4
Table 8.3: Comparison of preconditioners F−1KL, F−1
DW and F−1KW (γ = 2.0) for the
problem with highly discontinuous coefficients
96
F−1KW
N max(Hij/hij) γ = 0.5 γ = 1.0 γ = 2.0 γ = 10.0F−1KL
16 5.26e+1( 12 ) 1.09( 4 ) 1.03( 3 ) 1.04( 3 ) 1.04( 3 )
32 7.48e+1( 17 ) 1.15( 4 ) 1.04( 3 ) 1.04( 3 ) 1.04( 3 )
2 × 2 64 9.79e+1( 21 ) 1.22( 4 ) 1.05( 3 ) 1.05( 4 ) 1.05( 4 )
128 1.24e+2( 28 ) 1.30( 4 ) 1.06( 4 ) 1.07( 4 ) 1.07( 4 )
256 1.54e+2( 32 ) 1.39( 5 ) 1.08( 4 ) 1.08( 4 ) 1.08( 4 )
16 1.31e+1( 33 ) 1.25( 5 ) 1.05( 3 ) 1.06( 4 ) 1.06( 4 )
4 × 4 32 2.06e+2( 38 ) 1.42( 5 ) 1.08( 4 ) 1.09( 4 ) 1.09( 4 )
64 2.84e+2( 51 ) 1.62( 6 ) 1.12( 4 ) 1.13( 4 ) 1.13( 4 )
128 3.44e+2( 56 ) 1.85( 6 ) 1.17( 4 ) 1.17( 4 ) 1.17( 4 )
16 1.42e+2( 45 ) 1.28( 5 ) 1.05( 3 ) 1.05( 3 ) 1.05( 3 )
8 × 8 32 2.16e+2( 56 ) 1.48( 6 ) 1.08( 4 ) 1.09( 4 ) 1.09( 4 )
64 2.94e+2( 65 ) 1.72( 7 ) 1.12( 4 ) 1.12( 4 ) 1.12( 4 )
Table 8.4: Condition numbers (number of iterations) of F−1KW (γ = 0.5, 1.0, 2.0, 10.0)
and F−1KL for the problems with highly discontinuous coefficients
97
problem:
−4u + ∇p = f in Ω,
−∇ · u = 0 in Ω,
u = 0 on ∂Ω,
(8.6)
where f is chosen so that the exact solution of the problem becomes
u =
(sin3(πx)sin2(πy)cos(πy)
−sin2(πx)sin3(πy)cos(πx)
)and p = x2 − y2.
Let N denote the number of subdomains. We only consider the uniform parti-
tion of Ω as mentioned in Section 8.1.1. With this partition, we triangulate each
subdomain in the following manner. For all subdomains, we take the same number
of nodes n, including end points, in horizontal and vertical edges with n = 4k + 1
for some positive integer k. We solve (8.6) on matching and nonmatching grids
both. For matching grids, we make uniform triangulations in each subdomain with
(n − 1)/2 + 1 nodes on horizontal and vertical edges of the subdomain and denote
it by Ω2hii , a triangulation for the pressure. After bisecting each edge of triangles
in Ω2hii , we obtain Ωhi
i , a triangulation for the velocity. For nonmatching grids, we
take (n−1)/2+1 random quasi-uniform nodes on each horizontal and vertical edges
of subdomain, and generate nonuniform structured triangulations. We denote it by
Ω2hii . The triangulation Ωhi
i is obtained from Ω2hii similarly to matching grids. For
example, see Figure 8.4.
Now, we solve the FETI-DP operator with and without preconditioner varying
N and n. Those cases are denoted by PFETI-DP and FETI-DP, respectively. The
CG(Conjugate Gradient) iteration is stopped when the relative residual is less than
10−6.
In Tables 8.5-8.7, the number of CG iterations and condition numbers are shown
varying N and n. In Table 8.5, N = 4 × 4 and n − 1 increases by double. On
both matching and nonmatching grids, PFETI-DP performs well and the condition
numbers seem to behave log2-growth as n increases. Especially on nonmatching
grids, the CG iteration stops quickly with the preconditioner. In Tables 8.6 and 8.7,
N increases with n = 5 and n = 9. For both cases of FETI-DP and PFETI-DP, the
98
Figure 8.4: Triangulations Ωhii (left) and Ω2hi
i (right) when n = 5
Matching Nonmatchingn
FETI-DP PFETI-DP FETI-DP PFETI-DP
5 12(5.23) 9(2.62) 16(8.35) 12(3.75)
9 24(2.50e+1) 13(4.39) 50(1.15e+2) 15(5.79)
17 37(6.68e+1) 15(5.94) 86(5.01e+2) 17(7.93)
33 45(1.45e+2) 17(7.75) 119(1.31e+3) 20(9.88)
65 58(2.69e+2) 19(9.85) 153(3.29e+3) 22(1.20e+1)
Table 8.5: CG iterations(condition number) when N = 4 × 4
CG iteration becomes stable as N increases. From the results, we can see that the
developed preconditioner gives the condition number bound as confirmed in theory.
Moreover, we have observed the convergent behaviors of the approximated so-
lutions. The H1 and L2-errors for velocity and pressure are examined. uh and ph
denote the approximated solutions for the velocity and pressure, respectively, and
‖u− uh‖1,∗ means the square root of∑N
i=1 ‖u− uh‖21,Ωi
. The errors and reduction
factors are shown in Table 8.8 for various N and n with matching grids. Three cases
are considered: when n− 1 increases by double with N = 4 × 4, when N increases
by double in both edges of Ω with n = 5, and when N increases by double in both
edges of Ω with n = 9. For all cases, we can see that the H1-error for velocity,
‖u − uh‖1,∗, and L2-error for pressure, ‖p − ph‖0, reduce by half and L2-error for
99
Matching NonmatchingN
FETI-DP PFETI-DP FETI-DP PFETI-DP
4 × 4 12(5.23) 9(2.62) 16(8.35) 12(3.75)
8 × 8 12(5.42) 9(2.62) 16(9.18) 12(3.68)
16 × 16 10(5.54) 9(2.55) 16(9.57) 11(3.42)
32 × 32 10(5.61) 9(2.53) 16(10.88) 12(3.78)
Table 8.6: CG iterations(condition number) when n = 5
Matching NonmatchingN
FETI-DP PFETI-DP FETI-DP PFETI-DP
4 × 4 24(2.50e+1) 13(4.39) 50(1.15e+2) 15(5.79)
8 × 8 25(2.60e+1) 13(4.35) 53(1.19e+2) 15(6.21)
16 × 16 24(2.62e+1) 12(4.27) 57(1.34e+2) 16(6.27)
32 × 32 23(2.62e+1) 12(4.27) 56(1.25e+2) 16(6.24)
Table 8.7: CG iterations(condition number) when n = 9
100
N =
4 × 4n = 5 n = 9
‖u− uh‖1,∗ ‖u− uh‖0 ‖p− ph‖0
n N N
5 4 × 4 3.37e-1 3.75e-3 1.07e-1
9 8 × 8 4 × 4 1.72e-1 (0.510) 1.02e-3 (0.272) 5.99e-2 (0.559)
17 16 × 16 8 × 8 8.64e-2 (0.502) 2.64e-4 (0.258) 3.08e-2 (0.514)
33 32 × 32 16 × 16 4.32e-2 (0.500) 6.65e-5 (0.258) 1.55e-2 (0.503)
65 32 × 32 2.16e-2 (0.500) 1.66e-5 (0.249) 7.79e-3 (0.502)
Table 8.8: H1 and L2-errors(factor) on matching grids
n ‖u− uh‖1,∗ ‖u− uh‖0 ‖p− ph‖0
5 3.41e-1 3.79e-3 1.05e-1
9 1.78e-1 (0.521) 1.10e-3 (0.290) 6.08e-2 (0.579)
17 8.95e-2 (0.502) 2.85e-4 (0.259) 3.16e-2 (0.517)
33 4.48e-2 (0.500) 7.21e-5 (0.252) 1.58e-2 (0.500)
65 2.24e-2 (0.500) 1.81e-5 (0.251) 7.93e-3 (0.501)
Table 8.9: H1 and L2-errors(factor) on nonmatching grids: N = 4 × 4
velocity, ‖u−uh‖0, reduces by quarter. For the finite elements P1(h)−P0(2h), these
convergent behaviors are optimal.
For the case of nonmatching grids, the errors and reduction factors are shown
in Tables 8.9-8.11 with various N and n. In Table 8.9, we observe that the error
‖u−uh‖1,∗ and ‖p−ph‖0 reduce by half and the error ‖u−uh‖0 reduces by quarter
as n − 1 increases by double with N = 4 × 4. When n = 5 and n = 9, as N
increases, the errors also show the optimal convergent behaviors in Tables 8.10 and
8.11. These numerical results confirm that the stopping criterion for CG iteration
in Tables 8.5-8.7 is sufficient.
As mentioned in Section 7.1, if the inf-sup constant for the space V × P is
independent of N and n, then the optimality of approximation can be shown. Let
β∗ and β be the inf-sup constants for the space V × P and the P1(h) − P0(2h)
101
N ‖u− uh‖1,∗ ‖u− uh‖0 ‖p− ph‖0
4 × 4 1.78e-1 1.10e-3 6.08e-2
8 × 8 8.95e-2 (0.502) 2.94e-4 (0.269) 3.28e-2 (0.539)
16 × 16 4.49e-2 (0.501) 7.33e-5 (0.249) 1.63e-2 (0.496)
32 × 32 2.25e-2 (0.501) 1.84e-5 (0.251) 8.18e-3 (0.501)
Table 8.10: H1 and L2-errors(factor) on nonmatching grids: n = 5
N ‖u− uh‖1,∗ ‖u− uh‖0 ‖p− ph‖0
4 × 4 3.37e-1 3.75e-4 1.07e-1
8 × 8 1.72e-1 (0.510) 1.02e-3 (0.272) 5.99e-2 (0.559)
16 × 16 8.64e-2 (0.502) 2.64e-4 (0.258) 3.08e-2 (0.514)
33 × 32 4.32e-2 (0.500) 6.65e-5 (0.258) 1.55e-2 (0.503)
Table 8.11: H1 and L2-errors(factor) on nonmatching grids: n = 9
finite elements, respectively, and β0 be the inf-sup constant for the space V × Π0.
Then the constant β∗ depends on β and β0 from the trick conceived by Boland
and Nicolaides [12]. Hence, if the constant β0 is independent of n and N , then the
same holds for β∗. In [6], for V ×Π0 which is obtained from the Hood-Taylor finite
elements, it was shown that the constant β0 is independent of n, but not shown for
N . Following the proofs in [6], we can obtain the same results for the space V ×Π0
of the P1(h) − P0(2h) finite elements. We have no proof that β0 is independent of
N . Instead, we compute the constant β0 numerically as N increases. The results
are given in Table 8.12 both for matching and nonmatching grids when n = 5 and
n = 9. We observe that the constant β0 becomes stable as N increases. Table 8.13
gives the constant β0 as n increases with N = 4×4. This confirms that the constant
β0 is independent of n.
102
n = 5 n = 9N
Nonmatching Matching Nonmatching Matching
4 × 4 0.5780 0.5785 0.5921 0.5924
8 × 8 0.5293 0.5294 0.5352 0.5353
16 × 16 0.5008 0.5010 0.5041 0.5042
32 × 32 0.4827 0.4828 0.4854 0.4848
Table 8.12: Inf-sup constant β0 when n = 5 and n = 9
n Nonmatching Matching
5 0.5780 0.5785
9 0.5921 0.5294
17 0.5966 0.5967
33 0.5973 0.5979
65 0.5983 0.5983
Table 8.13: Inf-sup constant β0 when N = 4 × 4
103
Appendix
In the following, we show that how we approximate the inf-sup constant β0 of the
space V × Π0. By definition, the inf-sup constant β0 is
infq∈Π0
supv∈V
(∫Ω ∇ · vq dx
)2(∑N
i=1 ‖v‖21,Ωi
)‖q‖2
0,Ω
≥ β0. (A.1)
Since v ∈ V , there exist a constant C not depending on hi’s and Hi’s, such that
N∑
i=1
‖v‖21,Ωi ≤ C
N∑
i=1
|v|21,Ωi . (A.2)
Using the above relation, we replace the H1-norm in (A.1) by the semi H1-norm
and we will compute the constant β0 such that
infq∈Π0
supv∈V
(∫Ω ∇ · vq dx
)2(∑N
i=1 |v|21,Ωi
)‖q‖2
0,Ω
≥ β0. (A.3)
Our objective is to see that the constant β is independent of hi’s and Hi’s. Hence
it suffices to consider the above inequality to estimate the inf-sup constant. For
this purpose, we will give a matrix whose second smallest eigenvalue is the inf-sup
constant β0.
For v ∈ X, we split it into four parts, that is, the interior parts of subdomains,
the mortar sides of interfaces without end points, the nonmortar sides of interfaces
without end points and the global corners, and denote them by vI , vm, vn and vc,
respectively. Since q ∈ Π0 is constant in each subdomain, the denominator of L.H.S.
in (A.3) is independent of vI . We eliminate vI using
infvI
N∑
i=1
|vi|21,Ωi =
N∑
i=1
< Siwi,wi >,
∫
Ω∇ · vq dx =
N∑
i=1
∫
∂Ωi
wi · niq ds,
104
where vi = v|Ωi and wi = vi|∂Ωi .
Let us define
Z := w = (w1, · · · ,wN ) : wi = vi|∂Ωi for i = 1, · · · , N, ∀v ∈ V .
Similarly, we define wm, wc and wn for w ∈ Z. Then, we rewrite (A.3) into
infq∈Π0
supw∈Z
(∑Ni=1
∫∂Ωiw · niq ds
)2
(∑Ni=1 < Siwi,wi >
)‖q‖2
0,Ω
≥ β0. (A.4)
The space Wm,c is defined as a space with vectors wm,c =
(wm
wc
).
Since w ∈ Z satisfies
∫
Γij
(wi −wj) · λ ds = 0 ∀i = 1, · · · , N, ∀j ∈ mi, (A.5)
we can see that wn is determined by wm,c ∈Wm,c. We rewrite (A.5) into
Bnwn = Bmwm + (Bm,c −Bn,c)wc. (A.6)
Let
Bmc =(Bm Bm,c −Bn,c
).
Using (A.6), w ∈ V is obtained from wm,c ∈Wm,c
(wm,c
wn
)=
(I
B−1n Bmc
)wm,c.
More precisely, we have
w|Ωi =
(Lim,cwm,c
Linwn
),
where the maps Lim,c and Lin restrict wm,c and wn on the subdomain Ωi. Let us
define Eim : Wm,c → Z|∂Ωi by
Eim =
(Lim,c 0
0 Lin
)(I
B−1n Bmc
).
105
We may write
Si =
(Simm Simn
Sinm Sinn
),
where m and n denote the d.o.f. on mortar sides and corners ,and the d.o.f. on
nonmortar sides without end points, respectively. Then, we have
N∑
i=1
< Siwi,wi >=< Smwm,c,wm,c >, (A.7)
with
Sm =N∑
i=1
(Eim)tSiEim.
Let Gi be a matrix that gives
< Giwi, q >=
∫
∂Ωi
wi · niq ds.
We may consider the matrix Gi to be ordered as in Si and write
< Gmwm,c, q >=
N∑
i=1
∫
∂Ωi
wi · niq ds, (A.8)
with
Gm =N∑
i=1
GiEim.
In addition, a matrix M is defined as
< Mq, q >= ‖q‖20,Ω. (A.9)
Since q ∈ Π0 is constant in each subdomain, the matrix M is diagonal.
From (A.7), (A.8) and (A.9), we have the following identity:
(∑Ni=1
∫∂Ωiw · niq ds
)2
(∑Ni=1 < Siwi,wi >
)‖q‖2
0,Ω
=< Gmwm,c, q >
2
< Smwm,c,wm,c >< Mq, q >.
Hence, we consider
minq∈Π0
maxwm,c∈Wm,c
< Gmwm,c, q >2
< Smwm,c,wm,c >< Mq, q >(A.10)
106
to estimate the inf-sup constant β0. From (A.2), we can see that Sm is a s.p.d.
operator on Wm,c. Therefore, in (A.10), the maximum occurs when Smwm,c = Gtmq
and (A.10) is reduced into
minq∈Π0
< GmS−1m Gtmq, q >
< Mq, q >.
Since q ∈ Π0 is constant in each subdomain and∫Ω q dx = 0, the d.o.f. of Π0 is
exactly N − 1 and 1 ⊥ Π0. We may assume that there exist constants C1 and C2
independent of the number of subdomains and meshes such that
C1H2qtq ≤< Mq, q >≤ C2H
2qtq,
where H = maxi=1,··· ,N Hi. Let
Cm =1
H2GmS
−1m Gtm.
Hence, we consider
minq∈Π0
< Cmq, q > (A.11)
to estimate the constant β0. From the fact that Sm is s.p.d. and Null(Gtm) = 1 (see
[6]), the matrix Cm is symmetric and semi-positive definite and it has 0 eigenvalue
associated with the eigenvector 1. Therefore, to estimate the inf-sup constant β0 of
the space V × Π0, we compute the second smallest eigenvalue of the matrix Cm.
107
+, -/.0 12 345768:9<; 68 =<>?%@BACED87FG H IKJ)LMONPRQS T
FETI-DP UWVXZY)[\^]_preconditioner `badce fhg ijlknmoqpr
FETI(-DP) sutv suwxzy |~ | d ¢¡¤£¦¥§¨ª©«¬® ¯%° £ y |E±²³´¶µv"·¸xº¹» ¼¾½¿ ¼ÁÀ £ÃÂÄÆÅÇÈÊÉËuÌÎÍÏЪÑÒ suÓÔÕÖ×ÙØ , Ú ÇÜÛÝ Þàßâáãhä <åçæéèê ë ¹» ¼íìïîvñð¦òó Ëõô ¥§¨ ÕÖ ¢ö÷Oøù úüûhý ¸Ô Èuþ ÿ Ç ¼ ´x ¡¤£ Ç Ø ý ¸Ô Ç . ¡¤£ ±²³ "! #%$'&v | À £)(* | sutv suwx y |~,+.- ÿ 0/132 4 ¡¤£ sutv suwx 65.7v Ëõô +.- ÿ 98: <;>=?9@A BDCFEv ¹» ¼ $ ¸xHG>I ´x ÕÖ KJLNMOP ÅÇRQ £ S ÇUTVNW ²³ Q £ @X ZY[]\» ¼_^`bac ÅÇÈÊÉed §¨ G>I ´x ÕÖ "f gh÷ Èuþji | Ç øù 5lkm +.- ý ¸Ô Ç . ¡¤£ þ ÇUTVNW ²³ Q £ @X ZY[]\» ¼n^`bacpoqr ÕÖ ÿ FETI ±²³ ,+.- ÿ øù mixed problem ÕÖ ]s÷Ntu ^`wv¿ ¼ ¬® ¯bxy? ßâáã ±²³´¶µv"·¸xº¹» ¼ suÓz Ë| ÑÒ ×ÙØ ¡¤£ xy? ßâáã ±²³´¶µv"·¸x +.- ÿ ÇUTVNW ²³Q £ @X ZY[ ¡¤£ / Ç~ | ° £ Q £ Y[ èê ë ¹» ¼ ÅÇÈÊÉ ÇUTVNW ²³ Q £ @X ZY[ /1 ¯D>? ¬® ¯xy? ßâáã ±²³´¶µv"·¸x ¹» ¼suÓz øù Ç . ¡¤£ xy? ßâáã ±²³´¶µv"·¸x | ac,' ´x ÕÖ ill-conditioned ±²³´¶µv"·¸x ¡¤£Ö , Y[j £ ´x ÕÖ TV8: \» ¼ ¼ ´x ÕÖ 6 Ï À £ ÂÄ 8: ÿ øù ´xHG>I preconditioner
Ç 'Ö £Z c ÅÇ Ç . Ç~ | sutvsuwx^y |~,+.-j xy? ßâáã ±²³´¶µv"·¸x +.- £ 8: , ¡¤£ xy? ßâáã ±²³´¶µv"·¸x | Y[ oq ´x ÕÖ y | C Óx ÅÇ À £ Ç Ç . ¡¤£ þ Y[ oq ´x y | C Óx ÕÖ ]s÷Ntu preconditiner \» ¼ £ ´x Ë| ^`wv¿ ¼ 8:¡ £¢¤ Y[ Ç ý ¸Ô Ç . ¥` , ¡¤£ preconditioner øù À £)(* | sutv suwx y |~,+.- ÿ ;>=? JL ÑÒÈuþ¦§? 5lkmD¨©6ª « £ coarse
problem ¹» ¼ ÇRQ £ ý ¸Ô Q £ S ÕÖ Ö , ¬ ¯®°¯?,± £ ìïîvñð¦òó ©« ~ Y[ ý ¸ÔÕÖ×Ùس²´µ åçæ ·¶¹¸ º W)»³hä <åçæ\» ¼ ö÷Oøù úüûZý ¸Ô Èuþ ÿ TV ¼ ¼ CFEv ¡¤£ ac¾½ ÑÒ suÓÔ Ç .¿ÀÁ +.- FETI(-DP) ±²³ | £ :Z 7¡¤£¦¥§¨ª©«¬® ¯ ä <åçæ \» ¼ ö÷Oøù úüû ´x º ÑÒsuÓÔ Ç . ! # ± £ , Ã)ÄÅ Ã)ÄÅ sutv suwx +.- ÿ øù ÆJL CFEv ¡¤£ ÑÒÈuþ ý ¸Ô Q £Çb§¨ÉÈ §? G>I ÅÇ øù sutvsuwx èê ë d §¨ èê ë ¡¤£9ÊË BÌRÍ ´x ÕÖ 6Î÷ Èuþ ÿ ÐÏÑÒ ÓÕÔ §¨ £ ^ +.- ¶¹¸ 8: ;>=? JL Ç ¡¤£×Ö÷ Èuþ ØÙ Ú Ç . ¡¤£ /1Û2 4 | £ øù ! # ¡¤£ ´Ü ¡¤£ ý ¸Ô øù ° £ y |±²³´¶µv"·¸xº¹» ¼ ¡¤£¦¥§¨ª©«zÅÇøù 5.7v [ , 3 Ý ÇÜÛÝ Þ sutv suwx +.- ÿ 0/1·2 4 ¡¤£ \» ¼ Çb§¨ Ö øù húüû f S | £ d §¨ ¡¤£ ÑÒ øù 5.7v [ ,
G>I ~Þ ä <åçæ , multi-physics ä <åçæ ÊX ¹» ¼ Ç Ö÷Oøù úüûOý ¸Ô Èuþ ÿ ¨© ´x ¡¤£ À £àßá ä +.- f S ¡¤£ ;>=? JLÑÒÈuþ ØÙ Ú Ç . ¡¤£ þ £ í \» ¼ ¡¤£¦¥§¨ª©«:ÅÇ øù ¨© ´¶µv +.- ÿ sutv suwx "5.7v Ëõô +.- ÿ èê ë +.- ´xHG>I ãâ`åä.æ? ¹» ¼ Î÷Oøù úüû , ¡¤£ 5lkm ¹» ¼èç`êéë Ú Ç éë ac £ â`wä.æ? ¡¤£ Çì ÅÇ ×ÙØ ¡¤£ /1·2 4 |¡¤£¦¥§¨ª©« ±²³ ¹» ¼èç`êéë Ú Ç éë ±²³ ¡¤£ Çì Ç . ¡¤£ ±²³ | ÊË ac Ý Ç Y[ \» ¼ãí Ǻ 5.7v [ TV Á Hí Ç 8: ´¶µvïîð ñ ^` Ç í /1 ÊË ac ÅÇ Ç . ç`êéë Ú Ç éë ac £ â`åä.æ? ¡¤£
108
ÇUTVNW ²³ Q £ @X ZY[]\» ¼ !ò ÅÇÈÊÉ JLNMOP ÑÒåó? ¡¤£ þ ç`êéë Ú Ç éë ±²³ ÕÖ ]s÷Ntu mixed problem ¡¤£^`wv¿ ¼ ÑÒ ×ÙØ ¡¤£ ä <åçæ øù FETI ±²³ èê ë +.- ÿ ^`wv¿ ¼ ÑÒ øù mixed problem ¨© ÊË ac ÅÇ Ç .Q £ôõ öH÷ ÇRQ £ ç`êéë Ú Ç éë ±²³ ÕÖ ]s÷Ntu ^`wv¿ ¼ ¬® ¯ mixed problem +.- ¯ preconditioner ;>=? JL øù Y[ oq ´x 8: C Óx Èuþ Ø,øù ú ÕÖ È §? 8: ÈÊÉ þ ÇRQ £ Ëõô ´Ü ¡¤£ ý ¸Ô suÓÔ Ç . FETI(-DP) ±²³ ¹» ¼ ¡¤£ º ÅÇÈÊÉ ¡¤£ þ mixed problem ¹» ¼ ö÷Oøù 5lkm | preconditioner \» ¼ JL CFEv ÅÇ øù 5lkm ¡¤£º ¡¤£ ÅÇ Ö ¡¤£ +.- ¯ Y[j £ ´x È §? Ëõô ¥§¨ | Stefanica [48], Rapetti [42] ÊX +.-û 8: ;>=? JL ÑÒsuÓÔÕÖ×ÙØ , ¿ÀÁ +.- Widlund
/1Dryja [21, 22] +.-û 8: ÈÊÉ þ ÇRQ £ ßâáãýü : preconditioner èê ë
+.- ¶¹¸ Y[ oq ´x È §? 8: C Óx ¡¤£<¡¤£Ö÷ Èuþ ´þÔ Ç . TV þ ÿbÇTV èê ë Y[ oq ´x È §? 8: C Óx | Æ +.-åçæ Å ¡¤£ ý ¸Ô ac,' ´x È §? Ú ÇÜÛÝ ÞàßâáãOä <åçæ +.- ´x º ~ Y[ Ç øù Ëõô \» ¼ ÇRQ £ ý ¸Ô suÓÔ Ç . ¡¤£ø ä +.- ÿ øù ¡¤£ /1 2 4 | ä <åçæ \» ¼ Ç~ | ßâáãýü : preconditioner \» ¼ ^`bac ÅÇÈÊÉ 8: 5 ÅÇ suwÔÕÖ×ÙØ , 3 Ý ÇÜÛÝ Þ Ú ÇÜÛÝ Þ ä <åçæ TV « £ , Ö !ò # Ö ä <åçæ +.- ^` TV 5¾¨© \» ¼ îð ñ ÅÇ suwÔ Ç . ! # ± £ , Ëõô Y[ Ç¿ ¼ ;>=?9@A B ´x È §? Ú ÇÜÛÝ Þ ä åçæ 5.7v [ , À £)(* | +.- ËuÌÎÍÏÐ ¬® ¯ Ç~ | preconditioner èê ë Ç ¼ ´x ¡¤£ Ç øù 5lkm ¹» ¼ Y[j £ ´x Ëõô ¥§¨ ÕÖ ac¾½ ÅÇ suwÔ Ç .
109
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115
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EGFH IKJL1992. 3. – 1997. 2. MN o ²³POQ6R ¨© ¶¹¸ oq Y[ oq ¨© (B.S.)
1997. 3. – 1999. 2. TSU V ¨© oq À £@WX ë ÛÝ Þ Y[ oq ¨© (M.S.)
1999. 3. – 2004. 2. TSU V ¨© oq À £@WX ë ÛÝ Þ º ZY[ oq ¨© (Ph.D.)
YKZ[]\^`_bacedgfL1. Hi Jun Choe, Do Wan Kim, Hyea Hyun Kim and Yongsik Kim, Meshless
method for the stationary incompressible Navier-Stokes equations, Discrete and
Continuous Dynamical Systems. Series B, 1 (2001), no. 4, 495-526
2. Hyea Hyun Kim and Chang-Ock Lee, A Preconditioner for FETI-DP formu-
lation with mortar methods in two dimensions, Submitted.
3. Hyea Hyun Kim and Chang-Ock Lee, A preconditioner for the FETI-DP
formulation of the Stokes problem with mortar methods, Submitted.