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Korea Advanced Institute of Science and Technology

2004


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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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



6.2.1 FETI-DP operator . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . 62


7 Stokes problem in 2D 72



7.2.1 FETI-DP operator . . . . . . . . . . . . . . . . . . . . . . . . 76

7.2.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . 79


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 .


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,


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


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].



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.


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.


We have the following result as in 2D problems.


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

.


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.


‖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.


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 >,


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].


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.



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 .


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.


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.


< 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.


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


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.


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


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


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


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


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


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 £ÃÂÄÆÅÇÈÊÉËuÌÎÍÏÐªÑÒ 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×ÙØ³²´µ åçæ ·¶¹¸ º W)»³hä <åçæ\» ¼ ö÷Oøù úüûZý ¸Ô Èuþ ÿ TV ¼ ¼ CFEv ¡¤£ ac¾½ ÑÒ suÓÔ Ç .¿ÀÁÂ +.- 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 Çº 5.7v [ TV ÁÂ 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 , ¿ÀÁÂ +.- 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

References

[1] Y. Achdou, The mortar element method for convection diffusion problems, C.

R. Acad. Sci. Paris Ser. I Math., 321 (1995), pp. 117–123.

[2] Y. Achdou and O. Pironneau, A fast solver for Navier-Stokes equations in

the laminar regime using mortar finite element and boundary element methods,

SIAM J. Numer. Anal., 32 (1995), pp. 985–1016.

[3] R. A. Adams, Sobolev spaces, Academic Press, New York-London, 1975. Pure

and Applied Mathematics, Vol. 65.

[4] M. Amara, C. Bernardi, and M.-A. Moussaoui, Handling corner singular-

ities by the mortar spectral element method, Appl. Anal., 46 (1992), pp. 25–44.

[5] F. B. Belgacem, The mortar finite element method with Lagrange multipliers,

Numer. Math., 84 (1999), pp. 173–197.

[6] , The mixed mortar finite element method for the incompressible Stokes

problem: convergence analysis, SIAM J. Numer. Anal., 37 (2000), pp. 1085–

1100.

[7] F. B. Belgacem, A. Buffa, and Y. Maday, The mortar finite element

method for 3D Maxwell equations: first results, SIAM J. Numer. Anal., 39

(2001), pp. 880–901.

[8] F. B. Belgacem, P. Hild, and P. Laborde, The mortar finite element

method for contact problems, Math. Comput. Modelling, 28 (1998), pp. 263–

271. Recent advances in contact mechanics.

[9] F. B. Belgacem and Y. Maday, The mortar element method for three dimen-

sional finite elements, M2AN Math. Model. Numer. Anal., 31 (1997), pp. 289–

302.

110

[10] C. Bernardi and Y. Maday, Raffinement de maillage en elements finis par la

methode des joints, C. R. Acad. Sci. Paris Ser. I Math., 320 (1995), pp. 373–377.

[11] C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming ap-

proach to domain decomposition: the mortar element method, in Nonlinear par-

tial differential equations and their applications. College de France Seminar,

Vol. XI (Paris, 1989–1991), vol. 299 of Pitman Res. Notes Math. Ser., Long-

man Sci. Tech., Harlow, 1994, pp. 13–51.

[12] J. M. Boland and R. A. Nicolaides, Stability of finite elements under

divergence constraints, SIAM J. Numer. Anal., 20 (1983), pp. 722–731.

[13] D. Braess, Finite elements, Cambridge University Press, Cambridge, 1997.

Theory, fast solvers, and applications in solid mechanics.

[14] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on

Sobolev spaces with application to Fourier transforms and spline interpolation,

SIAM J. Numer. Anal., 7 (1970), pp. 112–124.

[15] , Bounds for a class of linear functionals with applications to Hermite

interpolation, Numer. Math., 16 (1970/1971), pp. 362–369.

[16] J. H. Bramble and J. E. Pasciak, A domain decomposition technique for

Stokes problems, Appl. Numer. Math., 6 (1990), pp. 251–261.

[17] J. H. Bramble, J. E. Pasciak, and J. Xu, The analysis of multigrid algo-

rithms with nonnested spaces or noninherited quadratic forms, Math. Comp.,

56 (1991), pp. 1–34.

[18] A. Buffa, Y. Maday, and F. Rapetti, A sliding mesh-mortar method for a

two dimensional eddy currents model of electric engines, M2AN Math. Model.

Numer. Anal., 35 (2001), pp. 191–228.

[19] P. Clement, Approximation by finite element functions using local regulariza-

tion, Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. Rouge

Anal. Numer., 9 (1975), pp. 77–84.

111

[20] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for

time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998),

pp. 2440–2463.

[21] M. Dryja and O. B. Widlund, A FETI-DP method for a mortar discretiza-

tion of elliptic problems, in Recent developments in domain decomposition

methods (Zurich, 2001), vol. 23 of Lect. Notes Comput. Sci. Eng., Springer,

Berlin, 2002, pp. 41–52.

[22] , A generalized FETI-DP method for a mortar discretization of elliptic

problems, in Domain decomposition methods in Science and Engineering (Co-

coyoc, Mexico, 2002), UNAM, Mexico City, 2003, pp. 27–38.

[23] C. Farhat, A saddle-point principle domain decomposition method for the solu-

tion of solid mechanics problems, in Fifth International Symposium on Domain

Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991),

SIAM, Philadelphia, PA, 1992, pp. 271–292.

[24] C. Farhat, P.-S. Chen, and J. Mandel, Scalable Lagrange multiplier based

domain decomposition method for time-dependent problems, Int. J. Numer.

Meth. Engng., 38 (1995), pp. 3831–3853.

[25] C. Farhat, P.-S. Chen, J. Mandel, and F. X. Roux, The two-level FETI

method. II. Extension to shell problems, parallel implementation and perfor-

mance results, Comput. Methods Appl. Mech. Engrg., 155 (1998), pp. 153–179.

[26] C. Farhat, M. Lesoinne, and K. Pierson, A scalable dual-primal domain

decomposition method, Numer. Linear Algebra Appl., 7 (2000), pp. 687–714.

[27] C. Farhat and J. Mandel, The two-level FETI method for static and dy-

namic plate problems. I. An optimal iterative solver for biharmonic systems,

Comput. Methods Appl. Mech. Engrg., 155 (1998), pp. 129–151.

[28] C. Farhat, J. Mandel, and F.-X. Roux, Optimal convergence properties

of the FETI domain decomposition method, Comput. Methods Appl. Mech.

Engrg., 115 (1994), pp. 365–385.

112

[29] C. Farhat and F.-X. Roux, A method of finite element tearing and intercon-

necting and its parallel solution algorithm, Int. J. Numer. Methods in Engrg.,

32 (1991), pp. 1205–1227.

[30] C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski, Multi-

plier spaces for the mortar finite element method in three dimensions, SIAM J.

Numer. Anal., 39 (2001), pp. 519–538.

[31] A. Klawonn and O. B. Widlund, FETI and Neumann-Neumann iterative

substructuring methods: connections and new results, Comm. Pure Appl. Math.,

54 (2001), pp. 57–90.

[32] A. Klawonn, O. B. Widlund, and M. Dryja, Dual-primal FETI methods

for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J.

Numer. Anal., 40 (2002), pp. 159–179.

[33] , Dual-primal FETI methods with face constraints, in Recent developments

in domain decomposition methods (Zurich, 2001), vol. 23 of Lect. Notes Com-

put. Sci. Eng., Springer, Berlin, 2002, pp. 27–40.

[34] J. Li, A dual-primal FETI method for incompressible Stokes equations, in Tech-

nical Report 816, Department of Computer Science, Courant Institute, New

York Unversity, 2001.

[35] , Dual-primal FETI methods for incompressible Stokes and linearlized

Navier-Stokes equations, in Technical Report 828, Department of Computer

Science, Courant Institute, New York Unversity, 2002.

[36] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems

and applications. Vol. I, Springer-Verlag, New York, 1972. Translated from the

French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften,

Band 181.

[37] J. Mandel and R. Tezaur, Convergence of a substructuring method with

Lagrange multipliers, Numer. Math., 73 (1996), pp. 473–487.

113

[38] , On the convergence of a dual-primal substructuring method, Numer.

Math., 88 (2001), pp. 543–558.

[39] J. Mandel, R. Tezaur, and C. Farhat, A scalable substructuring method

by Lagrange multipliers for plate bending problems, SIAM J. Numer. Anal., 36

(1999), pp. 1370–1391.

[40] L. Marcinkowski, Domain decomposition methods for mortar finite element

discretizations of plate problems, SIAM J. Numer. Anal., 39 (2001), pp. 1097–

1114.

[41] A. Quarteroni and A. Valli, Domain decomposition methods for partial

differential equations, Numerical Mathematics and Scientific Computation, The

Clarendon Press Oxford University Press, New York, 1999.

[42] F. Rapetti and A. Toselli, A FETI preconditioner for two-dimensional edge

element approximations of Maxwell’s equations on nonmatching grids, SIAM J.

Sci. Comput., 23 (2001), pp. 92–108.

[43] P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for

2nd order elliptic equations, Math. Comp., 31 (1977), pp. 391–413.

[44] D. J. Rixen, Extended preconditioners for the FETI method applied to con-

strainted problems, Internat. J. Numer. Methods Engrg., 54 (2002), pp. 1–26.

[45] H. A. Schwarz, Uber einige abbildungsaufgaben, Ges. Math. Abh., 11 (1869),

pp. 65–83.

[46] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth func-

tions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483–493.

[47] D. Stefanica, Poincare and Fridrichs inequalities for mortar finite element

methods, in Report 774, Department of Computer Science, Courant Institute,

New York Unversity, 1998.

[48] , A numerical study of FETI algorithms for mortar finite element methods,

SIAM J. Sci. Comput., 23 (2001), pp. 1135–1160.

114

[49] , FETI and FETI-DP methods for spectral and mortar spectral elements: a

performance comparison, in Proceedings of the Fifth International Conference

on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), vol. 17,

2002, pp. 629–638.

[50] R. Tezaur, Analysis of Lagrange multiplier based domain decomposition, in

Ph. D. Thesis, Applied Mathematics, Unversity of Colorado at Denver, 1998.

[51] A. Toselli, FETI domain decomposition methods for scalar advection-

diffusion problems, Comput. Methods Appl. Mech. Engrg., 190 (2001),

pp. 5759–5776.

[52] B. I. Wohlmuth, A residual based error estimator for mortar finite element

discretizations, Numer. Math., 84 (1999), pp. 143–171.

[53] , A mortar finite element method using dual spaces for the Lagrange mul-

tiplier, SIAM J. Numer. Anal., 38 (2000), pp. 989–1012.

[54] , Discretization methods and iterative solvers based on domain decompo-

sition, vol. 17 of Lecture Notes in Computational Science and Engineering,

Springer-Verlag, Berlin, 2001.

[55] J. Xu, Iterative methods by space decomposition and subspace correction, SIAM

Rev., 34 (1992), pp. 581–613.

[56] J. Xu and J. Zou, Some nonoverlapping domain decomposition methods,

SIAM Rev., 40 (1998), pp. 857–914.

115

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!v"%$&(') * ! U Tm 243 ¹$»> D GP KW ] ^ ËRQ km | ! FH +,¡ < = :< = ! O »½¢ ¿ DÀ _a¶Á n ³´ p _acbd e :< = fh S¸ÉÀ6Õs Ö KV ! . "%$&(') *z¢o p èé ! K ! , £ ³¤ :< = £ ³¥ !0! Lâ *dcfe ¿ DÀhgji R¦ ·¨§ KW ÎVÐª© ³¥ e :< =úfh S¸ÉÀ ),+m m\ D +, n ³¥«¬ :< = " Â%ÃÄ Z K ×û %¼ ¹32Ù ¯ KV ! . \m¯® èé Ñ° no p" 243M±² R µ¸· ! ¢´³ m ] K 7 D_a¶µ · Lâ * !d¸¹ e ÜßÞm àá §< = - K zñ %¼»º Ð X !½¼0! X !½¼0! ¿¾ÁÀ»ÃÂ K !ÅÄ6 Ò¸Ô fh S¸ÉÀ ),+m C D _a OQ R m\ D u4w _a-0/Æ I0Kx0z| ! ©«¬0®¯ KV ! . !T"%$& µ ¯ ïð ! Ë\ÌW \ÈÇÉ Ê 243 4 !Ë ! Ìn ³´ p - K çèé :< = Å ³Æ I0K X !^] K"Íl Î £8¥¦ ¹32Ù I0K !ÏÐÒÑo p mdÓÔ õ · qs t ïð àá qs t 243 ] x0z| ! +, ~ ! e :< = \m X ! FH; !#%T KV ! . gÕ ö i !^] K Tm 243Ön ³´ p _acbd e :< = fh S¸ÉÀ ),+m ! ÃÄ Z K×û 243Ø× z Lâ *ÚÙ û ! Ü m K , ÜßÞm àá N /1 \ÈÇÉ Ê :< = X ! Lâ *dcfe ¿ DÀhgji n ³´ p õ ·¨öß÷ §< =Ïµ¸·fhjilkm Û m 7 D m õ · Z K¨Ü ¾ mõ · u4wjx0z| ! ©«¬0®¯ KV ! . ÜßÞm àá N /1 +,ÞÝâ * Ýâ * £8¥¦ ÓÔ õ · qs t fh ×XW , p ,

-0/1 ½¢xß K , Z K OQ R , Ë\ÌW =@àA , JKxáãâ FH L K , ä =µ ¯ £8¥¦ ÜßÞm ¹,åW , Ê km , ÜßÞm Ñ¢ ,

JKML K FHçæ8zñ >A@ qs t Ñ° ! , bd I0KRèaé¢ Z K ×û 243n ³´ I0K × z :< =ª¹32Ù ¯ KV ! . êë ãå æ£8¥¦ àá qs t Í ÜßÞm , Ë\ÌW ÜßÞm ,êë 243 ] _a x0z| ! +, ~ ! e :< = \m X ! FH N /Æ Õs Ö KV ! .üþ ÿh X ! Lâ *pìß÷îí« L K +, h rs "%$ï :< = x0z>ð ! fh - K FH , n ³´ p _acbd e :< = fh S¸ÉÀ ),+m´- K h èa D qs tvu4w ì i L KPñò b gÕx0z| ! ©«¬0®¯ KV ! . ¢×ØÙ(ÚÛ ! _a , gjiîó ¢ SVUW I0K §< = qô tRõ fh S¸ÉÀ ),+m gji ì i D u4w -0/Æ I0K x0z| ! ©«¬0®¯ KV ! . JKML KFH , §< = " Â%ÃÄ ] Köv÷øõ fh - K FHúù DE gji fh S¸ÉÀ ),+m ,

! ÃÄ òô p ¹,åWYX ! Lâ * h èa D u4w x0z| ! ©« L K ìß÷ , I0K zñ p¹4º» N /1 :< =ª©«¬0®¯ KV ! . ~ !^] Kûæ8zñ q« øÛ , | !½ü X ! Lâ * ! qs tP ¢ SVUW I0KW¶Ýâ * Ýâ * £8¥¦ ÓÔ Z\ < ! Q km × zýûþm 243 ] I0K LN * ') *q« øÛ x0z| ! +, ~ ! e ïð K½ÿ èé "%$ï :< = \m X ! FH; !%T KV ! .

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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.

FETI-DP preconditionerhhk2/publish/w995113.pdfA preconditioner BDC E FG5IHKJ L MON P Q R...

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