A Domain Decomposition Method With Lagrange Multipliers and

A DOMAIN DECOMPOSITION METHOD WITH LAGRANGEMULTIPLIERS AND INEXACT SOLVERS FOR

LINEAR ELASTICITY∗

AXEL KLAWONN† AND OLOF B. WIDLUND‡

SIAM J. SCI. COMPUT. c© 2000 Society for Industrial and Applied MathematicsVol. 22, No. 4, pp. 1199–1219

Abstract. A new domain decomposition method with Lagrange multipliers for elliptic problemsis introduced. It is based on a reformulation of the well-known finite element tearing and intercon-necting (FETI) method as a saddle point problem with both primal and dual variables as unknowns.The resulting linear system is solved with block-structured preconditioners combined with a suitableKrylov subspace method. This approach allows the use of inexact subdomain solvers for the positivedefinite subproblems. It is shown that the condition number of the preconditioned saddle point prob-lem is bounded independently of the number of subregions and depends only polylogarithmically onthe number of degrees of freedom of individual local subproblems. Numerical results are presentedfor a plane stress cantilever membrane problem.

Key words. domain decomposition, Lagrange multipliers, finite element tearing and intercon-necting, preconditioners, elliptic systems, finite elements

AMS subject classifications. 65F10, 65N30, 65N55

PII. S1064827599352495

1. Introduction. In the past decade a great deal of research has been carriedout on nonoverlapping domain decomposition methods using Lagrange multipliers. Inthese methods the original domain is decomposed into nonoverlapping subdomains.The continuity is then enforced by using Lagrange multipliers across the interfacedefined by the subdomain boundaries. A computationally quite efficient memberof this class of domain decomposition algorithms is the finite element tearing andinterconnecting (FETI) method introduced by Farhat and Roux [7]. In its originalversion, a Neumann problem is solved on each subdomain and the method is knownto be scalable in the sense that its rate of convergence is independent of the numberof subproblems. In a variant of the FETI method introduced in Farhat, Mandel, andRoux [6] an additional Dirichlet problem is solved exactly on each subdomain, in eachiteration. This makes the rate of convergence of the iteration even less sensitive tothe number of unknowns of the local problems. The use of inexact Dirichlet solvers ispossible without a radical change of the FETI method. However, the use of inexactNeumann solvers does require a redesign of these algorithms; this is the topic of thepresent work.

In this paper, a new domain decomposition method with Lagrange multipliers isintroduced by first reformulating the system of the FETI algorithm as a saddle pointproblem with both primal and dual variables. The resulting system is then solved

∗Received by the editors February 26, 1999; accepted for publication (in revised form) May 1,2000; published electronically October 18, 2000.

http://www.siam.org/journals/sisc/22-4/35249.html†SCAI—Institute for Algorithms and Scientific Computing, GMD—German National Research

Center for Information Technology, Schloss Birlinghoven, D–53754 Sankt Augustin, Germany([email protected], http://www.gmd.de/SCAI/people/klawonn). The work of this author was sup-ported in part by the DAAD (German Academic Exchange Service) within the program HSP III(Gemeinsames Hochschulsonderprogramm von Bund und Landern).

‡Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York,NY 10012 ([email protected], http://www.cs.nyu.edu/cs/faculty/widlund). The work of this au-thor was supported in part by the National Science Foundation under grant NSF-CCR-9732208 andin part by the US Department of Energy under contract DE-FG02-92ER25127.

1199

1200 AXEL KLAWONN AND OLOF B. WIDLUND

using block-structured preconditioners and a suitable Krylov subspace method. Wecan then avoid potentially quite costly direct solvers relying instead on any of a numberof well-tested preconditioners for positive definite subproblems, such as incomplete LUmethods, (algebraic) multigrid, etc. The good features of the FETI method such asscalability and efficiency are preserved. Essentially, the storage and factorization costsrelated to use of exact solvers in the FETI algorithm are eliminated while the newmethod will incur new costs for the preconditioners on the subdomains and additionalstorage of primal variable components in the iteration; we also have to expect thatthe iteration count will increase in comparison to when exact subdomain solvers areemployed. A complete theory, based quite directly on the pioneering work of Mandeland Tezaur [16], and on earlier work by Klawonn [11], is also developed in this paper;see Theorems 9 and 15. In addition, we show how these results can be improved fora different family of preconditioners developed in [14]; see Theorems 10 and 16.

The remainder of this article is organized as follows. In section 2, we present theequations of linear elasticity and a finite element discretization thereof. In section3, we review the FETI method and we develop our new method in section 4. Insubsection 5.1, the convergence analysis of Mandel and Tezaur [16] is extended fromscalar, second order elliptic equations to the system of equations of linear elasticity.A convergence analysis and condition number estimates for the block-diagonal pre-conditioner are given in subsection 5.2. The paper concludes with section 6, in whichwe report on some of our numerical experiments.

We note that a short conference paper [13] prepared previously describes anddiscusses a slightly different version of our main algorithm.

2. The elliptic problem. In this section, we introduce our model problem,the elliptic system arising from the displacement formulation of compressible, linearelasticity and its discretization by conforming finite elements.

2.1. The equations of linear elasticity. The equations of linear elasticitymodel the displacement of a linear elastic material under the action of external andinternal forces. We denote the elastic body by Ω ⊂ Rd, d = 2, 3, and its boundary by∂Ω and assume that one part of the boundary, Γ0, is clamped, i.e., with homogeneousDirichlet boundary conditions, and that the rest, Γ1 := ∂Ω\Γ0, is subject to a surfaceforce g, i.e., a natural boundary condition. We can also introduce a body force f , e.g.,gravity. The appropriate space for a variational formulation is the Sobolev spaceH1

Γ0(Ω) := v ∈ H1(Ω)d : v|Γ0

= 0. The linear elasticity problem consists of

finding the displacement u ∈ H1Γ0

(Ω) of the elastic body Ω, such that

G

∫Ω

ε(u) : ε(v)dx + Gβ

∫Ω

divu divv dx = 〈F,v〉 ∀ v ∈ H1Γ0

(Ω).(1)

Here G and β are material parameters depending on the Poisson ratio ν and Young’smodulus E with ν ∈ (0, 1/2] and E > 0. In the case of plane stress, we have G =E/(1+ν), β = ν/(1−ν), and for plane strain and three-dimensional elasticity we have

G = E/(1 + ν), β = ν/(1− 2ν). Furthermore, εij(u) := 12 (

∂ui

∂xj+

∂uj

∂xi) is the linearized

strain tensor, and

ε(u) : ε(v) =

d∑i,j=1

εij(u)εij(v), 〈F,v〉 :=

d∑i=1

∫Ω

fivi dx +

d∑i=1

∫Γ1

gividσ.

DOMAIN DECOMPOSITION WITH LAGRANGE MULTIPLIERS 1201

The associated bilinear form of linear elasticity is

a(u,v) = G

∫Ω

ε(u) : ε(v)dx + Gβ

∫Ω

divu divv dx.

In this article, we consider only the case of compressible elasticity. This meansthat the Poisson ratio ν is bounded away from 1/2. We will also focus on the case ofd = 2 in order to simplify our notation. We note that all our work extends easily tothe case of d = 3 and to many other elliptic problems.

We will also use the standard Sobolev space norm

‖u‖H1(Ω) :=(|u|2H1(Ω) + ‖u‖2

L2(Ω)

)1/2

with ‖u‖2L2(Ω) :=

∫Ω|u|2dx, and |u|2H1(Ω) := ‖∇u‖2

L2(Ω). It is obvious that the bilinear

form a(·, ·) is continuous with respect to ‖ · ‖H1(Ω). However, proving ellipticity is farless trivial but it can be established from Korn’s first inequality; see, e.g., Ciarlet [3].

Lemma 1 (Korn’s first inequality). Let Ω ⊂ Rd, d ≥ 2, be a Lipschitz domain.Then, there exists a positive constant c = c(Ω,Γ0), such that∫

Ω

ε(u) : ε(u)dx ≥ c |u|2H1(Ω) ∀ u ∈ H1Γ0

(Ω).

The well-posedness of the linear system (1) follows immediately from the conti-nuity and ellipticity of the bilinear form a(·, ·).

It follows from Korn’s first inequality that ‖ε(u)‖L2(Ω) is equivalent to ‖u‖H1(Ω)

on H1Γ0

(Ω). This result is not directly valid for the case of pure natural boundary

conditions when we are working with the space (H1(Ω))d. This case is of interestwhen considering interior subregions, i.e., those that do not touch Γ0. However, aGarding inequality is provided by Korn’s second inequality

Lemma 2 (Korn’s second inequality). Let Ω ⊂ Rd, d ≥ 2, be a Lipschitz domainwith diameter one. Then, there exists a positive constant c = c(Ω), such that∫

Ω

ε(u) : ε(u)dx+ ‖u‖2L2(Ω) ≥ c ‖u‖2

H1(Ω) ∀ u ∈ (H1(Ω))d.

There are several proofs; see, e.g., Nitsche [19].We can now derive a Korn inequality on the space u ∈ (H1(Ω))d : u ⊥ ker (ε).

The null space ker (ε) is the space of rigid body motions. Thus, for d = 2, thelinearized strain tensor of u and its divergence vanish only for the elements of thespace spanned by the two translations

r1 :=

[10

], r2 :=

[01

]and the single rotation

r3 :=

[x2

−x1

].

In three dimensions the corresponding space is spanned by three translations andthree rotations.


For convenience, we also introduce the notation

(ε(u), ε(v))L2(Ω) :=

∫Ω

ε(u) : ε(v)dx

and introduce two inner products on (H1(Ω))2, for a region Ω with diameter one,

(u,v)E1:= (ε(u), ε(v))L2(Ω) + (u,v)L2(Ω)

and

(u,v)E2 := (ε(u), ε(v))L2(Ω) +

3∑i=1

li(u)li(v),

where the li(·) are defined by

li(u) :=

∫Ω

(ri)tudx.

By using Lemma 2, ‖ · ‖E1 , given by the inner product (·, ·)E1 , is a norm and not justa seminorm, and so, by construction, is ‖ · ‖E2

.The norms, just introduced, are equivalent.Lemma 3. There exist constants 0 < c ≤ C <∞, such that

c ‖u‖E1 ≤ ‖u‖E2 ≤ C ‖u‖E1 ∀ u ∈ (H1(Ω))2.

Proof. The proof of the right inequality follows immediately from the Cauchy–Schwarz inequality. The left inequality is proven by contradiction and by using Rel-lich’s theorem as in a proof of generalized Poincare–Friedrichs inequalities, cf., e.g.,Necas [18, Chap. 2.7].

We obviously have

‖ε(u)‖L2(Ω) ≤ ‖∇u‖L2(Ω) ∀ u ∈ (H1(Ω))2.(2)

Using (2) and Lemmas 2 and 3, we obtain the following.Lemma 4. There exists a constant 0 < c, such that

c ‖∇u‖L2(Ω) ≤ ‖ε(u)‖L2(Ω) ≤ ‖∇u‖L2(Ω) ∀ u ∈ (H1(Ω))2,u ⊥ ker (ε).

2.2. Finite elements and the discrete problem. Since we consider onlycompressible elastic materials, it follows from Lemma 1 that the bilinear form a(·, ·)is uniformly elliptic. We can therefore successfully discretize the system (1) withlow-order, conforming finite elements, such as linear or bilinear elements.

We assume that a triangulation τh of Ω is given which is shape regular and hasa typical element diameter of h. We denote by Wh(Ω) ⊂ H1

Γ0(Ω) the corresponding

conforming space of finite element functions, e.g., piecewise linear or bilinear contin-uous functions. Thus, it is our goal to solve the discrete problem

a(uh,vh) = 〈F,vh〉 ∀ vh ∈ Wh(Ω).(3)

In what follows, we work exclusively with the discrete problem and we drop thesubscript h from now on.


3. A review of the FETI method. In this section, we give a brief review ofthe original FETI method introduced in Farhat and Roux [7] and the variant witha Dirichlet preconditioner introduced in Farhat, Mandel, and Roux [6]. For moredetailed descriptions and proofs, we refer to [4, 5, 17, 22] and the references therein.

Let the domain Ω ⊂ R2 be decomposed into N nonoverlapping subdomainsΩi, i = 1, . . . , N, each of which is the union of elements and such that the finiteelement nodes on the boundaries of neighboring subdomains match across the inter-face Γ := (

⋃Ni=1 ∂Ωi) \ ∂Ω. Let the corresponding conforming finite element spaces

be Wi = Wh(Ωi), i = 1, . . . , N, and let W :=∏N

i=1 Wi be the associated productspace. When it is necessary to use vectors of nodal values, which define the elementsof a finite element space, we underline; e.g., W is the product space that correspondsto W. Analogously, we denote the vector of nodal values associated with the finiteelement function u by u.

For each subdomain Ωi, i = 1, . . . , N , we assemble the local stiffness matrices Ki

and local load vectors f i by integrating the appropriate expressions over individualsubdomains. We denote the local vectors of nodal values by ui.

We can now formulate a minimization problem with constraints given by theintersubdomain continuity conditions.

Find u ∈ W, such that

J(u) := 12u

tKu− f tu → minBu = 0

,(4)

where

u =

u1...uN

, f =

f1...fN

, and K =

K1 O · · · O

O K2. . .

......

. . .. . . O

O · · · O KN

.The matrix B = [B1, . . . , BN ] is constructed from 0, 1,−1 such that the values ofthe solution u, associated with more than one subdomain, coincide when Bu = 0.The local stiffness matrices Ki are positive semidefinite. The problem (4) is uniquelysolvable if and only if ker (K) ∩ ker (B) = 0, i.e., K is invertible on the null spaceof B. This condition holds since the original finite element model is elliptic.

By introducing a vector of Lagrange multipliers λ to enforce the constraint Bu =0, we obtain a saddle point formulation of (4):

Find (u, λ) ∈ W ×U, such that

Ku + Btλ = fBu = 0

.(5)

We note that the solution λ of (5) is in general only unique up to an additive vec-tor from ker (Bt). The space of Lagrange multipliers U is therefore chosen as therange (B); see also discussion below.

We will also use a full rank matrix, built from the rigid body motions of thesubdomains

R =

R1 O · · · O

O R2. . .

......

. . .. . . O

O · · · O RN

,


such that range (R) = ker (K); the subdomains with nonsingular stiffness matrices donot contribute any columns to R.

The solution of the first equation in (5) exists if and only if f −Btλ ∈ range (K);this constraint will lead to the introduction of a projection P . We obtain

u = K†(f −Btλ) +Rα if f −Btλ ⊥ ker (K),(6)

where K† is a pseudoinverse of K and α has to be determined; see the discussionbelow.

We assume, for the time being, that B has full rank; i.e., the constraints arelinearly independent. Substituting u into the second equation of (5) gives

BK†Btλ = BK†f +BRα.

By considering the component orthogonal to BR, we find that

PFλ = PdGtλ = e

(7)

with G := BR,F := BK†Bt,d := BK†f , P := I −G(GtG)−1Gt, and e := Rtf . Thesecond equation in (7) is a consequence of the orthogonality condition of (6). We notethat P is an orthogonal projection from U onto ker (Gt).

Any solution λ of (5) and (7) yields the same solution u of (4) and (5) if α :=−(GtG)−1Gt(d− Fλ) is chosen; see Mandel, Tezaur, and Farhat [17, Thm. 2.4].

We define the space of admissible increments by

V := µ ∈ U : µ ⊥ Bw ∀ w ∈ ker (K) = ker (Gt).

The original FETI method is a conjugate gradient method in the space V applied to

PFλ = Pd, λ ∈ λ0 +V(8)

with an initial approximation λ0 chosen such that Gtλ0 = e. To introduce pre-conditioned variants, let D be a diagonal matrix. Then, the preconditioner M−1,introduced in Farhat, Mandel, and Roux [6], is of the form

M−1 = B

[O OO D−1SD−1

]Bt

with

S =

S1 O · · · O

O S2. . .

......

. . .. . . O

O · · · O SN

.The matrix S is the Schur complement of K obtained by eliminating the interiordegrees of freedom of each of the subdomains. This computation clearly can becarried out in parallel and results in the block-diagonal matrix S which operates onlyon the degrees of freedom on the subdomain boundaries. In the application ofM−1 toa vector, N independent Dirichlet problems have to be solved in each iteration step;M−1 is therefore often called the Dirichlet preconditioner. The simplest choice for D


is the identity matrix; this choice is made for the original Dirichlet preconditioner asintroduced in Farhat, Mandel, and Roux [6]. Another possibility, which leads to fasterconvergence, is to choose D as a diagonal matrix, where the diagonal elements equalthe number of subdomains to which the interface node belongs. This multiplicityscaling (MS) is discussed in Rixen and Farhat [22, 23]; see also further discussion in[14].

To keep the search directions of this preconditioned conjugate gradient method inthe space V, the application of the preconditionerM−1 has to be followed by anotherapplication of the projection P . Hence, the Dirichlet variant of the FETI method isthe conjugate gradient algorithm applied to the preconditioned system

PM−1PF λ = PM−1P d, λ ∈ λ0 +V,(9)

with an initial approximation λ0 chosen such that Gλ0 = e. We note that, for λ ∈ V,PM−1PF λ = PM−1P tP tFP λ and that we can therefore view the operator on theleft-hand side of the preconditioned FETI system as the product of two symmetricmatrices as required for the conjugate gradient algorithm.

4. The block-diagonal preconditioner. Using the decomposition λ = λ0 +µ,with µ ∈ V , we can rewrite (8) as

PBK†Btµ = PBK†(f −Btλ0).(10)

Since u = K†(f −Btλ) +Rα and Bu = 0, we see that the solution of (10) satisfies[K Bt

PB O

] [uµ

]=

[f −Btλ0

0

].

We note that the elimination of the displacement variables of this system, by applyingK† to the first equation, gives us (10). Using that µ ∈ V, i.e., Pµ = µ, and thatP t = P, we can make the system matrix symmetric such that[

K (PB)t

PB O

] [uµ

]=

[f −Btλ0

0

].(11)

The displacement variable is not uniquely defined by this system; we are enforcingPBu = 0 but not Bu = 0. Given any solution u of (11), we obtain a solution thatsatisfies all the requirements by computing u−R(GtG)−1GtBu; this is quite similarto the choice of the null space component of (6).

For the solution of the saddle point problem (11), we propose a preconditionedconjugate residual method with a block-diagonal preconditioner. For a detailed de-scription of this algorithm, see Hackbusch [9] or Klawonn [11, 12]. We note that thisalgorithm will be designed such that the first component of each iterate belongs torange (K).

Our preconditioner has the form

B =

[K O

O M

].

Here K is assumed to be symmetric and a good preconditioner for K + DH Q,


where

Q =

Q1 O · · · O

O Q2. . .

......

. . .. . . O

O · · · O QN

,with Qi the mass matrices associated with the mesh on Ωi and DH = diagNi=1(H

−2i Ii)

is a diagonal matrix. Here Hi denotes the diameter of the subdomain Ωi. We furtherassume that M is symmetric and a good preconditioner for M ; i.e., we assume thereexist constants k0, k1,m0,m1 with 0 < c ≤ m0 ≤ m1 ≤ C < ∞ and 0 < c ≤ k0 ≤k1 ≤ C <∞, such that

k0 ut(K +DHQ)u ≤ utKu ≤ k1 u

t(K +DHQ)u ∀u ∈ W,

m0 λtMλ ≤ λtMλ ≤ m1 λ

tMλ ∀λ ∈ V.(12)

Here c and C are generic constants independent of, or only weakly dependent on, themesh size. Because of the block-diagonal structure of K andM and the precondition-ers, c and C do not depend on the number of subdomains.

A preconditioner is often said to be optimal when the constants c and C canbe chosen to be independent of the mesh size and the number of subdomains. Wealso note that all that is required here are preconditioners for quite benign positivedefinite problems on individual subregions and that the bounds are independent ofthe number of subdomains.

From these assumptions it is clear that our preconditioner B is symmetric positivedefinite and thus it can be used with the preconditioned conjugate residual method.In order to have a computationally efficient preconditioner, we must also assume thatK−1 and M−1 can be applied to a vector at a low cost.

To guarantee that the iterates belong to range (K), we make use of the orthogonalprojection PR onto range (K) which is given by

PR := I −R(RtR)−1Rt.

Consequently, we have that range (R) = ker (K) and we note that PR is a block matrixwith a 3×3 block for each interior subdomain; the expense of applying PR to a vectoris therefore very modest.

Our domain decomposition method is now the conjugate residual algorithm ap-plied to the preconditioned system

B−1Ax = B−1F

with

A =

[K (PB)t

PB O

], B−1 =

[PRK

−1P tR O

O PM−1P t

],(13)

x =

[uµ

], F =

[f −Btλ0

0

].

We note that it is easy to see that only two matrix-vector products with the projectionP and one with the projection PR are required in each step. We note that the iteratesof the conjugate residual method belong to WR ×V with WR := range (K).


5. Analysis. In this section, we will work with both finite element functionsand vectors of nodal values representing them. We will make no distinction betweenoperators and their matrix representation.

We will use the spaces W, U, and V, and we begin by defining a norm ‖ · ‖Wand a seminorm | · |W on W by

‖v‖2W :=

N∑i=1

‖vi‖2H1(Ωi)

, |v|2W :=

N∑i=1

|vi|2H1(Ωi),

where ‖vi‖2H1(Ωi)

:= |vi|2H1(Ωi)+ 1

H2i

‖vi‖2L2(Ωi)

for v = [v1, . . . ,vN ] ∈ W. We also

need the orthogonal decomposition of W into WR := range (K) and its orthogonalcomplement W⊥

R := ker (K), i.e.,

W = WR ⊕W⊥R .

For the analysis we need to consider the trace space WΓ of W. We equip thespace WΓ with the seminorm and norm

|wΓ|2WΓ:=

N∑i=1

|wΓ,i|2H1/2(∂Ωi), ‖wΓ‖2

WΓ:= |wΓ|2WΓ

+

N∑i=1

1

Hi‖wΓ,i‖2

L2(∂Ωi).

We also define

BΓ : WΓ → U BΓ := B|WΓ.

From the construction of the jump operator B it is clear that

Bw = BΓwΓ for w ∈ W where wΓ = w|Γ ,

since continuity is enforced only across the interface. As a consequence, we obtainker (Bt) = ker (Bt

Γ). A direct computation yields

F = BK†Bt = BΓS†Bt

Γ.

Let us now introduce a norm on V by ‖λ‖V := |BtΓλ|WΓ

. This defines a norm,and not only a seminorm, since range (Bt

Γ) ⊥ ker (S). We will also use the dual space

V′with a norm defined by ‖λ‖V ′ := supµ∈V

(λ,µ)‖µ‖V

. For the sake of simplicity, we will

identify the space V′with V, but we will use both norms. The above definitions are

essentially the same as in [16].Let us also define the product space X := W×V which we equip with the graph

norm ‖x‖X :=√‖v‖2

W + ‖λ‖2V ′ for x = (v, λ) ∈ X. We also need the subspace

XR := WR ×V with the same norm as X.

5.1. FETI for linear elasticity. For scalar, second order elliptic equations, ithas been shown by Mandel and Tezaur [16] that the condition number of the FETImethod with the original Dirichlet preconditioner (D = I) satisfies

κ(PM−1PF ) ≤ C (1 + log(H/h))3.

We note that (H/h)2 is proportional to the number of degrees of freedom of a subdo-main. An improved result for a family of Dirichlet preconditioners using multiplicity


scaling, which leads to a C (1+log(H/h))2 estimate, is given in Klawonn and Widlund[14]; this work is also extended to problems with discontinuous coefficients.

In this section, we extend the results of Mandel and Tezaur [16] from scalar ellipticequations to the system of linear elasticity. To be able to use the results of [16], wenow introduce the vector-valued Laplacian to be used only in our analysis. We denoteby

K∆ :=

K∆,1 O · · · O

O K∆,2. . .

......

. . .. . . O

O · · · O K∆,N

a block-diagonal matrix, where the local stiffness matrices K∆,i, i = 1, . . . , N, areobtained from the discretization of the inner product (∇u,∇u)H1(Ωi) using the samefinite element space as for a(·, ·). We denote by S∆ the Schur complement of K∆

obtained by eliminating the interior variables of each Ωi, just as in the case of K andS.

It is well known that

(S∆uΓ,uΓ) = minv∈W

v|Γ=uΓ

(K∆v,v) = (K∆vharm,vharm)

for uΓ ∈ WΓ. Here vharm ∈ W is the discrete harmonic extension of uΓ ∈ WΓ

defined as the unique solution of

(K∆vharm,v) = 0 ∀ v ∈ W0 with vharm|Γ = uΓ,

where W0 is the subspace of W with elements that vanish on the interface Γ.Returning to the elasticity case, it can also easily be seen that

(SuΓ,uΓ) = minv∈W

v|Γ=uΓ

(Kv,v) = (Kvelast,velast)

for uΓ ∈ WΓ. Here velast ∈ W is the extension of uΓ ∈ WΓ with the smallest elasticenergy defined as the unique solution of

(Kvelast,v) = 0 ∀ v ∈ W0 with velast|Γ = uΓ.

Using these formulas and the continuity of a(·, ·), we find that for uΓ ∈ WΓ,

(SuΓ,uΓ) = minv∈W

v|Γ=uΓ

(Kv,v)

≤ (Kvharm,vharm)

≤ C (K∆vharm,vharm)

= C minv∈W

v|Γ=uΓ

(K∆v,v)

= C(S∆uΓ,uΓ).

To bound (SuΓ,uΓ) from below, we restrict ourselves to a subspace. Using theleft inequality in Lemma 4, we obtain for uΓ ∈ range (S),


(S∆uΓ,uΓ) = minv∈W

v|Γ=uΓ

(K∆v,v)

≤ (K∆velast,velast)

≤ C (Kvelast,velast)

= C minv∈range (K)v|Γ=uΓ

(Kv,v)

= C(SuΓ,uΓ).

Since (S∆uΓ,uΓ) and |uΓ|2WΓare equivalent for uΓ ∈ range (S), we have proven

the following.Lemma 5. There exist constants 0 < c ≤ C < ∞, independent of the mesh size

and the subdomain diameters, such that

c |uΓ|2WΓ≤ (SuΓ,uΓ) ≤ C |uΓ|2WΓ

∀ uΓ ∈ range (S).

The next lemma follows from Mandel and Tezaur [16, proof of Lem. 3.11].Lemma 6.

infλ∈V

supwΓ∈WΓ

(λ,BΓwΓ)

‖λ‖V ′‖wΓ‖WΓ

≥ C (1 + log(H/h))−1/2,

supλ∈V

supwΓ∈WΓ

(λ,BΓwΓ)

‖λ‖V ′‖wΓ‖WΓ

≤ C (1 + log(H/h)).

The next result is essentially an extension of that lemma to the system of equationsof linear elasticity.

Lemma 7. There exist constants 0 < c ≤ C < ∞, independent of the mesh sizeand the number of subdomains, such that

c(1 + log(H/h))−1‖λ‖2V ′ ≤ (Fλ, λ) ≤ C(1 + log(H/h))2‖λ‖2

V ′ ∀ λ ∈ V.Proof. As shown in [16, proof of Lem. 3.11], we have for λ ∈ V

(Fλ, λ) = supwΓ∈range (S)

(λ,BΓwΓ)2

(SwΓ,wΓ).

Using Lemma 5 and that BtΓλ ⊥ ker (S) for λ ∈ V, we obtain

(Fλ, λ) ≤ C supwΓ∈range (S)

(λ,BΓwΓ)2

|wΓ|2WΓ

≤ C supwΓ∈range (S)

(λ,BΓwΓ)

infzΓ∈ker (S)

‖wΓ + zΓ‖2WΓ

= C supwΓ=wΓ+zΓ∈WΓ

wΓ∈range (S),zΓ∈ker (S)

(λ,BΓwΓ)

‖wΓ + zΓ‖2WΓ

= C supwΓ∈WΓ

(λ,BΓwΓ)2

‖wΓ‖2WΓ

.

Analogously, we obtain

(Fλ, λ) ≥ c supwΓ∈WΓ

(λ,BΓwΓ)2

‖wΓ‖2WΓ

.

The bounds of (Fλ, λ) now follow from Lemma 6.Combining the definitions of the (exact) Dirichlet preconditioner M−1 and of the

norm ‖ · ‖V with Lemma 5, we obtain the next lemma.


Lemma 8. There exist constants 0 < c ≤ C < ∞, independent of the mesh sizeand the number of subdomains, such that

c ‖λ‖2V ≤ (M−1λ, λ) ≤ C ‖λ‖2

V ∀ λ ∈ V.A condition number estimate of PM−1PF follows easily from these estimates;

cf. also [16]. The proof for the algorithm using an inexact Dirichlet solver proceedsalong very similar lines.

Theorem 9. There exists a positive constant C, independent of the mesh sizeand the number of subdomains, such that

κ(PM−1PF ) ≤ C (1 + log(H/h))3,

where κ(PM−1PF ) := σmax/σmin is the spectral condition number defined by theratio of the largest and the smallest eigenvalue σmax and σmin of PM

−1PF .Similarly, there exists a positive constant C, independent of the mesh size and the

number of subdomains, such that

κ(PM−1PF ) ≤ C (1 + log(H/h))3,

for any preconditioner M−1 that is spectrally equivalent to the exact Dirichlet precon-ditioner.

We note that Tezaur [27] established a condition number estimate of the formC(1 + log(H/h))2 for an algebraic FETI method developed by Park and Felippa [20]and Park, Justino, and Felippa [21]; see also the discussion in Rixen et al. [24].

Recently, a modified family of Dirichlet preconditioners, which includes the onewith multiplicity scaling for both redundant and nonredundant Lagrange multiplierswas introduced and analyzed by the authors in [14]. Using redundant Lagrange multi-pliers amounts to choosing the maximal number of constraints, pairwise connecting alldegrees of freedom which belong to the same point x, and using a Lagrange multiplierfor each possible pair. In contrast, nonredundant Lagrange multipliers are obtainedby choosing the minimal number which ensures continuity of the displacements ateach point on the interface.

For the case of nonredundant Lagrange multipliers, our preconditioner is of theform

M−1nr := (BBt)−1B

[O OO S

]Bt(BBt)−1 = (BΓB

tΓ)

−1BΓSBtΓ(BΓB

tΓ)

−1

and for the redundant case, we have

M−1r :=M−1 = B

[O OO D−1SD−1

]Bt = BΓD

−1SD−1BtΓ,

where D is defined by multiplicity scaling; cf. section 3. We note that in the caseof discontinuous coefficients, the preconditioners Mnr and Mr have to be enhancedusing a more elaborate scaling; we refer to [14] for a more detailed discussion.

In [14], condition number estimates of C (1 + log(H/h))2 are given for second

order scalar elliptic equations for both the preconditioners Mnr and Mr. We notethat the construction and analysis in [14] also hold for problems with discontinuouscoefficients. In [14, proof of Thm. 1], we show in the nonredundant case

supwΓ∈range (S)

〈λ,BΓwΓ〉|wΓ|2S

≥ 〈Mnrλ, λ〉,

supwΓ∈range (S)

〈λ,BΓwΓ〉|wΓ|2S

≤ C (1 + log(H/h))2 〈Mnrλ, λ〉.(14)


Similar inequalities can be obtained using Mr. In that paper, [14, Lem. 3], we define

a norm on V in terms of the preconditioner Mnr, i.e., ‖λ‖V := 〈M−1nr λ, λ〉; the roles

of ‖λ‖V and ‖λ‖V ′ are reversed in [14] compared to this paper. The dual norm ‖λ‖V ′

can now be defined, as before, using the new V -norm. A simple computation givesλ‖2

V ′ = 〈Mnrλ, λ〉. The same arguments are equally valid for Mr in the redundantcase.

Using the inf-sup and sup-sup estimates (14), Lemma 5, and the definition of‖λ‖V ′ , we obtain a condition number estimate for the FETI preconditioners for linearelasticity, using the same arguments as in the proof of Theorem 9. We note thatthe improved condition number estimate is a result of the optimality of the inf-supconstant in the first inequality in (14).

Theorem 10. There exists a positive constant C, independent of the mesh sizeand the number of subdomains, such that

κ(PM−1PF ) ≤ C (1 + log(H/h))2,

where M is either Mr or Mnr.

5.2. Analysis of the block-diagonal preconditioner. In this section, we givea condition number estimate for the block-diagonal preconditioner for the systems ofequations arising from linear elasticity. This results in a convergence estimate for thepreconditioned conjugate residual method.

As shown in section 4, the system of equations (11) involves an operator fromW×V onto itself. The component of u in ker(K) is determined after the completionof the iteration. It is therefore appropriate to consider the restriction of the operatorA to the subspace XR = WR ×V. Similarly, we can view the preconditioner B−1 asa mapping from XR onto itself; see (13).

An upper bound for the convergence rate of the conjugate residual method canbe given in terms of the condition number κ(B−1A) of the preconditioned system. Atheory of block-diagonal preconditioners for saddle point problems of different originshas been developed by several authors; see Rusten and Winther [25], Silvester andWathen [26], Kuznetsov [15], and Klawonn [12]. To the best of our knowledge, thefirst proof for block-preconditioners applied to saddle point problems with a singularblock K is given in Klawonn [10, 11] and independently in Arnold, Falk, and Winther[1]. In order to obtain a condition number estimate for B−1A, we follow the shortargument given in [1] which is in the same spirit as that given by Mandel and Tezaur[16, Lem. 3.1] for the positive definite case. For completeness, we include the shortproof here using our notation and the norms of XR and its dual. Denoting by ρ(·)the spectral radius of a matrix, we find

κ(B−1A) = ρ(B−1A)ρ((B−1A)−1)

≤ ‖B−1A‖XR→XR‖(B−1A)−1‖XR→XR

≤ ‖B−1‖X′R→XR

‖A‖XR→X′R‖B‖XR→X

′R‖A−1‖X′

R→XR

.

Hence, we need estimates of the norms of the operators B,A, and their inverses.The next lemma is due to Brezzi [2] and is given here in our notation.Lemma 11. Let B : WR → V satisfy an inf-sup and a sup-sup condition, i.e.,

infλ∈V

supw∈WR

(λ,Bw)

‖λ‖V ′‖w‖W ≥ β0,

supλ∈V

supw∈WR

(λ,Bw)

‖λ‖V ′‖w‖W ≤ β1,(15)


where β0, β1 > 0.Furthermore, let K : WR → WR be a symmetric operator satisfying

|(Kw,v)| ≤ α1‖w‖W ‖v‖W ∀ w,v ∈ WR,(Kw,w) ≥ α0‖w‖2

W ∀ w ∈ WR,(16)

where α0, α1 > 0.Then, A : XR → XR is an isomorphism with ‖A‖XR→X

′R

≤ C(α1, β1) and

‖A−1‖X′R→XR

≤ C(α0, α1, β0), where C(α1, β1) := α1 + β1 and C(α0, α1, β0) :=

max(α−10 + β−1

0 (1 + α1/α0)), (β−10 + α1β

−20 )(1 + α1/α0).

The uniform boundedness and ellipticity of K on WR follow directly from thedefinition of the norm ‖ · ‖W . Thus, we obtain constants α0, α1 > 0 which areindependent of h,H.

We are left with showing the inf-sup and sup-sup conditions for B.Lemma 12.

infλ∈V

supw∈WR

(λ,Bw)

‖λ‖V ′‖w‖W ≥ C (1 + log(H/h))−1/2 =: β0,

supλ∈V

supw∈WR

(λ,Bw)

‖λ‖V ′‖w‖W ≤ C (1 + log(H/h)) =: β1.

Proof. For λ ∈ V, let us now consider

supw∈WR

(λ,Bw)

‖w‖W ≤ supw∈WR

(λ,Bw)

|w|W≤ C sup

w∈WR

(λ,Bw)

(Kw,w)1/2

= C supw∈WR

(λ,BΓwΓ)

infv ∈W

v|Γ=wΓ

(Kv,v)1/2

= C supwΓ∈range (S)

(λ,BΓwΓ)

(SwΓ,wΓ)1/2

≤ C supwΓ∈WΓ

(λ,BΓwΓ)

‖wΓ‖WΓ

≤ β1 ‖λ‖V ′

with β1 := C (1+log(H/h)). The last inequality follows from Lemma 6. Analogously,we obtain

supw∈WR

(λ,Bw)

‖w‖W ≥ β0 ‖λ‖V ′

with β0 := C (1 + log(H/h))−1/2.Combining these results with Lemma 11, we obtain estimates of the norm of A

and A−1 as follows.Lemma 13. The operator A : XR → XR is an isomorphism and satisfies

‖A‖XR→X′R≤ C (1 + log(H/h)),

‖A−1‖X′R→XR

≤ C (1 + log(H/h)),


where C > 0 is a generic constant independent of the mesh size and the subdomaindiameters.

The next lemma provides bounds for ‖B‖XR→X′R

and ‖B−1‖X′R→XR

.

Lemma 14. There exist constants 0 < c ≤ C < ∞, which depend only onk0, k1,m0, and m1, such that

‖B‖XR→X′R≤ C, ‖B−1‖X′

R→XR

≤ 1

c.

These bounds are uniform with respect to the mesh size and the number of subdomainsif B is an optimal preconditioner, i.e., if the constants k0, k1,m0,m1 are uniformlybounded.

Proof. From the first inequalities of (12) and Korn’s second inequality (Lemma

2), we obtain by a standard scaling argument that (Ku,u) and ‖u‖2W are spectrally

equivalent u ∈ WR. From the second inequalities in (12) and Lemma 5, it followsthat for µ ∈ V

(M−1µ, µ) ≤ C (M−1µ, µ) = C (BΓSBtΓµ, µ) ≤ C |Bt

Γµ|2WΓ= C ‖µ‖2

V .

In the last inequality, we have used that µ ∈ V implies BtΓµ ∈ range (S). Analogously,

we obtain

(M−1µ, µ) ≥ c ‖µ‖2V .

By using these inequalities, we get for λ ∈ V

‖λ‖2V ′ = sup

µ∈V

(λ, µ)2

‖µ‖2V

≤ C supµ∈V

(λ, µ)2

(M−1µ, µ)= C sup

ν∈V

(λ, M1/2ν)2

(ν, ν)= C (Mλ, λ)

and analogously

‖λ‖2V ′ ≥ c (Mλ, λ).

From the definition of B follows

(Bx,x) = (Ku,u) + (Mλ, λ) ≤ C(‖u‖2W + ‖λ‖2

V ′ ) = C ‖x‖2X

with x = (u, λ) ∈ XR and

(Bx,x) ≥ c ‖x‖2X .

The boundedness of B and B−1 follows by using the following formulas:

‖B‖XR→X′R

= supx∈XR

(Bx,x)‖x‖2

X

,

‖B−1‖−1

X′R→XR

= infx∈XR

(Bx,x)‖x‖2

X

.

From Lemmas 13 and 14 the next theorem follows.Theorem 15.

κ(B−1A) ≤ C (1 + log(H/h))2


1.0E5

1.0E5

H

H

Fig. 1. Sample domain decomposition of the cantilever with 16 subdomains.

with a constant C independent of H,h.We can also improve the results of Theorem 15 by using the results of [14], just

as Theorem 10 represents an improvement of Theorem 9. We see that the inf-supestimate used in the proof of Lemma 12 can be replaced with the improved constantbound given by the first inequality of (14). Revising the arguments in the proof ofTheorem 15 in this respect, we obtain the following.

Theorem 16. There exists a constant C > 0 independent of H,h such that

κ(B−1A) ≤ C (1 + log(H/h)).

It is known that the convergence rate of the conjugate residual method dependslinearly on the condition number, whereas that of the conjugate gradient algorithmdoes depend on its square root. Thus, from the improved estimates in Theorems 10and 16, we see that both methods have an asymptotic convergence rate on the orderof (1 + log(H/h)). This is also reflected in the numerical experiments in section 6.

6. Numerical results. We have applied our domain decomposition method toa plane stress problem described in section 2. The Poisson ratio is ν = 0.3 and theelasticity modulus E = 2.1 ·1011N/m2, which models steel. The domain Ω is the unitsquare fixed on the left-hand side and free on the remainder of the boundary exceptfor the upper-right corner element. The only nonzero components of the load vectorare associated with the node at that corner; cf. Figure 1.

All our computations have been performed in MATLAB 5.3. Our Krylov subspacemethod is the preconditioned conjugate residual method with a zero initial guess. Thestopping criterion is ‖rn‖2/‖r0‖2 < 10−6, where rn and r0 are the nth and initialresiduals.

Our domain Ω is decomposed into N × N square subdomains with H := 1/N ;see Figure 1. In our implementation, we use redundant Lagrange multipliers; see theend of subsection 5.1 for a definition. This is known to yield a smaller number ofiterations. An explanation from a mechanical viewpoint is given in Rixen and Farhat[23]; see also Klawonn and Widlund [14] for an analysis and connections between thecases of redundant and nonredundant Lagrange multipliers.


Table 1Iteration counts for an increasing number of subdomains of constant size. (I) K = K+1/H2MQ

and M = M ; (II) K = K+1/H2MQ and M using ILU(0); (III) K ILU(10−3) of K+1/H2MQ and

M = M ; (IV) K using ILU(10−3) of K + 1/H2MQ and M using ILU(0); (V) FETI using Dirichletpreconditioner. MS : D = multiplicity scaling, Id : D = Identity.

H/h = 8 Iter (I) Iter (II) Iter (III) Iter (IV) Iter (V)1/h 1/H MS Id MS Id MS Id MS Id MS Id16 2 11 19 11 19 23 36 24 37 9 1432 4 17 27 17 27 33 67 33 70 13 2464 8 21 33 23 33 41 84 41 85 17 2996 12 21 39 23 39 47 89 49 93 20 32128 16 21 39 25 41 51 91 55 96 21 35

Table 2Iteration counts for a constant number of subdomains of increasing size. (I) K = K+1/H2MQ

and M = M ; (II) K = K+1/H2MQ and M using ILU(0); (III) K ILU(10−3) of K+1/H2MQ and

M = M ; (IV) K using ILU(10−3) of K + 1/H2MQ and M using ILU(0); (V) FETI using Dirichletpreconditioner. MS : D = multiplicity scaling, Id : D = Identity.

H=1/4 Iter (I) Iter (II) Iter (III) Iter (IV) Iter (V)1/h H/h MS Id MS Id MS Id MS Id MS Id16 4 15 25 15 25 28 61 28 60 12 2232 8 17 27 17 27 33 67 33 70 13 2464 16 19 29 19 29 51 99 58 102 15 27128 32 19 33 23 47 69 140 79 145 18 31

We report on two different series of experiments for different combinations ofpreconditioners K and M in order to analyze the numerical scalability of the method.In our first set of runs, we keep the dimension of the subproblems, and H/h, fixed andincrease the number of subdomains and thus the overall problem size. In a secondseries, we keep a fixed number of subdomains and increase the value of H/h resultingin a smaller h.

In order to see how our method behaves in the best possible case, we first reporton results for K = K + 1

H2Q and M = M ; cf. Tables 1 and 2 and Figures 2 and

3. For both cases, we present results for M constructed using D = I as well aswith the multiplicity scaling D = MS; cf. section 3. We also compare the iterationcounts of our new method with those of the original FETI method using the Dirichletpreconditioner M ; see Tables 1 and 2 and Figures 2 and 3. In the experiments withthe FETI algorithm, we stop the iteration when the preconditioned projected residualis smaller than 10−6‖f‖2; see Farhat and Roux [8]. In all cases, the convergence isconsiderably faster with MS; using this scaling the asymptotic convergence rate isalso reached much earlier than for D = I. For both choices of D, we obtain scalabledomain decomposition methods in both series of experiments.

To gain insight into the convergence behavior with inexact blocks K and M ,we use preconditioners based on an incomplete Cholesky factorization (ILU). In thefollowing, ILU(0) stands for an incomplete Cholesky factorization with no fill in whileILU(tol) is a threshold ILU factorization, with a threshold of tol, as provided inMATLAB 5.3; any entry in a column of the Cholesky factor L is dropped if itsmagnitude is smaller than the drop tolerance tol times the norm of its column. Wedenote by S the matrix that replaces S when ILU(0) is used to solve the Dirichlet

problems in each subdomain. Three different combinations are considered: 1. K =


20 40 60 80 100 120 1400

5

10

15

20

25

30

35

40

45

DIAG MS =

DIAG Id =

FETI MS = . .

FETI Id = .

1/h

ITE

RA

TIO

NS

20 40 60 80 100 120 1400

5

10

15

20

25

30

35

40

45

DIAG MS =

DIAG Id =

1/h

ITE

RA

TIO

NS

20 40 60 80 100 120 1400

10

20

30

40

50

60

70

80

90

100

DIAG MS =

DIAG Id =

1/h

ITE

RA

TIO

NS

20 40 60 80 100 120 1400

10

20

30

40

50

60

70

80

90

100

DIAG MS =

DIAG Id =

1/h

ITE

RA

TIO

NS

Fig. 2. Iteration counts for an increasing number of subdomains of constant size. H/h = 8.

Upper left: K = K + 1H2 Q, M = M . Upper right: K = K + 1/H2MQ and M using ILU(0). Lower

left: K ILU(10−3) of K+1/H2MQ and M = M . Lower right: K using ILU(10−3) of K+1/H2MQ

and M using ILU(0).

K + 1H2Q and

M−1 = B

[O O

O D−1SD−1

]Bt;

2. K is built with ILU(10−3) applied to K + 1H2Q and M−1 = M−1; 3. K is built

with ILU(10−3) applied to K + 1H2Q and M−1, as in combination 1., by using the

inexact Schur complement S. The computational results are given in Tables 1 and 2;cf. also Figures 2 and 3.

We also present results with a more accurate ILU decomposition based on athreshold tolerance tol = 10−6 in Table 3. We see that a more accurate preconditionerK improves the overall rate of convergence significantly. Comparing these results andthe number of iterations obtained for the new method and the original FETI algo-rithm, both with the exact Dirichlet preconditioner, there is a strong indication thatthe increase in the iteration counts, which results when using the ILU(10−3) precon-ditioner, would be largely eliminated if a better block-preconditioner, e.g., (algebraic)

multigrid, were used as K.


20 40 60 80 100 120 1400

5

10

15

20

25

30

35

40

DIAG MS =

DIAG Id =

FETI MS = . .

FETI Id = .

1/h

ITE

RA

TIO

NS

20 40 60 80 100 120 1400

5

10

15

20

25

30

35

40

45

50

55

DIAG MS =

DIAG Id =

1/h

ITE

RA

TIO

NS

20 40 60 80 100 120 1400

50

100

150

DIAG MS =

DIAG Id =

1/h

ITE

RA

TIO

NS

20 40 60 80 100 120 1400

50

100

150

DIAG MS =

DIAG Id =

1/h

ITE

RA

TIO

NS

Fig. 3. Iteration counts for a constant number of subdomains of increasing size. H = 1/4.

Upper left: K = K + 1H2 Q, M = M . Upper right: K = K + 1/H2MQ and M using ILU(0). Lower

left: K ILU(10−3) of K+1/H2MQ and M = M . Lower right: K using ILU(10−3) of K+1/H2MQ

and M using ILU(0).

Table 3Iteration counts. (I) K based on ILU(10−6) of K + 1/H2MQ and M = M ; (II) K based on

ILU(10−6) of K +1/H2MQ and M using ILU(0). MS : D = multiplicity scaling, Id : D = Identity.

H/h = 8 Iter (I) Iter (II)1/h 1/H MS Id MS Id16 2 18 30 20 3032 4 17 27 17 2764 8 21 33 23 35128 16 21 39 25 41

Acknowledgments. The first author would like to acknowledge the hospitalityof and many discussions with Xiao-Chuan Cai, Charbel Farhat, and Daniel Rixenat the University of Colorado at Boulder and also wishes to thank Daniel Rixen forproviding parts of his FETI MATLAB code.

REFERENCES

[1] D. N. Arnold, R. S. Falk, and R. Winther, Preconditioning discrete approximations of theReissner–Mindlin plate model, RAIRO Model. Math. Anal. Numer., 31 (1997), pp. 517–557.


[2] F. Brezzi, On the existence, uniqueness, and approximation of saddle-point problems arisingfrom Lagrange multipliers, RAIRO Model. Math. Anal. Numer., 8 (1974), pp. 129–151.

[3] P. G. Ciarlet, Mathematical Elasticity Volume I: Three-Dimensional Elasticity, North–Holland, Amsterdam, 1988.

[4] C. Farhat, P. Chen, and J. Mandel, A scalable Lagrange multiplier based domain decomposi-tion method for time-dependent problems, Internat. J. Numer. Methods Engrg., 38 (1995),pp. 3831–3853.

[5] C. Farhat, P. Chen, J. Mandel, and F.-X. Roux, The two-level FETI method—Part II:Extensions to shell problems, parallel implementation and performance results, Comput.Methods Appl. Mech. Engrg., 155 (1998), pp. 153–179.

[6] C. Farhat, J. Mandel, and F.-X. Roux, Optimal convergence properties of the FETI domaindecomposition method, Comput. Methods Appl. Mech. Engrg., 115 (1994), pp. 367–388.

[7] C. Farhat and F.-X. Roux, A method of finite element tearing and interconnecting and itsparallel solution algorithm, Internat. J. Numer. Methods Engrg., 32 (1991), pp. 1205–1227.

[8] C. Farhat and F.-X. Roux, Implicit parallel processing in structural mechanics, Comput.Mech. Adv., 2 (1994), pp. 1–124.

[9] W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Springer, New York,1994.

[10] A. Klawonn, An Optimal Preconditioner for a Class of Saddle Point Problems with a PenaltyTerm, Part II: General Theory, Technical report 14/95, Westfalische Wilhelms-UniversitatMunster, Germany, April 1995. Also available as Technical report 683, Courant Instituteof Mathematical Sciences, New York University, New York, 1995.

[11] A. Klawonn, Preconditioners for Indefinite Problems, Ph.D. thesis, Westfalische Wilhelms-Universitat Munster, 1995; Technical report 716, Courant Institute of Mathematical Sci-ences, New York University, New York, 1996.

[12] A. Klawonn, An optimal preconditioner for a class of saddle point problems with a penaltyterm, SIAM J. Sci. Comput., 19 (1998), pp. 540–552.

[13] A. Klawonn and O. B. Widlund, A domain decomposition method with Lagrange multipli-ers for linear elasticity, in Proceedings of the 11th International Conference on DomainDecomposition Methods, Greenwich, UK, 1998, C. H. Lai, P. E. Bjørstad, M. Cross, andO. B. Widlund, eds., Domain Decomposition Press, Bergen, Norway, 1998, pp. 49–56. Alsoavailable online from http://www.ddm.org/DD11/Klawonn.ps.gz.

[14] A. Klawonn and O. B. Widlund, FETI and Neumann–Neumann iterative substructuringmethods: Connections and new results, Comm. Pure Appl. Math., to appear.

[15] Yu. A. Kuznetsov, Efficient iterative solvers for elliptic finite element problems on nonmatch-ing grids, Russian J. Numer. Anal. Math. Modelling, 10 (1995), pp. 187–211.

[16] J. Mandel and R. Tezaur, Convergence of a substructuring method with Lagrange multipliers,Numer. Math., 73 (1996), pp. 473–487.

[17] J. Mandel, R. Tezaur, and C. Farhat, A scalable substructuring method by Lagrange mul-tipliers for plate bending problems, SIAM J. Numer. Anal., 36 (1999), pp. 1370–1391.

[18] J. Necas, Les methodes directes en theorie des equations elliptiques, Academia, Prague, 1967.[19] J. A. Nitsche, On Korn’s second inequality, RAIRO Model. Math. Anal. Numer., 15 (1981),

pp. 237–248.[20] K. C. Park and C. A. Felippa, A Variational Framework for Solution Method Developments

in Structural Mechanics, Technical report CU-CAS Report 96-21, University of Coloradoat Boulder, 1996.

[21] K. C. Park, M. R. Justino, and C. A. Felippa, An algebraically partitioned FETI method forparallel structural analysis: Algorithm description, Internat. J. Numer. Methods Engrg.,40 (1997), pp. 2717–2737.

[22] D. Rixen and C. Farhat, Preconditioning the FETI and balancing domain decompositionmethods for problems with intra- and inter-subdomain coefficient jumps, in Proceedingsof the Ninth International Conference on Domain Decomposition Methods in Science andEngineering, Bergen, Norway, 1996, P. Bjørstad, M. Espedal, and D. Keyes, eds., Do-main Decomposition Press, Bergen, Norway, 1998, pp. 472–479. Also available online fromhttp://www.ddm.org/DD9/Rixen.ps.gz.

[23] D. Rixen and C. Farhat, A simple and efficient extension of a class of substructure based pre-conditioners to heterogeneous structural mechanics problems, Internat. J. Numer. MethodsEngrg., 44 (1999), pp. 489–516.

[24] D. Rixen, C. Farhat, R. Tezaur, and J. Mandel, Theoretical comparison of the FETI andalgebraically partitioned FETI methods, and performance comparisons with a direct sparsesolver, Internat. J. Numer. Methods Engrg., 46 (1999), pp. 501–534.

[25] T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems,


SIAM J. Matrix Anal. Appl., 13 (1992), pp. 887–904.[26] D. Silvester and A. Wathen, Fast iterative solutions of stabilised Stokes systems Part II:

Using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), pp. 1352–1367.[27] R. Tezaur, Analysis of Lagrange Multiplier Based Domain Decomposition, Ph.D. the-

sis, University of Colorado at Denver, 1998; also available online from http://www-math.cudenver.edu/graduate/ thesis/rtezaur.ps.gz.

A Domain Decomposition Method With Lagrange Multipliers and

Documents

Transcript of A Domain Decomposition Method With Lagrange Multipliers and