Lagrangian Formulation of Domain Decomposition Methods a Unified Theory

Applied Mathematical Modelling 30 (2006) 593–615

www.elsevier.com/locate/apm

Lagrangian formulation of domain decomposition methods:A unified theory

Frederic Magoules a,*, Francois-Xavier Roux b

a Institut Elie Cartan de Nancy, Universite Henri Poincare, BP 239, 54506 Vandoeuvre-les-Nancy Cedex, Franceb High Performance Computing Unit, ONERA, 29 av. de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France

Received 15 April 2005; received in revised form 14 June 2005; accepted 16 June 2005Available online 22 December 2005

Abstract

Domain decomposition methods based on one Lagrange multiplier have been shown to be very efficient for solving ill-conditioned problems in parallel. Several variants of these methods have been developed in the last ten years. These vari-ants are based on an augmented Lagrangian formulation involving one or two Lagrange multipliers and on mixed typeinterface conditions between the sub-domains. In this paper, the Lagrangian formulations of some of these domain decom-position methods are presented both from a continuous and a discrete point of view.� 2005 Elsevier Inc. All rights reserved.

Keywords: Domain decomposition; Lagrange multipliers; Saddle point problem; Augmented Lagrangian; Interface conditions

1. Introduction

The development of high-performance computing algorithms for partial differential equations has been amajor event in Applied Mathematics and Engineering for the last ten years. Driven by the availability of pow-erful parallel processors and computer networks, the field of high-performance computing has matured.Beyond the possible algorithms well suited for parallel computers with distributed memory, the domaindecomposition methods [1–3] are very efficient. Research in domain decomposition methods is a very activearea, see for example [4–9] and references therein. These methods mainly consist to split a global domain intoseveral (overlapping or non-overlapping) sub-domains. The solution of the initial global problem is thenachieved through the solution of local sub-problems.

The domain decomposition methods can mainly be classified into two families: the overlapping [10–12] andthe non-overlapping [13,14] domain decomposition methods. The non-overlapping domain decomposition

0307-904X/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2005.06.016

* Corresponding author. Tel.: +33 383 684564; fax: +33 383 684534.E-mail addresses: [email protected] (F. Magoules), [email protected] (F.-X. Roux).

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594 F. Magoules, F.-X. Roux / Applied Mathematical Modelling 30 (2006) 593–615

methods usually introduce some new variables on the interface between the sub-domains and lead either to thePrimal Schur complement methods [13,15–17] or to the Dual Schur complement methods [18–20]. In the DualSchur complement methods, these new variables on the interface are called Lagrange multipliers. The domaindecomposition methods based on Lagrange multipliers have been shown to be very efficient for solving ill-con-ditioned problems in parallel [18–21].

The first domain decomposition method based on Lagrange multipliers involved one Lagrange multiplierand Neumann interface conditions on the interface between the sub-domains [18]. Recent developments haveextended the initial method to two Lagrange multipliers and mixed interface conditions on the interface [22].For this purpose an augmented Lagrangian formulation has been considered, as first introduced in [23]. Usingsuitable mixed type interface conditions on the interface between the sub-domains ensure the well-posedness ofthe local sub-problems. Several mixed interface conditions have then been investigated for acoustics or linearelasticity problems [22]. Similar work have been performed for the overlapping Schwarz method [24–27], andfor the non-overlapping Schwarz method [28–32]. These interface conditions used on the interface between thesub-domains have similar behavior as the absorbing boundary conditions defined on the boundary of the glo-bal domain.

This paper presents some domain decomposition methods based on Lagrange multipliers. The Lagrangianformulations with one or two Lagrange multipliers, and the Neumann or mixed type interface conditions aredescribed. The mathematical properties of the discrete linear system issue from these methods are presentedtoo. The discretization of these methods, the algorithm considered and their parallel implementation aredescribed.

This paper does not pretend to perform a review of all the domain decomposition methods, but is ratherdevoted to presenting a nice framework with (augmented) Lagrangian of some domain decomposition meth-ods. It does not bring any new results on the convergence analysis of these methods, but presents an originalelegant and unique way to formulate these domain decomposition methods. For the sake of clarity, the anal-ysis is performed on the Laplace equation only, but the results can be extended to other (real) elliptic prob-lems, like linear elasticity for example.

The scope of this paper is the following. In Section 2, the way to introduce one Lagrange multiplier on theinterface between the sub-domains is presented. The saddle point problem and the variational hybrid formu-lation derived from this saddle point problem are analyzed. Finally the case of possible cross points in themesh partitioning, i.e. points where more than two sub-domains intersect, is discussed. Section 3 analyzesthe interface operator and the interface problem arising from the variational hybrid formulation. The casesof ill-posed local problems are investigated and the projected condensed interface problem is explained. In Sec-tion 4 the discrete analysis is performed. Section 5 describes the parallel implementation of this algorithm withmessage passing environment. Section 6 presents the implementation of a coarse grid correction and its inter-pretation as a global preconditioning technique. Section 7 presents the way to define mixed type interface con-ditions on the interface between the sub-domains, through an augmented Lagrangian formulation. Section 6presents the two Lagrange multipliers method, with Neumann or mixed type interface conditions. Finally, Sec-tion 9 contains the conclusions of this paper.

2. Introducing a Lagrange multiplier on interface of non-overlapping domain splitting

2.1. Non-overlapping domain splitting

Let us consider a partial differential equation on a spatial open bounded domain X split into a set of non-overlapping sub-domains Xi. The intersection of the boundaries of the sub-domains defines the interfacebetween sub-domains:

C ¼[

ij

Cij; with Cij ¼ oXi \ oXj; i 6¼ j.

The boundary of each sub-domain is split in two parts, the ‘‘external’’ boundary:

Cexti ¼ oXi \ oX

F. Magoules, F.-X. Roux / Applied Mathematical Modelling 30 (2006) 593–615 595

and the sub-domain interface:

Ci ¼ oXi \ C ¼[

j

Cij.

In this paper the boundaries and interfaces of the sub-domains are assumed to have the Lipschitz regularity. Inthe sequel of the paper, the restriction on each interface Cij of a field m defined on C is denoted as mij, and itsrestriction on Ci is denoted as mi. Although Cij and Cji are identical, mij and mji may be different. So, the restric-tion of mi on Cij, mij, may be different from the restriction of mj on Cji, mji.

2.2. Introduction of Lagrange multipliers on interfaces

The Laplace equation is considered in a domain X, with homogeneous Dirichlet boundary conditions onthe boundary. This problem can be expressed as

� Du ¼ f in X; ð2:1Þu ¼ 0 on oX. ð2:2Þ

Since this problem is a second order coercive elliptic problem, it can be shown that the solution u is the fieldthat minimizes the energy:

JðuÞ ¼ 1

2

ZXruru dV �

ZX

fudV ð2:3Þ

in the space of admissible fields, that is in this case H 10ðXÞ, where dV denotes the volume integral. The follow-

ing Sobolev spaces and subspaces are considered: H1(X) denotes the space of functions belonging to L2ðXÞ andwith their gradient belonging to L2ðXÞ; H

12ðoXÞ denotes the space of the trace on oX of functions belonging to

H 1ðXÞ; H1200ð� Þ denotes the space of the restriction on of the trace on oX of functions belonging to H1(X) and

with zero value on oXn� . The interested reader could find more explanations and details about these spaces inreferences [33,34].

Lemma 2.1. The functions ui defined on each sub-domain Xi which minimizes the sum of the local energies:
Xi
J iðuiÞ ¼X

i

1

2

ZXi

ruirui dV �Z

Xi

fiui dV� �

ð2:4Þ

under the interface continuity constraint:

ui � uj ¼ 0 on Cij 8ij; ð2:5Þ
where ui is the restriction on each sub-domain Xi of the solution u of the global problem (2.1), (2.2).
Proof. The functions ui 2 H 10Cext

iðXiÞ (the functions belonging to H1(Xi) equal to zero on Cext

i ) which satisfy theconstraint (2.5) are the restriction in Xi of a function u 2 H 1

0ðXÞ, and Eq. (2.3) is equal to Eq. (2.4). h

Lemma 2.2. The continuity constraint (2.5) may be written in a variational form, by introducing a space of

Lagrange multipliers on each interface Cij.
ZCij
ðui � ujÞkij dS ¼Z

Cji

ðuj � uiÞkji dS ¼ 0 8kij 2 H�12ðCijÞ; ð2:6Þ

where kji = �kij and where dS denotes the surface integral.

Proof. The proof is obvious since the space of Lagrange multipliers is the dual space of the jumps on the inter-face of local admissible fields. When there is no cross-points on the interface, as supposed Eq. (2.6)—as illus-trated on one example Fig. 1—this space is H�

12ðCÞ, the dual space of H

1200ðCÞ. A cross-point is a point that

belongs to more than two sub-domains, and so to more than one interface Cij. When cross-points exist onthe interface, the space of Lagrange multipliers must be the dual space of the jumps on the interface of local

admissible fields which is H12locðCÞ. h

Ω1

Γ

Ω2 Ω

12 23

3

Γ

Fig. 1. Decomposition in three sub-domains without cross-points.


If a sub-domain of the domain decomposition is in contact with a Dirichlet boundary condition i.e.Cext

i 6¼ ;, Ji(ui) is coercive on H 10Cext

iðXiÞ, and therefore

PiJ iðuiÞ is coercive on

QiH

10Cext

iðXiÞ. In the general case,

when some sub-domains are in contact or not with a Dirichlet boundary condition,P

iJ iðuiÞ ¼ JðuÞ on thesubspace of functions of

QiH

10Cext

iðXiÞ that satisfy the continuity condition (2.5) because this subspace is

H 10ðXÞ. The coercivity of

PiJ iðuiÞ is then always satisfied on the subspace of functions of

QiH

10Cext

iðXiÞ that

satisfy the continuity condition (2.5), even if Ji(ui) was not coercive in one sub-domain Xi.

Lemma 2.3. The following conditions are satisfied.

• The bilinear form:
Xi
ZXi

ruirui dV� �

is coercive on the subspace ofQ

iH10Cext

iðXiÞ of fields satisfying the continuity condition (2.5).

• The Ladyzhenskaya Babuska Brezzi condition is satisfied:

infl2H�

12ðCÞ

supv2Q

iH1

0Cexti

ðXiÞ

12

Pij

RCijðui � ujÞlij dS

h ikvkQ

i

H10Cext

i

ðXiÞklkH�12ðCÞ

P a > 0. ð2:7Þ

Proof. As discuss previously, the first property is satisfied, since the subspace of functions ofQ

iH10Cext

iðXiÞ that

satisfy the continuity condition (2.5) is identical to H 10ðXÞ. The second property is equivalent to

infl2H�

12ðCÞ

supv2Q

iH1

0Cexti

ðXiÞ

Pi

RCi

uili dSh i

kvkQi

H1

0Cexti

ðXiÞklkH�12ðCÞ

P a > 0;

because each interface Cij is shared by two sub-domains and because kij = �kji. Using this equivalent formu-lation, Eq. (2.7) can be checked, since the trace on Ci of functions belonging to H 1

0CextiðXiÞ is H

1200ðCiÞ whose dual

space is H�12ðCiÞ. h

Theorem 2.1. Finding the solution of the minimization problem (2.4) under the constraint (2.6) is equivalent to

finding the saddle point of the following Lagrangian:

Lðu; kÞ ¼X

i

1

2

ZXi

ruirui dV �Z

Xi

fiui dV� �

þ 1

2

Xij

ZCij

ðui � ujÞkij dS

" #

¼X

i

1

2

ZXi

ruirui dV �Z

Xi

fiui dV� �

þX

i

ZCi

uiki dS� �

. ð2:8Þ

The second form of L derives from the fact that each interface Cij is shared by two sub-domains and that kij = �kji.

Proof. Lemma 2.1, Lemma 2.2, and Lemma 2.3 give the hypothesis required for the proof of this theorem.The interested reader can find the complete detailed proof in [34]. h


The variational principle derived from this saddle point problem is the following hybrid system ofequations:
Z
Xi

ruirvi dV ¼Z

Xi

fivi dV þZ

Ci

kivi dS 8vi 2 H 10Cext

iðXiÞ; ð2:9ÞZ

Cij

ðui � ujÞlij dS ¼ 0 8ij 8lij 2 H�12ðCijÞ. ð2:10Þ

Theorem 2.2. The hybrid system of equations (2.9), (2.10) has a unique solution that is also the solution of the

saddle point problem of Lagrangian (2.8).

Proof. The classical theory [34] on mixed or hybrid formulation shows that the hybrid system of equations(2.9), (2.10) has a unique solution that is also to find the couple solution of the saddle point problem ofLagrangian (2.8) if both conditions of Lemma 2.3 are satisfied, which is indeed the case. h

2.3. Interpretation of the saddle point problem

The saddle point of the Lagrangian (2.8) is the couple of fields (u, k) that satisfies:

Lðu; lÞ 6 Lðu; kÞ 6 Lðv; kÞ 8v 2Y

i

H 10Cext

iðXiÞ 8l 2 H�

12ðCÞ.

The left inequality means:

1

2

Xij

ZCij

ðui � ujÞlij dS

" #6

1

2

Xij

ZCij

ðui � ujÞkij dS

" #8l 2 H�

12ðCÞ.

The left hand side of this inequality can be bounded only if:

1

2

Xij

ZCij

ðui � ujÞlij dS

" #¼ 0 8l 2 H�

12ðCÞ;

which means that the set of fields ui satisfies the interface continuity constraint (2.5). Hence, this set forms afield u belonging to H 1

0ðXÞ.From the right inequality, it results that for any field v in H 1

0ðXÞ:
JðuÞ 6 JðvÞ.
Hence, u is the field of H 10ðXÞ that minimizes the energy, and so it is the solution of the global Laplace problem

(2.1), (2.2).It should be noted that the geometrical interpretation for the saddle point is no longer valid if the method is

applied to problems with complex-valued functions such as in scattering problems.

2.4. Interpretation of the variational hybrid formulation derived from the saddle point problem

Eq. (2.9) means that in each sub-domain Xi, ui is solution of the Neumann problem:

� Dui ¼ fi on Xi; ð2:11Þui ¼ 0 on Cext

i ; ð2:12Þoui

oni¼ ki on Ci. ð2:13Þ

Since kji = �kij, both Eqs. (2.11)–(2.13) and (2.5) mean that the restrictions of the global solution on the sub-domains satisfy local Neumann problems with two continuity relations:


ui ¼ uj on Cij 8ij;ouioni¼ � ouj

onjon Cij 8ij.

(

The first continuity relation is an admissibility conditions: fields ui belonging to H 10Cext

iðXiÞ form a field of

H 10ðXÞ only if they are continuous along the interfaces between sub-domains.The second continuity relation is an equilibrium condition: a field u in H 1

0ðXÞ whose restriction in each sub-domain Xi satisfying the Laplace equation does satisfy the global Laplace equation only if the fluxes on theinterfaces are matching. This is easily proved using the variational formulation of the Laplace equation in eachsub-domain:
Z
Xi

ruirvi dV ¼Z

Xi

fivi dV þZ

Ci

oui

onivi dS 8vi 2 H 1

0CextiðXiÞ.

Adding all these local equations gives the global variational equation satisfied by u:
ZXrurvdV ¼
ZX

fv dV þ 1

2

Xij

ZCij

oui

oniþ ouj

onj

� �vdS

" #8v 2 H 1

0ðXÞ;

that shows that u is solution of the global Laplace problem, whose variational formulation is
ZXrurvdV ¼
ZX

fv dV 8v 2 H 10ðXÞ

if and only if the equilibrium condition is satisfied:

oui

oni¼ � ouj

onjon Cij 8ij. ð2:14Þ

2.5. Cross-points

When the mesh decomposition has cross-points, like in Fig. 2, then, although the trace of any function vi inH 1

0CextiðXiÞ belongs to H

1200ðCiÞ, the jump of the traces defined by vi � vj on each interface Cij does not always

belong to H1200ðCÞ. This comes from the fact that although the jump is regular on each interface Cij, it may

be discontinuous at the cross-points. So the jump belongs to a space less regular than H1200ðCÞ whose dual space

is included in H�12ðCÞ. The same issue exists with the Neumann–Neumann preconditioner for the Schur com-

plement method. This preconditioner is the discretization of the mapping of functions in H1200ðCÞ onto the aver-

age value of the traces of the solution of the associated Neumann problems in each sub-domain. Whatever the

Ω1

Ω2

Ω3

23Γ

Γ12

Γ13

Fig. 2. Decomposition in three sub-domains with cross-points.


averaging operator is, the averaged trace is not always continuous at the cross-point, and so it does not alwaysbelong to H

1200ðCÞ.

For a sake of simplicity, in the sequel of the paper, the theoretical analysis of the continuous problem willbe made for the case of a mesh splitting without cross-points.

3. Interface problem

3.1. Condensed interface operator

Let us consider a splitting of the sub-domains without cross-points similar to the previous section. Now letus note D the mapping of H�

12ðCÞ onto H

1200ðCÞ defined by

Dk ¼ wi � wj on Cij 8ij;
where wi is the solution of the homogeneous Laplace equation in Xi with Neumann interface conditions on theinterface defined by ki:
� Dwi ¼ 0 on Xi; ð3:1Þwi ¼ 0 on Cext

i ; ð3:2Þowi

oni¼ ki on Ci. ð3:3Þ

Theorem 3.1. D is a compact coercive self-adjoint operator in H�12ðCÞ.

Proof. Consider a field l belonging to H�12ðCÞ. Define vi in each sub-domain Xi as the solution of the local

problem with Neumann interface conditions on the interface defined by li:

�Dvi ¼ 0 on Xi;

vi ¼ 0 on Cexti ;

ovi

oni¼ li on Ci.

8>><>>:

The variational formulation of this equation is
ZXi
rvirwi dV ¼Z

Ci

liwi dS 8wi 2 H 10Cext

iðXiÞ. ð3:4Þ

Since each interface Cij is shared by two sub-domains and lij = �lji, the following relations hold:
ZC
DkldS ¼ 1

2

Xi;j

ZCij

ðwi � wjÞlij dS

¼ 1

2

Xij

ZCij

wilij dS þ 1

2

Xij

ZCji

wjlji dS

¼X

i

ZCi

wili dS. ð3:5Þ

So, Eqs. (3.4) and (3.5) finally give the fundamental property:
ZC
DkldS ¼X

i

ZXi

rwirvi dS. ð3:6Þ

This means that D is a self-adjoint operator. Eq. (3.6) shows that it is coercive, since, by taking l = k:
ZC
DkkdS ¼X

i

ZXi

rwirwi dS.


The coerciveness can be easily derived from the properties of the mapping of ki onto the solution wi of theNeumann problem (3.1)–(3.3) that is bi-continuous from H�

12ðCiÞ on H 1

0CextiðXiÞ.

Furthermore, since the trace operator defines a continuous mapping from H10Cext

iðXiÞ onto H

12

00ðCiÞ, operatorD is a continuous mapping of H�

12ðCÞ onto H

12

00ðCÞ. From the compactness of the injection of H12

00ðCÞ inH�

12ðCÞ, the theorem holds. h

Operator D may be decomposed as the sum of local operators. In each sub-domain Xi, Di is a mapping ofH�

12ðCiÞ on H

1200ðCiÞ defined by
Z
Ci

Dikili dS ¼Z

Xi

rwirvi dS;

where wi and vi are the solutions of local Neumann problems like (3.1)–(3.3) associated with ki and li.So, if ci is the trace operator on Ci of functions belonging to H

1200ðCiÞ, and if wi is the solution of the

Neumann problem (3.1)–(3.3) associated with ki, then:

Diki ¼ ciðwiÞ on Ci:

Conversely, wi is the unique solution of Dirichlet problem associated with ci(wi):

� Dwi ¼ 0 on Xi; ð3:7Þwi ¼ 0 on Cext

i ; ð3:8Þwi ¼ ciðwiÞ on Ci. ð3:9Þ

Hence, Di is the inverse of Steklov–Poincare operator Si defined from H1200ðCiÞ into H�

12ðCiÞ by

SiðciðwiÞÞ ¼owi

onion Ci;

where wi is the solution of Dirichlet problem (3.7)–(3.9).Define Vi as the subset of H

1200ðCiÞ of functions wi on Xi that satisfy:

�Dwi ¼ 0 on Xi;

wi ¼ 0 on Cexti .

�

This leads to the following lemma:

Lemma 3.1. Si is the first order pseudo-differential operator that maps the trace on Ci of a function wi in Vi onto

its normal derivative on Ci.

Di is the first order pseudo-integral operator that maps a flux ki on Ci onto the trace on Ci of function wi in Vi

whose normal derivative on Ci is equal to ki, and Di ¼ S�1i . In other words, Di is the inverse of a first order pseudo-

differential operator.

More details on these operators can be found in [2]. It can be shown that the conditioning number of thestiffness matrix, which is associated to a second-order operator is in O(h2), where h denotes the mesh size. Theconditioning number of the discrete operator associated to Si and Di, which are first-order operators, is in O(h)[2,13]. This is one reason why the domain decomposition methods with Lagrange multipliers are so efficient[14].

Furthermore, since Di is a compact operator, and since Si is the inverse of Di, the eigenvalue repartitionof the operator Di presents some cluster around zero, rather than the eigenvalue repartition of the operatorSi has a dense spectrum for high eigenvalue [13,14]. This is one reason why the iterative solution with aKrylov method of the condensed linear system issue from the domain decomposition methods withLagrange multipliers converges faster than the primal Schur complement method as already investigatedin [13,14].


3.2. Condensed interface problem

Let us call u0i the solutions of the local problems with homogeneous Neumann interface conditions:

�Du0i ¼ fi on Xi;

u0i ¼ 0 on Cext

i ;ou0

ioni¼ 0 on Ci.

8<:

Then, if ui is the restriction of the solution of global problem (2.1), (2.2) in Xi, then wi ¼ ui � u0i satisfies the

local Neumann problem (3.1)–(3.3) with

ki ¼owi

oni¼ oui

oni� ou0

i

oni.

The continuity relation (2.5) satisfied by the set of fields ui gives the following relation between the jumps of wi

and u0i on the interface:

wi � wj ¼ u0j � u0

i on Cij 8ij: ð3:10Þ
Finally, for the set of fields wi, Eq. (3.5) combined with relation (3.10) gives:Z Z
CDkldS ¼ 1

2

Xij Cij

ðwi � wjÞlij dS 8l 2 H�12ðCÞ ð3:11Þ

¼ 1

2

Xij

ZCij

ðu0j � u0

i Þlij dS 8l 2 H�12ðCÞ. ð3:12Þ

So, k is the solution of a condensed interface problem:
Dk ¼ d; ð3:13Þ
where d is defined as
dij ¼ u0
j � u0i on Cij 8ij. ð3:14Þ

Conversely, consider k, the solution of the condensed interface problem (3.13) with d defined as in Eq. (3.14).If wi is the set of solutions of homogeneous Laplace equations (3.1)–(3.3) with Neumann interface conditionsdefined by k on the interfaces, then, fields ui ¼ u0

i þ wi satisfy local Laplace equations (2.11)–(2.13). Further-more, Eq. (3.12) and definition of d in Eq. (3.14) give the following relation:
Z
CðDk� dÞldS ¼ 1

2

Xij

ZCij

ððwi � wjÞ þ ðu0i � u0

j ÞÞlij dS 8l 2 H�12ðCÞ

¼ 1

2

Xij

ZCij

ðui � ujÞlij dS 8l 2 H�12ðCÞ

¼ 0 8l 2 H�12ðCÞ;

which means that the set of functions ui satisfy also the continuity relation (2.5) on the interface. So u is thesolution of the global problem (2.1), (2.2). So, the condensed interface problem (3.13) defines the same k as thehybrid system of equations (2.9), (2.10), fields ui being defined as the solutions of the Neumann problems(2.11)–(2.13).

This demonstration of equivalence between both problems finally proves the following theorem:

Theorem 3.2. The solution of the condensed interface problem Dk = d, with right-hand-side defined by

dij ¼ u0j � u0

i on Cij is the unique field in H�12ðCÞ for which the solution of the associated Neumann problem (2.11)–

(2.13) in each sub-domain is the restriction of the solution of the global Laplace equation (2.1), (2.2).

An equivalent formulation is: the solution of the condensed interface problem is the unique field for which the

solutions of the associated local Neumann problems are continuous along the interfaces.

3.3. The case of ill-posed local problems

It may happen that for some sub-domains Xi, Cexti ¼ ;, and so the local Neumann problems are ill-posed

[14,23]. Nevertheless, for such a sub-domain, the restriction of the solution of the global problem does satisfythe local Neumann problem (2.11)–(2.13) with ki ¼ oui

oni. The variational formulation of this problem is


ZXi

ruirvi dV ¼Z

Xi

fivi dV þZ

Ci

kivi dS 8vi 2 H 1ðXiÞ. ð3:15Þ

This problem has a unique solution in H 1ðXiÞ=R, if and only if its right-hand-side satisfies the compatibilityrelation derived from the variational equation (3.15) with $vi = 0:
Z
Xi

fivi dV þZ

Ci

kivi dS ¼ 0. ð3:16Þ

Since the global problem has a solution, k defined by ki ¼ ouioni

on each sub-domain interface does satisfy theconstraint (3.16). Each solution of problem (3.15) in H1(Xi) differs from the restriction ui of the global problemby a constant.

The same development as in Section 3.1 can be made to define the condensed interface operator, by replac-ing H 1

0CextiðXiÞ by H 1ðXiÞ=R in each sub-domain where Cext

i ¼ ;, except that Dk is not defined for any k inH�

12ðCÞ, but for the fields that satisfy the homogeneous compatibility condition (3.17) in each sub-domain

Xi where Cexti ¼ ;:
Z
Ci

kivi dS ¼ 0. ð3:17Þ

Thanks to compatibility condition (3.17), local problem (3.4) can be replaced by any of the two following var-iational formulations, since two solutions of the Neumann problem just differ by a constant value:
Z
Xi

rwirwi dV ¼Z

Ci

kiwi dS 8wi 2 H 1ðXiÞ=R; ð3:18ÞZXi

rwirwi dV ¼Z

Ci

kiwi dS 8wi 2 H 1ðXiÞ. ð3:19Þ

Eq. (3.6) completely defines D in K, the subset of H�12ðCÞ consisting of the fields that satisfy the homogeneous

compatibility condition (3.17) in each sub-domain Xi where Cexti ¼ ;. The question is which solution of local

Neumann problem (3.18) and (3.19) to choose in H1(Xi), for defining properly the condensed interface prob-lem like in Section 3.2? If an arbitrary solution is taken, then, for k = 0, the solutions of the homogeneouslocal problems may be non-zero, and so these jumps on the interface may be piecewise non-zero constants.To get the right local solution, these piecewise constant jumps must be eliminated.

3.4. Projected condensed interface problem

Define G, the subspace of H1200ðCÞ consisting of the jumps of all solutions of local homogeneous Neumann

problems. In the case of Laplace equations, G is made of the jumps of possibly non-zero constant functions insub-domains in which Cext

i ¼ ;, and zero constant functions in the other ones. In particular, if Xi is a sub-domain where Cext

i ¼ ;, the jump of the fields defined as vi = 1 in Xi and vj = 0 in all the other sub-domainsis equal to 1 on Ci and equal to 0 on CnCi. Let us call gi this jump, then G is a finite dimensional subspace ofH

1200ðCÞ, and the set of functions gi, Xi being a sub-domain where Cext

i ¼ ;, is a basis of G.For a given k satisfying admissibility condition (3.17) in each sub-domain Xi where Cext

i ¼ ;, the solution ofNeumann problem (3.18) and (3.19) is undetermined in H1 (Xi). The difference between two solutions satisfieshomogeneous Neumann problem. So, given two sets of solutions of all local Neumann problems associatedwith k, the jump of the differences belongs to G. Hence, if the jump is orthogonal to G, it is null.

So, a projected condensed interface operator PD is defined by

PDk ¼ wi � wj on Cij 8ij;
where wi is the solution of the homogeneous Laplace equation in Xi with Neumann interface conditions on theinterface defined by k. In sub-domains Xi where Cext
i 6¼ ;, wi is uniquely defined in H 10Cext

iðXiÞ. In sub-domains

where Cexti ¼ ;, wi is uniquely defined in H1(Xi) thanks to the orthogonality condition:
Z
CPDkg dS ¼ 0 8g 2 G.


Furthermore, admissibility condition for k is also an orthogonality condition to G, since relation (3.17) isequivalent to:
Z
Ckgi dS ¼ 0 8i Cext

i ¼ ;. ð3:20Þ

So the subset of H�12ðCÞ consisting of the fields that satisfy the homogeneous compatibility condition (3.20) in

each sub-domain Xi where Cexti ¼ ;, is the orthogonal of G in H�

12ðCÞ.

Finally, the extension of the definition of the condensed interface problem defined in Section 3.2 can bemade as follows. Choose any k0 such that:
Z
Xi

fivi dV þZ

Ci

k0i vi dS ¼ 0

in each sub-domain Xi where Cexti ¼ ;, then k defined as ki ¼ oui

oni� k0

i on Ci does satisfy compatibility condition(3.20).

Define u0i the solutions of the local problems with Neumann interface conditions associated with k0:

�Du0i ¼ fi on Xi;

u0i ¼ 0 on Cext

i ;

ou0i

oni¼ k0

i on Ci.

8>>><>>>:

In sub-domains Xi where Cexti 6¼ ;, u0

i is uniquely defined in H 10Cext

iðXiÞ. In sub-domains where Cext

i ¼ ;, u0i is

uniquely defined in H1(Xi) thanks to the orthogonality condition:

1

2

Xij

ZCij

ðu0i � u0

j Þg dS ¼ 0 8g 2 G.

Then the restriction of the solution of the global Laplace equation (2.1), (2.2) is the solution of Neumannproblem (3.21)–(3.23) in each sub-domain:

� Dui ¼ fi on Xi; ð3:21Þ

ui ¼ 0 on Cexti ; ð3:22Þ

oui

oni¼ k0

i þ ki on Ci. ð3:23Þ

Theorem 3.3. The solution of the projected condensed interface problem PDk = d, with right-hand-side defined

by dij ¼ u0j � u0

i on Cij is the unique field in the orthogonal subset of G in H�12ðCÞ for which the restriction of the

solution of the global Laplace equation (2.1), (2.2) is the solution of Neumann problem (3.21)–(3.23) in each sub-

domain.

It must be noted that the property of orthogonality to G is not local, since if two sub-domains Xi and Xj areneighboring, kij = �kji on Cij. So, ki and kj are not independent.

In the same way, if k + k0 is known, the local Neumann problem (3.21)–(3.23) does not define a unique ui inH1(Xi), if Cext

i ¼ ;. The uniqueness of ui is ensured by the G orthogonality:

1

2

Xij

ZCij

ðui � ujÞg dS ¼ 0 8g 2 G;

which enforces the continuity of the correct local solutions on the interfaces.


4. Discretization

4.1. Discrete trace operator

With a finite element discretization, the variational equations (2.9), (2.10) give a hybrid system of algebraicequations:

Kiui ¼ bi þ Btik; ð4:1ÞX

i

Biui ¼ 0; ð4:2Þ

where Ki is the local stiffness matrix, bi is the volume right hand side and Bi is a signed trace operator. Thesame notations ui and k have been taken for representing the vectors of discrete representations in Eq.(4.1), (4.2) and the continuous fields in the initial variational formulation (2.9), (2.10).

The Bi operator must be a signed discretization of the duality:
ZCi
vili dS vi 2 H 10Cext

iðXiÞ li 2 H�

12ðCiÞ:

On each interface Cij, there are only two trace operators with non-zero restrictions, Bi and Bj, and they haveopposite signs. The restriction of Bi on Cij is denoted as Bij, and the restriction of Bj, Bji.

Given a discrete approximation space for primal fields vi, defining Bi is equivalent to defining the discret-ization space for the Lagrange multiplier k. The standard theory on mixed and hybrid formulation advocatesthe choice of such a discrete space that the Ladyzhenskaya Babuska Brezzi condition (2.7) is uniformly sat-isfied by the discrete spaces, with constant a independent upon the mesh size, in order to ensure the consistencyof the discretization.

In the present context, the Lagrange multiplier is just an auxiliary unknown, used to divide the problem persub-domain, and only the primal variable is of interest and the following theorem holds:

Theorem 4.1. If the discrete trace operator is defined in such a way that:

Bijui þ Bjiuj ¼ 0 on Cij 8ij: ð4:3Þ
is equivalent to u is continuous, then vectors ui satisfying the discrete hybrid system of equations (4.1), (4.2) are
the restrictions of the solution of global system:

Ku ¼ b. ð4:4Þ

Proof. Under the assumption of the theorem, vectors ui satisfying the discrete hybrid system of equations(4.1), (4.2) are the restrictions of a continuous vector u. The restriction ui of u in Xi satisfies:

Kiui ¼ bi þ Btik.

Assembling these equations over all sub-domains gives:

Ku ¼ bþ c;

where c is the vector resulting of the assembling of all vectors Btik. Consider any continuous v, then:

vtc ¼X

i

vtiB

tik ¼

1

2

Xij

ktijðBijvi þ BjivjÞ ¼ 0.

So, c is null and u is solution of (4.4). This means that the discrete continuity condition (4.3) is not required tobe such that the discrete hybrid system of equations (4.1), (4.2) satisfies the discrete Ladyzhenskaya BabuskaBrezzi condition. So, the discrete trace operator Bi can just be taken as the restriction operator. h

Theorem 4.1 shows that solving the discrete sub-problems is similar to solve the global discrete problem.No additional discretization error appears, and the a priori error estimate between the continuous and the dis-crete problem are the same as that of between the continuous and the discrete Laplace equation [33,35]. This is

Fig. 3. Redundant restriction operator for a decomposition into three sub-domains.


not the case if a different trace operator is used, like in the Mortar element method in the case of non-conform-ing meshes for example [36–38]. Furthermore, when the solution is not smooth enough, like for domain withsingularities, hp-version with mortars should be used [39,40].

4.2. Redundant restriction operator

When more than two sub-domains intersect in one point, the Lagrange multipliers defined on this point areredundant. Fig. 3 presents such a configuration for a global domain meshes with regular quadrangles, andsplit into three sub-domains. In this case, the trace operator described in the previous Section may just bethe restriction operator. This choice is usually performed for any conforming meshes. In the case of non-con-forming meshes [37], the trace operator must be computed in a different way as investigated in [41,42].

The Finite Element Tearing and Interconnecting (FETI) [18] method is based on using an Uzawa method[43] for solving the discrete hybrid system (4.1), (4.2). The elimination of the local solutions ui in (4.1), (4.2)gives a condensed problem on the interface related to k:
X
i

BiK�1i Bt

ik ¼ �X

i

BiK�1i bi.

This system can be solved by the conjugate gradient algorithm [43].

5. Parallel implementation with message passing programming environment

This kind of method is very well suited for parallel implementation on distributed memory MIMDmachines with message passing programming environment.

Given a splitting of the computational domain, each sub-domain Xi is allocated to one processor. Thedescription of the sub-domain is the classical description of a single computational domain for any standardfinite element code. The only non-standard data are related to the description of the interface. For each inter-face between Xi and another sub-domain Xj, the list of the nodes of Xi that belong to this interface must beknown. In fact, such a list can be described in the same way as a classical boundary area, associated withDirichlet or Neumann boundary conditions, except that it corresponds to a new ‘‘interface’’ type of boundary.

Each processor can assemble and factorize the local stiffness matrix associated with its sub-domain, as wellas the local right hand side arising from the prescribed values on boundary and loadings. The only non-localoperations are related to the computation of the jump of local fields along the interface.

The practical implementation consists in gathering in each processor the values of the local field vi on eachinterface Cij = oXi \ oXj, sending them to the processor treating sub-domain Xj, then receiving the data arriv-ing from neighboring sub-domains and computing for each interface the jump of ‘‘inner’’ values (local) and‘‘outer’’ values (received). This requires that each sub-domain knows the lists of equations located on nodesbelonging to the interfaces with its neighbors (see Fig. 4). Such lists are similar to the standard description ofboundary conditions in finite element codes.

Ω

Γ

i

j2iΓ

j1iΓ

j4i

j3iΓ

Fig. 4. Description of interface.


Thus, just a purely local description of the interfaces is required. The values of the jumps are computed atthe same time for all sub-domains, with opposite signs for the jumps computed within two neighboring sub-domains Xi and Xj along their interface Cij. This means that each processor computes the jump according tothe outer normal derivative of its assigned sub-domain, and thus that it computes the interaction forces k onits interfaces as external forces. So, there is no need to define a global orientation of the normal vectors on theinterfaces: each sub-domain considers its natural outer normal.

The solution of the global problem through the domain decomposition method with one Lagrange multi-plier goes as follows.

• Initialization, computation of the local initial solution:

Kiu0i ¼ bi. ð5:1Þ

• Computation of the initial residual, equal to the jump of the local solution fields along the interfaces:

g0ij ¼ u0

i � u0j on Cij. ð5:2Þ

• Computation of the starting direction vector:

w0 ¼ g0. ð5:3Þ
Given the values of fields kp, up, gp and wp, iteration p + 1 consists of the following steps:
• Computation of the local field satisfying the Neumann problem:

Kivpi ¼ Bt

iwp. ð5:4Þ

• Computation of product of w by the Dual Schur complement matrix D:

Dwpij ¼ vp

i � vpj on Cij. ð5:5Þ

• Updating of k, g and w:

qp ¼ �ðgp;wpÞ=ðDwp;wpÞ;

kpþ1 ¼ kp þ qpwp;

gpþ1 ¼ gp þ qpDwp; ð5:6Þcp ¼ �ðgpþ1;DwpÞ=ðDwp;wpÞ;wpþ1 ¼ gpþ1 þ cpwp.

• Updating of the local solution fields:

upþ1i ¼ up

i þ qpvpi . ð5:7Þ


Most of these computations are purely local and therefore parallelizable. The computation of the jumps alongthe interface (5.5) requires data transfers, according to the scheme presented above: each processor sends itscontributions on its interfaces to the processors in charge of its neighboring sub-domains, then receive theircontributions, and compute the jump that is equal to the inner contribution minus the outer contributionon each interface. These data transfers are made according to the topology of the splitting of the domain. Theyjust require gather operations and node to node transfers.

The second kind of data transfers are related to the computation of the descent and re-conjugation coef-ficients q and c in (5.6). Each processor computes the associated dot products for its own interfaces, and thenthe local contributions are globally assembled and broadcasted. These data transfers involve all the proces-sors, independent of the topology of the domain splitting. They can be realized by calling the global datatransfers functions available in any message passing library that are optimized for the network topology ofthe target machine.

So, the programming effort for the parallel implementation of this method with a message passing program-ming environment is very small. In fact, it is even more natural than within a shared memory environment.With a distributed memory environment, each processor runs the same code, which is essentially a classicalsequential mono-domain code. The only non-standard routines are the computation of the jumps along theinterfaces, and the global assembly of local contributions to the dot products. Hence, the only real implemen-tation problem lies in the preprocessing phase.

6. Implementation of a coarse grid correction

6.1. Case of floating sub-domains

Consider a general splitting in a given number of sub-domains for a linear elasticity problem.The local Neu-mann problem in sub-domain Xi can be written:

Kiui ¼ bi þ Btik ð6:1Þ

and the continuity relation along interfaces:
XBiui ¼ 0. ð6:2Þ
With such a formulation, matrices Bi are signed trace matrices, in such a way that Eq. (6.2) means that on eachinterface Cij between sub-domains Xi and Xj: ui � uj = 0.

The pseudo-inverse Kþi and the kernel Ni of the stiffness matrix of sub-domain Xi can be computed locally.The local Neumann equation (6.1) is equivalent to the two following equations:

ui ¼ Kþi ðbi þ BtikÞ þ Niai; ð6:3Þ

Ntiðbi þ Bt

ikÞ ¼ 0. ð6:4Þ

In the case of elasticity, the kernel consists of rigid body motions. There are at most three of them per sub-domain in two dimension and six of them in three dimension.

By substitution in the continuity relation (6.2), the condensed interface problem can be written as follows:
XBiKþi Bt
ikþX

BiNiai ¼ �X

BiKþi bi. ð6:5Þ

Eqs. (6.5) and (6.4) represent an hybrid system of equation satisfied by k and the local rigid body motion coef-ficients a:

D E

Et 0

� �k

a

� �¼

c

d

� �; ð6:6Þ

where D ¼P

BiKþi Bti is the Dual Schur complement. Given (ai), the coefficients of a rigid body motion in each

sub-domain, Ea ¼P

BiNiai represents the jump at interface of the associated local rigid body motions. So, allthe matrix blocks of the hybrid system (6.6) can be computed via local operations and interface data transfers.


6.2. Coarse grid projection

The solution of the hybrid problem (6.6) can be computed via a projected gradient algorithm. The initial-ization process requires to find a starting k0 satisfying the constraint:

Etk0 ¼ c.

Then at each iteration, the search direction vector is computed from the projection of the gradient into thekernel of Et.

The orthogonal projection in the kernel of Et can be written algebraically:

P ¼ I � EðEtEÞ�1Et. ð6:7Þ
So given a vector g, computing the orthogonal projection of g requires mainly the solution of the problem:
ðEtEÞa ¼ �Etg; ð6:8Þ
that is a global ‘‘coarse grid’’ problem whose unknowns are all the coefficients of the rigid body motions in thefloating sub-domains.
For a given approximation kp of the interaction forces between sub-domains along their interface, the resid-ual of the condensed interface problem:

gp ¼X

BiKþi Btik

p þX

BiKþi bi ¼ gp ¼X

Biupþi ð6:9Þ

is equal to the jump of the particular solutions upþi of the local Neumann problems, computed using the pseu-

do-inverses:

upþi ¼ Kþi ðbi þ Bt

ikpÞ. ð6:10Þ

From Eqs. (6.7) and (6.8), the projected gradient Pgp is of form:

Pgp ¼ gp þ Eap ¼X

Biupþi þ

XBiNiap

i . ð6:11Þ

Hence, the projected gradient Pgp is equal to the jump of the local displacement fields given by the particularsolutions of Neumann problems upþ

i completed by rigid body motions with coefficients api :

upi ¼ upþ

i þ N iapi . ð6:12Þ

The definition of the constraint (6.4) associated with the orthogonal projector (6.8) entails that the rigid bodycomponents of jump of the displacement fields up

i are minimal in the sense that this jump is orthogonal to allthe traces of rigid body motions:

EtPgp ¼ 0() ðBiNiÞtPgp ¼ 0 8i. ð6:13Þ

Hence, computing the projected gradient Pgp consists in fact in computing the coefficients ap of optimal localrigid body motions. The underlying process is a kind of ‘‘coarse grid’’ smoothing of approximate solutionwhich entails the convergence rate for the overall FETI process to be asymptotically independent upon thenumber of sub-domains [14]. Hence, the FETI method with floating sub-domain is a kind of two-level optimalsolver.

6.3. Implementation of the coarse grid projection

Applying the projected conjugate gradient algorithm to the hybrid condensed interface problem (6.6) givesthe complete FETI procedure.

• Initialization, computation of admissible starting k0:

ci ¼ �N tibi;

k0 ¼ EðEtEÞ�1c. ð6:14Þ


• Initialization, computation of the local particular solution:

u0þi ¼ Kþi ðbi þ Bt

ik0Þ. ð6:15Þ
• Computation of the initial gradient:
g0 ¼X

Biu0þi . ð6:16Þ

• Computation of the projected initial gradient:

a0 ¼ �ðEtEÞ�1Etg0;

Pg0 ¼ g0 þ Ea0. ð6:17Þ

• Computation of the preconditioned projected initial gradient:

w0 ¼ MPg0. ð6:18Þ
• Computation of the starting direction vector:
Pw0 ¼ I � EðEtEÞ�1Etw0. ð6:19Þ

Given the values of fields kp, up+, gp and Pwp, iteration p + 1 consists of the following steps:

• Solution of the local Neumann problem:

vpi ¼ Kþi Bt

iPwp. ð6:20Þ
• Computation of product of Pwp by the Dual Schur complement matrix D:
DPwp ¼X

Bivpi . ð6:21Þ

• Updating of k and g:

qp ¼ �ðgp; PwpÞ=ðDPwp; PwpÞ;kpþ1 ¼ kp þ qpPwp; ð6:22Þgpþ1 ¼ gp þ qpDPwp.

• Projection of gradient:

apþ1 ¼ �ðEtEÞ�1Etgpþ1;

Pgpþ1 ¼ gpþ1 þ Eapþ1. ð6:23Þ

• Preconditioning:

zpþ1 ¼ MPgpþ1. ð6:24Þ
• Projection of preconditioned projected gradient:
Pzpþ1 ¼ I � EðEtEÞ�1Etzpþ1. ð6:25Þ
• Computation of the new direction vector by conjugation:
cp ¼ �ðPzpþ1;DPwpÞ=ðDPwp; PwpÞ;Pwpþ1 ¼ Pzpþ1 þ cpPwp. ð6:26Þ

• Updating of the local displacement fields:

upþ1þi ¼ upþ

i þ qpvpi ;

upþ1i ¼ upþ1þ

i þ Niapþ1i . ð6:27Þ


The projected gradient is equal to the jump of complete local displacement fields along the interface. So itsnorm has the same order of magnitude as the norm of the error on displacement fields. Hence the convergenceis controlled by monitoring the norm of the projected gradient.

Except the solution of global coarse grid problems required by the initialization of k and the computationof projected vectors, all operations associated to the various steps described above just require local compu-tations or data exchange along interfaces or global reduction for the dot products. So, the only new questionwith the parallel implementation of the overall procedure lies in the parallelization of the solution of globalcoarse grid problems:

ðEtEÞa ¼ b. ð6:28Þ
This problem has a small dimension, equal to the total number of local rigid body motions, but it is globallycoupled between all sub-domains. So, assembling and factorizing the complete (EtE) matrix would require todefine the complete sub-domain connectivity and to design a master process in charge of treating the coarsegrid solution.
Another approach consists in solving the coarse grid problems using an iterative procedure, namely theconjugate gradient algorithm. Then, only matrix vector products with blocks E or Et, that can be performedvia local computation and standard interface data exchange, are required. The drawback of this approach isits overall computational cost: each coarse grid problem must be solved with very high precision, requiring anumber of iterations close to the dimension of the problem.

Hence, computing the projected gradient Pgp consists in fact in computing the coefficients ap of optimallocal zero energy fields as illustrated in Fig. 5 for a two-dimensional cantilever beam problem split into slices.In this case, the zero energy fields are the rigid body motions. The dimension of the set of rigid body motionsin each sub-domain is equal to 3.

7. Mixed type interface conditions

7.1. Augmented Lagrangian

Consider a set of quadratic forms Qij defined on Cij for fields belonging to L2ðCijÞ. The Lagrangian (2.8)may be augmented, using these quadratic forms in two ways:

LQþðu; kÞ ¼ Lðu; kÞ þ 1

4

Xij

Qijðui � ujÞ� �

; ð7:1Þ

LQðu; kÞ ¼ Lðu; kÞ þ 1

4

Xij

QijðuiÞ � QijðujÞ� �

: ð7:2Þ

The saddle point of both augmented Lagrangians of Eqs. (7.1) and (7.2) define the same u as the Lagrangian ofEq. (2.8), since in every case, the continuity condition (2.5) must be satisfied at the saddle point [23]. As theadded terms of LQ+ and LQ vanish when u satisfies the continuity condition (2.5), u satisfies the same mini-mization problem for the three Lagrangians defined in (2.8), (7.1) and (7.2).

The same result still holds even if Qij is not positive. In particular, if Sij is a set of continuous self-adjointmappings of H

1200ðCijÞ onto H�

12ðCijÞ,

QijðuiÞ ¼Z

Cij

SijðuiÞðuiÞdS

then the two following augmented Lagrangians satisfy the same property:

LSþðu; kÞ ¼ Lðu; kÞ þ 1

4

Xij

ZCij

Sijðui � ujÞðui � ujÞdS

" #; ð7:3Þ

LSðu; kÞ ¼ Lðu; kÞ þ 1

4

Xij

ZCij

ðSijðuiÞui � SijðujÞujÞdS

" #: ð7:4Þ

Fig. 5. Optimal approximation of solution using rigid body motions.


In the same way than in Section 2.2, minimizing the energy (2.3) is equivalent to minimizing the sum of thelocal energies
X
i

J QiþðuiÞ; orX

i

J QiðuiÞ

with the following quantities:

J QiþðuiÞ ¼1

2

ZXi

ruirui dV �Z

Xi

fiui dV þ 1

2

Xj

ZCij

Sijðui � ujÞðui � ujÞdS

" #;

J QiðuiÞ ¼

1

2

ZXi

ruirui dV �Z

Xi

fiui dV þ 1

2

Xj

ZCij

ðSijðuiÞui � SijðujÞujÞdS

" #:

Since the last term added to J QiþðuiÞ or to J QiðuiÞ is not always positive, it may appears that J QiþðuiÞ or that

J QiðuiÞ is not coercive in one sub-domain Xi. Because

PiJ QiþðuiÞ ¼ JðuÞ and

PiJ QiðuiÞ ¼ JðuÞ on H 1

0ðXÞ, thecoercivity is still achieved and the Ladyzhenskaya Babuska Brezzi condition is still ensure. The same Lemmaand Theorems than in Section 2.2 can be derived which prove the existence and uniqueness of the solution.Once again, from the Ladyzhenskaya Babuska Brezzi condition, the equivalence between the global problem,the minimization problem and the saddle point problem can be proved.

7.2. Hybrid formulation derived from the augmented Lagrangian

The variational principle derived from the saddle point of Lagrangian LS+ in Eq. (7.3) is the followinghybrid system of equations:
R
Xiruirvi dV þ

Pj

RCij

Sijðui � ujÞvi dS ¼R

Xifivi dV þ

RCi


iðXiÞ;

RCijðuj � uiÞlij dS ¼ 0 8ij 8lij 2 H�

12ðCijÞ:

8<: ð7:5Þ

Eq. (7.5) means that in each sub-domain Xi, ui is solution of the following problem with mixed interface con-ditions on interface:

�Dui ¼ fi on Xi;

ui ¼ 0 on Cexti ;

ouioniþ Sijðui � ujÞ ¼ kij on Cij 8j:

8><>: ð7:6Þ


This approach is not very well suited for defining a domain decomposition solver, since the mixed interfacecondition (7.6) introduces a coupling between ui and uj in local problem in Xi.

The Lagrangian LS defined in Eq. (7.4) does not lead to the same drawback, since it leads to the followinghybrid system of equations:
R
Xiruirvi dV þ

Pj

RCij�ijSijðuiÞvi dS ¼

RXi

fivi dV þR

Cikivi dS 8vi 2 H 1

0CextiðXiÞ;

RCijðuj � uiÞlij dS ¼ 0 8ij 8lij 2 H�

12ðCijÞ:

8<: ð7:7Þ

where �ij is the sign of interface Cij, �ij = ��ji. Each interface Cij is shared by two sub-domains, and there is onlyone augmented term for it in LS:
Z
Cij

ðSijðuiÞui � SijðujÞujÞdS: ð7:8Þ

When deriving the term in Eq. (7.8) with respect to ui a plus sign appears in variational formulation (7.7) insub-domain Xi. Deriving the same term with respect to uj gives a minus sign in variational formulation (7.7) insub-domain Xj.

Eq. (7.7) means that in each sub-domain Xi, ui is solution of the following problem with mixed interfaceconditions on interface:

�Dui ¼ fi on Xi;

ui ¼ 0 on Cexti ;

ouioniþ �ijSijðuiÞ ¼ kij on Cij 8j:

8><>:

The FETI method issue from the solution of this hybrid system of equations is usually called the FETI-Hmethod [22,44] since this method has first been introduced for the Helmholtz equation.

8. Two-Lagrange multiplier methods

8.1. Three-field formulation

The Lagrange multipliers introduced in Section 2 are coupled by relation:

kji ¼ �kij on Cij 8ij
This coupling may be removed by a so-called three-field formulation [2,22] that consists in introducing a thirdtrace function on each interface as an auxiliary variable. The continuity condition (2.5) may be replaced by acondition of existence of a trace function t in H
1200ðCiÞ such that, in each sub-domain Xi:

ui � tij ¼ 0 on Cij 8j: ð8:1Þ

The variational formulation of this constraint is
ZCij
ðui � tijÞkij dS ¼ 0 8kij 2 H�12ðCijÞ: ð8:2Þ

This means that there are two independent Lagrange multipliers defined on each interface, kij, which is themultiplier of condition ui � tij, and kji, the multiplier of condition uj � tji, with tji = tij. Thanks to this decou-pling, the continuity relation may be written independently in each sub-domain:
Z
Ci

ðui � tiÞki dS ¼ 0 8ki 2 H�12ðCiÞ: ð8:3Þ

The following theorem shows the equivalence of the problems.

Theorem 8.1. Finding the solution of the minimization problem (2.3) under the constraint (8.3) is equivalent to

finding the saddle point of the following Lagrangian:


Lðu; t; kÞ ¼X

i

1

2

ZXi

ruirui dV �Z

Xi

fiui dV� �

þX

i

ZCi

ðti � uiÞki dS: ð8:4Þ

Thanks to the decoupling between sub-domain, the demonstration of the Ladyzhenskaya Babuska Brezzicondition and the interpretation of the saddle point problem are easier than for the one-Lagrange multipliercase in Section 2.

8.2. Variational formulation

The variational principle derived from this saddle point problem is the following hybrid system ofequations:
Z
Xi

ruirvi dV ¼Z

Xi

fivi dV þZ

Ci


iðXiÞ; ð8:5ÞZ

Ci

ðti � uiÞli dS ¼ 0 8li 2 H�12ðCiÞ; ð8:6Þ

1

2

Xij

ZCij

ðkj þ kiÞsij dS

" #¼ 0 8s 2 H

12ðCÞ: ð8:7Þ

The third equation (8.7) results from the derivation of Lagrangian defined in (8.4) with respect to t, since

Xi

ZCi

ðti � uiÞki dS� �

¼ 1

2

Xij

ZCij

ðtij � uiÞkij þ ðtji � ujÞkji dS

" #:

The first equation (8.5) means that in each sub-domain Xi, ui is solution of the Neumann problem (2.11)–(2.13), like in Section 2. The second equation enforces the continuity relation (8.1), and, since tji = tij, it impliesthat ui = uj on Cij. The third equation results from the introduction of auxiliary variable t. Since the first equa-tion entails ki ¼ oui

oni, the third one is equivalent to equilibrium condition (2.14).

So, each function ui in Eq. (8.5)–(8.7) is the restriction on sub-domain Xi of the solution of global problem(2.1), (2.2), since it satisfies the Laplace equation:

�Dui ¼ fi on Xi;

ui ¼ 0 on Cexti :

�

and both continuity conditions of the trace and the flux on each interface:

ui ¼ uj on Cij 8j;ouioni¼ � ouj

onjon Cij 8j:

(

8.3. Augmented Lagrangian with two-Lagrange multiplier formulation

The same kind of condensed interface problem as in Section 3.2 may be derived from the two-Lagrangemultiplier approach introduced in Sections 8.1 and 8.2. But the solution algorithm that could be derived fromit does not present any new interesting feature. The nature of interface operator is similar with one or twoLagrange multipliers. The only difference lies in the fact that with two Lagrange multipliers, the local problemsare more decoupled, and the dimension of the discrete problem is twice more with two Lagrange multipliersthan with one. In fact, using two Lagrange multipliers just implies to change the computation of the jump inthe conjugate gradient used for the iterative solution of the interface problem. The other steps of the algorithmkeep the same.


9. Conclusion

In this paper, an original elegant and unique interpretation with augmented Lagrangian of some domaindecomposition methods has been presented. Both a continuous and a discrete analysis have been performedfor the Laplace equation. In these analyses, the key theorems required for the correct solution of the initialglobal problem split into several sub-problems have been presented. The possible limitations arising in the caseof general mesh partitioning with cross-point have been indicated. The condensed interface problem issuefrom these methods has clearly been analyzed.

The first domain decomposition method introduced by Farhat et al. [18,19] involves one Lagrange multi-plier and Neumann interface conditions on the interface between the sub-domains. Recent extensions of thismethod to two Lagrange multipliers and mixed type interface conditions have been analyzed in the literature.A suitable choice of these mixed interface conditions ensures the well-posedness of the local sub-problems ineach sub-domain. In this paper, an interpretation framework of some of these domain decomposition methodswith one and two Lagrange multipliers and with Neumann and mixed type interface conditions has been pre-sented and analyzed. Parallel implementation of these methods with a message passing environment has beenanalyzed.

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Lagrangian Formulation of Domain Decomposition Methods a Unified Theory

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