A BEM–FEM overlapping algorithm for the Stokes equation

Applied Mathematics and Computation 182 (2006) 691–710

www.elsevier.com/locate/amc

A BEM–FEM overlapping algorithm for the Stokes equation

Vıctor Domınguez a,*, Francisco-Javier Sayas b

a Dep. Matematica e Informatica, Universidad Publica de Navarra, Campus de Arrosadıa s/n, Pamplona 31006, Spainb Dep. Matematica Aplicada, Universidad de Zaragoza, C.P.S., C./ Marıa de Luna, no 3, Zaragoza 50018, Spain

Abstract

In this work, we study the numerical solution of the Dirichlet problem for the Stokes equation by overlapping boundaryand finite elements. The domain where we state our problem is the intersection of a polyhedron with the exterior of astrictly contained obstacle with smooth boundary. The solution is then decoupled as a sum of an incident flow, definedon the polyhedron domain plus the response of the interior obstacle expressed as a single layer potential and constructedonly on the exterior of the obstacle. The numerical algorithm follows closely this ansatz by replacing the continuous termsof this decomposition by a finite and a boundary element, respectively. We prove that, under not very restrictive assump-tions, the method is well defined and converges to the exact solution with the same order as the best approximation of thesolution by the discrete spaces in the natural norms of the problem. Finally, in the last section we show how this methodcan be implemented, overcoming some of the difficulties appearing in the implementation, and demonstrating its applica-bility to practical problems.� 2006 Elsevier Inc. All rights reserved.

Keywords: Boundary element methods; Finite element methods; Stokes equation; Overlapping

1. Introduction

The Stokes system is a well-known simplified model for laminar viscous fluids near equilibrium. In additionto its importance in applications, which can be assessed from the huge literature on it, this system of partialdifferential equations has attracted the attention of numerical analysts as a sufficiently difficult test problemwhere some new numerical methods can be tried and analyzed before moving to more complicated and real-istic situations. The literature on numerical discretizations of the Stokes equations has developed in manydirections, from finite differences and finite volumes to finite element methods. There has also been someresults on boundary element methods [1] and also of coupled boundary and finite element procedures(BEM–FEM) to deal with Stokes flows in unbounded domains [2–5]. Each of the methods has its advantagesand disadvantages and because of that has its supporters and detractors. This work is somewhat related to the

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.04.031

* Corresponding author.E-mail addresses: [email protected] (V. Domınguez), [email protected] (F.-J. Sayas).

mailto:[email protected]

mailto:[email protected]

692 V. Domınguez, F.-J. Sayas / Applied Mathematics and Computation 182 (2006) 691–710

BEM–FEM realm, although the proposal of using both methods together stems from a very different point ofview and has some potential applications which we will comment later on.

We are concerned with the numerical discretization of the Stokes problem in a two or three dimensionaldomain with obstacles inside. The basic idea is to decompose the velocity and pressure fields into two differentparts, one that does not take into account the obstacles plus another that goes exterior to these, ignoring theboundary that encloses the whole domain. Figs. 1 and 2 sketch this geometrical situation. Both fields satisfytheir own equations, with a term coupling one to each other. Because of the very different nature of the purelyinterior and purely exterior fields it seems natural to use finite and boundary element methods to approximatethem respectively. In fact we will show that anything that works separately for each of the problems, works (atleast when discretizations are fine enough) for the coupled problem.

This kind of decomposition appears often in the theoretical and numerical discretization of time-harmonic(acoustic, elastic or electromagnetic) waves: the interior part takes the role of an incident wave and the exteriorof a scattering term. The fact that the method proposes the sum of two fields as solution implies that its prac-tical use is a priori limited to linear problems. Nevertheless it is also possible to apply these ideas in linearizediterations of a non-linear flow problem and, especially, in inverse and optimal control problems, where theobstacles (or some of their properties) are the unknowns and we want to avoid a full remeshing of the problemin each iteration of a numerical scheme for the inverse problem.

In [6,7], there has been a first experience of this kind of discretizations applied to the very simple Laplaceequation, proving optimal convergence estimates in terms of the BEM and FEM discretization parametersand the convergence of some natural Neumann–Schwarz iteration, consisting of solving alternatively the finiteand boundary element systems that appear in the coupled system. These ideas were extended in [8] to trans-mission problems for the Helmholtz equations, where ellipticity has dissapeared and the integral equation hasevolved to a boundary integral system that has to be stabilized by using adequate boundary element spaces.This work is therefore an extension of the preceding papers, including the additional difficulties of having totackle with a mixed problem (the Stokes problem) instead of and elliptic or strongly elliptic one. We alsoapproach the discretization in a more general fashion and give new theoretical and practical hints on the effi-cient solution of the coupled system in such a way that the whole system is never assembled.

It is important to remark that this is not a traditional BEM–FEM coupling for an exterior Stokes problems.Here the BEM mesh is immersed in the FEM mesh, making the method more apparently similar to animmersed boundary or fictitious domain method. The nature of the system we obtain is however completelydifferent to those two families of methods, and this can be seen in the happy instance that we are not obliged to

Fig. 1. Domain of the problem.

Fig. 2. Auxiliary domains.

V. Domınguez, F.-J. Sayas / Applied Mathematics and Computation 182 (2006) 691–710 693

have compatibility of the FEM and BEM meshes to make the method stable. In this sense, the method resem-bles more the use of Chimera meshes with overlapping [9].

The structure of the paper is as follows. In Section 2, we state the problem and propose the decompositionof the solution as the already mentioned superposition of two fields. We then write down the variational formof the equation for the interior fields (in the domain without obstacles) and of the boundary integral equationstemming from the use of a single layer potential representation of the fields. In Section 3, we propose anumerical method for each of the given variational problems, including the way of handling the couplingterms.

In Section 4, we formulate the coupled variational system in a very different way, by using some operators(and adequate unknowns) that allow us to write the whole problem as an operator equation of the secondkind, that is, of the form ðIþKÞf ¼ g, where I is the identity operator in some Hilbert space and K iscompact. The Fredholm alternative is then used to prove that injectivity of the equation (which is simpleto verify) proves invertibility and well-posedness of the problem. Then in Section 5 we use similar ideas toprove that the discretization we have proposed can be understood as a discrete version of the operator equa-tion. In fact, by using some simple tricks we can understand the discrete equation to be held in the full space(as opposed to in a finite-dimensional one). We profit from this rewriting to transfer convergence properties ofthe finite and boundary element methods to convergence properties of discretization of operator equations.

In Section 6, we will show a numerical example demonstrating how the method can be implemented. Thelinear system is not assembled but solved by an iterative method, specifically, by GMRES. Moreover, the goodproperties of the continuous equations are preserved after discretized, in such a way that the number of iter-ations remains independent of the discretization level. The numerical results show also that the discretizationlevel of both the boundary and finite element schemes can be taken independently.

We leave for the appendix the study of some properties of the single layer operator for the Stokes system,which plays an essential role in the analysis of this paper. We also prove here a simple result related to theinvertibility of this operator in the two dimensional case. This result, which was unknown to the authors(and does not appear in the literature we have consulted) runs in parallel to the concept of logarithmic capac-ity of a curve (for the Laplace operator) [10] and to the existence of at most three critical values for the bihar-monic single layer operator [11].

Remark 1.1. Throughout this work we will use boldface for vectors and vector functions. The operators, suchas integrals, differential operators, etc., are then applied element-wise. The Frobenius product

A : B ¼X

aijbij

defined between matrices of the same size, will appear also in this paper.We will write

a K b

if there exists C, independent of any discretization parameter involved, such that a 6 Cb.For H � Rd (d = 2,3) Lipschitz, H1(H) denote the classical Sobolev space of order one. The product space

is

H1ðHÞ ¼ fðx1; . . . ;xdÞjxj 2 H 1ðHÞ; j ¼ 1; . . . ; dg:
The subspace of the elements of H1(H) vanishing on the boundary, denoted as usual by H1
0ðXÞ, will appearalso in this work.

If C is the boundary of H, H1/2(C) is the subspace of the traces of the elements of H1(H) and H�1/2(C) is thedual space, with the duality product, extension itself of the L2(C) product, denoted by (Æ, Æ)C.

2. Statement of the problem

Let Q � Rd (d = 2,3) be a Lipschitz polygonal/polyhedral domain containing a simply connected domainwith sufficiently smooth boundary C. The annular domain defined by the intersection of the exterior of C andinterior of Q (see Fig. 1) will be denoted by X. Let


H1=2n ðRÞ :¼ gR 2 H1=2ðRÞ

ZR

gR � nR ¼ 0

�� ; ð1Þ

nR being the outward normal vector of R.Given any pair ðgR; gCÞ 2 H1=2

n ðRÞ �H1=2n ðCÞ the problem we want to solve is

x 2 H1ðXÞ; q 2 L2ðXÞ s:t:

�Dxþrq ¼ 0; on X;

r � x ¼ 0 on X;

cRx ¼ gR; cCx ¼ gC:

�� ð2Þ

This problem models the velocity (x) and pressure (q) of a viscous and incompressible fluid when the velocityis prescribed at the boundary.

Note that the Dirichlet data are required to satisfy a more restrictive assumption than the usual
ZR
gR � nR �Z

CgC � nC ¼ 0

(notice that nC is outward oriented, and therefore points inward to the interior of X) which ensures the exis-tence and uniqueness of solution. This restriction on the data will be fully explained when we propose adecomposition of the solution on which the numerical procedure is based.

Let (U,p) be the fundamental solution for the Stokes system:

UðzÞ ¼1

4p � log jzjI2 þ 1jzj2 zz>

h iif d ¼ 2;

18p � 1

jzj I3 þ 1jzj3 zz>

h iif d ¼ 3;

8><>: pðzÞ ¼z

2pjzj2 if d ¼ 2;

z

4pjzj3 if d ¼ 3:

(

In this definition, Id is d · d identity matrix. Hence, U is a d · d matrix functions and p a (column) vector of d

components. The single layer potentials for the velocity and pressure are then defined as

SCkC ¼Z

CUð� � yÞkCðyÞ dy; PCkC :¼

ZC

pð� � yÞ � kCðyÞ dy: ð3Þ

The integrals appearing in the definition should be understood in a weak sense if the density is not an integra-ble function. Differentiating under the integral sign, one checks easily that the pair ðSCkC;PCkCÞ provides asolution of the homogeneous Stokes equation in Rd n C. The single layer operator

VC :¼ cCSC

is the common interior and exterior trace of the single layer potential, which defines an integral operator onthe boundary C.

Let L20ðQÞ be the space of square integrable functions with null average on Q. We propose to solve the var-

iational problem

u 2 H1ðQÞ; p 2 L20ðQÞ s:t:

cRu ¼ gR � cRSCkC;RXru : rw�

RX pðr � wÞ ¼ 0 8w 2 H1

0ðQÞ;RX rðr � uÞ ¼ 0 8r 2 L2

0ðXÞ

�� ð4Þ

and the boundary integral equation

kC 2 H�1=2n ðCÞ;

VCkC ¼ gC � cCu

�� ð5Þ

with

H�1=2n ðCÞ :¼ fgC 2 H�1=2ðCÞjðgC; nCÞC ¼ 0g:

Note that if C is C2, then nC 2 C1ðCÞ � H1ðCÞ and therefore H�1=2n ðCÞ is well defined.


Problem (4) is a non-homogeneous Dirichlet problem for the Stokes equation on Q, whereas (5) is theboundary integral formulation (via a single layer potential) of a non-homogeneous Dirichlet problem forthe Stokes equation exterior to C. The coupling between both problems happens precisely in the Dirichletcondition.

Moreover, we remark that the Dirichlet data of (4) and the right-hand side of (5) belong to H1=2n ðRÞ and

H1=2n ðCÞ. This is because both u and SCkC have zero flux through any closed surface (see [1], Chapter 4)

not intersecting the boundaries of their respective domains of definition.Alternatively, if we can invert VC (we will deal with this issue in the appendix), (5) can be solved and its

solution inserted in the Dirichlet condition in (4) which becomes a non-standard non-local boundarycondition.

Once the coupled problem is solved, the functions (defined on X)

x ¼ uþSCkC; q ¼ p þPCkC ð6Þ
solve (2).
By reasons we will also clarify in Section 4, (5) can be equivalently written in variational form

kC 2 H�1=2n ðCÞ;

ðgC;VCkCÞC ¼ ðgC; gC � cCuÞC 8gC 2 H�1=2n ðCÞ:

�� ð7Þ

This form is better suited for Galerkin schemes applied to solve (5).

3. Numerical approximation

Let fTh;T0hg be shape-regular meshes of Q [12]. On these grids we construct {Vh,Ph} a pair of stable finite

element spaces of continuous piecewise polynomials for the velocity and the pressure. That is, Vh � H1(Q) andP h � L2

0ðQÞ are piecewise polynomial on Th and T0h, respectively, which satisfy the infimum–supremum

condition:

supvh2Vh

RQ phðr � vhÞkvhk1;Q

J kphk0;Q 8ph 2 Vh:

Examples of such pairs are the Taylor–Hood spaces and the modified Taylor–Hood spaces [13–15]. Let r 2 N

be such that the space of the continuous piecewise polynomial functions of degree r is contained in Vh. Thenthe following property holds

infvh2Vh

ku� vhk1;Q K hrQkukrþ1;Q; ð8Þ

hQ being the maximum of the diameters of the elements of Th.Regarding the problem on C, we consider a sequence of discrete spaces XC

h � H�1=2n ðCÞ. We denote hC a dis-

crete parameter directing the sequence of spaces in such a way

limhC!0

infnC2XC

h

kkC � nCk�1=2;C ¼ 0 8kC 2 H�1=2n ðCÞ:

We point out that, unlike the discrete space on Q, hC may have no geometric meaning at all.The solution of the Stokes equation (4) is approximated by that given by the scheme

uh 2 Vh; ph 2 P h;

cRuh ¼ ghR � IR

hSCkhC;R

Xruh : rwh �R

X phðr � whÞ ¼ 0 8wh 2 Vh \H10ðQÞ;R

X rhðr � uhÞ ¼ 0 8rh 2 P h;

�� ð9Þ

where cRVh 3 ghR � gR and

IRh : C0ðRÞ ! cRVh ð10Þ


is the Lagrange interpolation operator on the finite element space on the boundary. How ghR is chosen is imma-

terial for defining the method, although it will affect convergence as will be clear later (see Theorem 5.6 andCorollary 5.7).

The numerical approximation of (5) is the solution of

khC 2 XC

h ;RC gh

C �VCkhC ¼

RC gh

C � ðgC � cCuhÞ 8ghC 2 XC

h :

�� ð11Þ

Again, we observe that both problems are coupled with uh and khC appearing simultaneously in the right-hand

sides of both systems.We remark that the method works simultaneously with two different discrete parameters, one related with

the finite element method and therefore having a geometric meaning (the size of the mesh), and the other onewith the boundary element method on C. Each of them can be freely chosen, which allows a great flexibility ofour method.

The numerical approximation of problem (2) is the pair of functions defined on X

xh :¼ uh þSCkhC; qh :¼ ph þPCkh

C: ð12Þ
This kind of decompositions was introduced in [6] for the Laplace equation. Numerical methods exploitingthis decomposition have been proposed and studied in [7,8] for the Laplace equation and, in a more generalsetting, for Helmholtz transmission problems. The decomposition given above can be understood as an inci-dent flow (u) plus the corresponding response of the obstacle C. However, this interpretation would require tobe fully valid that the scattered flow was negligible far away from the obstacle, i.e. that the flow vanishes atinfinity. In the end of Section 4 we will study under what hypotheses this occurs.
4. Theoretical analysis

We assume in what follows that VC : H�1=2n ðCÞ ! H1=2

n ðCÞ is invertible. This is well known in R3 where theoperator is actually elliptic. However in R2 the single layer operator can fail to be invertible for some specialsizes of C. We refer to the appendix, and references therein for a discussion on this issue.

Remark 4.1. Assume that C is Crþ2. Then, both the Sobolev spaces Hs(C) and their subspaces HsnðCÞ can be

univocally defined for all real |s| 6 r + 1. Moreover, it can be proven (see [10], Theorem 7.17) thatVC : H�1=2þs

n ðCÞ ! H1=2þsn ðCÞ for |s| 6 r is Fredholm of index zero with the kernel independent also of s. As a

simple byproduct, we derive the invertibility for the single layer operator in a wider range of Sobolev spaces.

For any fR 2 H1/2(R), consider the solution of the problem

v 2 H1ðQÞ; q 2 L20ðQÞ s:t:

cRv ¼ fR;RQrv : rw�

RQ qðr � wÞ ¼ 0 8w 2 H1

0ðQÞ;RQ rðr � vÞ ¼ 0 8r 2 L2

0ðQÞ:

��ð13Þ

It is easy to check that (v,q) is just the weak solution of

�Dvþrq ¼ 0; r � v ¼ 1

jQj

ZR

fR � nR; cRv ¼ fR:

That is, (13) has full sense for any fR 2 H1/2(R), although it only provides a solution for the Stokes system ifthe Dirichlet data belong to the correct subspace, i.e, to H1=2

n ðRÞ. In any case, we can define the solutionoperators

T R : H1=2ðRÞ ! H1ðQÞ PR : H1=2ðRÞ ! L20ðQÞ

fR 7! v fR 7! qð14Þ


On the other hand, we introduce the projection QC : H1=2ðCÞ ! H1=2n ðCÞ defined by

QCgC :¼ gC �1

jCj

ZC

gC � nC

� �nC: ð15Þ

It is straightforward to check now that the solution

nC 2 H�1=2n ðCÞ;

ðgC;VCnCÞC ¼ ðgC; fCÞC 8gC 2 H�1=2n ðCÞ;

�� ð16Þ

for any fC 2 H1/2(R) is just

nC ¼V�1C QCfC:

Therefore, both (13) and(16) can deal with right-hand sides belonging to the full trace space. This makespossible to move the analysis of the method to H1/2(R) · H�1/2(C), which will result in a simpler analysisfor the numerical method.

With these operators, it is convenient to reconsider (4) and (5) by taking fR = cRu as unknown instead of u.Therefore, finding decomposition (6) is equivalent to the following algorithm

1. Solve

fR 2 H1=2ðRÞ; kC 2 H�1=2n ðCÞ s:t:

fR þ cRSCkC ¼ gR;

V�1C QCcCT RfR þ kC ¼V�1

C QCgC:

�� ð17Þ

2. Construct

x :¼ T RfR þSCkC; q :¼ PRfR þPCkC: ð18Þ

Eq. (17) can be written as the equation of the second kind (I stands for the identity operator)

ðIþKÞfR

kC

� �¼

gR

V�1C QCgC

� �;

where

K :¼0 cRSC

V�1C QCcCT R 0

� �:

It is easy to see that the existence and uniqueness of the decomposition is equivalent to the invertibility ofIþK : H1=2ðRÞ �H�1=2ðCÞ ! H1=2ðRÞ �H�1=2ðCÞ. Clearly

K : H1=2ðRÞ �H�1=2ðCÞ ! H1=2ðRÞ �H�1=2ðCÞ

is well defined, compact, and RðKÞ � H1=2n ðRÞ �H�1=2

n ðCÞ, R(K) being the range of K.

Theorem 4.2. The mapping IþK : H1=2ðRÞ �H�1=2ðCÞ ! H1=2ðRÞ �H�1=2ðCÞ is an isomorphism.

Proof. Using the Fredholm alternative, it suffices to prove the injectivity of IþK. Let (fR,kC) belongs to thekernel of IþK. Then necessarily ðfR; kCÞ 2 H1=2

n ðRÞ �H�1=2n ðCÞ. Consider now

u :¼ T RfR; p ¼ PRfR; v :¼ SCkC; q ¼ PCkC:

Note that the first pair is defined on Q whereas the second one is a solution of the homogeneous Stokes equa-tion in Rd n C.


Since (fR,kC) is in the kernel of IþK, it follows that

uþ v ¼ 0; on Q; p þ q ¼q0 on X;

q1 on intC;

�
with q0, q1 constants. Recall now the jump relation of the normal stress for the single layer potential (cf. [1])
½tCðv; qÞ� :¼ tþC ðv; qÞ � t�C ðv; qÞ ¼ �kC; ð19Þ
where tþC ðv; qÞ (respectively t�C ðv; qÞ) is the normal stress on C taken from the exterior (respectively interior) ofC. Hence,
ðq1 � q0ÞnC ¼ ½tCðuþ v; p þ qÞ� ¼ ½tCðv; qÞ� ¼ �kC:

Since kC 2 H�1=2n ðCÞ, it follows that q1 = q0 and kC = 0. Now it is very easy to see that u = 0 which finishes the

proof. h

In the rest of this section we will investigate under what circumstances

limjzj!1ðSCkCÞðzÞ ¼ 0:

This property clearly holds in R3 because the fundamental solution of the Stokes system decreases like Oðjzj�1Þas z grows to infinity. However, in R2 we have to impose in addition that
Z
CkC ¼ 0

(obviously this restriction must be understood in a week sense if kC is not an integrable function).Recall that
Z
RtRðx; qÞ �

ZC

tCðx; qÞ ¼ 0: ð20Þ

This is a simple byproduct of second Green identity for the Stokes system (see [16], Chapter 4)
ZR
tRðx; qÞ �Z

CtCðx; qÞ ¼

ZX

T ½x; q� : rw

(T[x, q] denotes the stress tensor) by simply taking w constant.

Proposition 4.3. Let (x,q) be the solution to (2) and ðSCkC;PCkCÞ the term of the decomposition appearing in

(6). ThenR

C kC ¼ 0 if and only if
ZC
tCðx; qÞ ¼Z

RtRðx; qÞ ¼ 0:

Proof. Since (u,p) and ðSCkC;PCkCÞ solve the homogeneous Stokes equation in the interior of C, then
ZC
tCðu; pÞ ¼ 0 ¼Z

Ct�C ðSCkC;PCkCÞ

(note that by regularity tþC ðu; pÞ ¼ t�C ðu; pÞ). Applying the jump relations (19) it follows that
ZC
kC ¼ �Z

C½tCðSCkC;PCkCÞ� ¼ �

ZC

tþC ðSCkC þ u;PCkC þ pÞ ¼Z

CtCðx; qÞ:

Taking into account (20) the proof is finished. h

5. Numerical analysis

As we did in the previous section, we first rewrite the numerical method as a equation of the second kind. Akey feature of the analysis is that we will rewrite the discrete equations (9) and (11) as an operator equation inthe full infinite-dimensional space. We start by introducing the discrete mappings associated to the numericalmethod.


Let thus define

T hR : cRVh ! Vh P h

R : cRVh ! P h

fhR 7! vh fh

R 7! qh

associate to fhR the solution of

vh 2 Vh; qh 2 P h s:t:

cRvh ¼ fhR;R

Xrvh : rwh �R

X qhðr � whÞ ¼ 0 8wh 2 Vh \H10ðQÞ;

�R

X rhðr � vhÞ ¼ 0 8rh 2 P h:

�� ð21Þ

Note that the mappings T hR, P h

R are the discrete version of those introduced in (14). We are implicitly assumingthat fh

R is an approximation of the exact Dirichlet condition fR 2 H1/2(R). We do not consider here how this isactually computed, since it is not relevant for the correct definition of the method, although it has obviously agreat influence on the convergence of the method. One way, at least theoretical, to approximate the Dirichletdata is by means of a projection

phR : H1=2ðRÞ ! cRVh ð22Þ

satisfying

phRfh

R ¼ fhR for all fh

R 2 cRVh; ð23aÞkph

RfRk1=2;R K kfRk1=2;R; ð23bÞkph

RfR � fRk0;R 6 1ðhÞkfRk1=2;R with 1ðhÞ ! 0 as h! 0: ð23cÞ

A mapping fulfilling conditions (23) can be constructed by using a Scott–Zhang type projection. Hence, ifPQ

h : H1ðQÞ ! Vh is a uniformly bounded projection on Vh satisfying

kPQh u� uk0;Q K hQkuk1;Q; cRPQ

h u ¼ 0; if u 2 H10ðQÞ;

we can simply take phR :¼ cRPQ

h cþR . Here, cþR is the pseudoinverse of the trace operator, or more in general, anycontinuous lifting operator. Examples of such projections can be found in [17,18] for triangular and quadri-lateral elements (see also [19]). Again we must emphasize that all the analysis of the full method (Theorem 5.6and Corollary 5.7) is immaterial of the choice of this approximation of the Dirichlet data. Theorem 5.1 uses ph

R

only for theoretical purposes.

Theorem 5.1. Let fR 2 H1/2(R), fhR 2 cRVh and take (u, p) :¼ (TRfR,PRfR), ðuh; phÞ :¼ ðT Rfh

R; PhRfh

RÞ. Then

ku� uhk1;Q þ kp � phk0;Q K kfR � fhRk1=2;R þ inf

wh2Vh

ku� whk1;Q þ infrh2P h

kp � rhk0;Q:

Moreover, if fhR :¼ ph

RfR, with phR satisfying (23), then there exists u1 with u1(hQ)! 0 as hQ! 0 such that

ku� uhk0;Q 6 u1ðhQÞ infwh2Vh

ku� whk1;Q þ infrh2P h

kp � rhk0;Q

� �:

Proof. The first estimate for the problem

�Duþrp ¼ f; r � u ¼ 0; cRu ¼ 0

is well known (see for instance [17], Chapter 12). We refer to [19] to see how this proof can be adapted to han-dle non-homogeneous Dirichlet data.

The second estimate is derived by duality estimates (Aubin–Nitsche trick) [17,20]. h

The boundary element method (11) is just a Galerkin scheme. Therefore, the solution of the followingboundary integral method

khC 2 XC

h ;RC gh

C �VCkhC ¼

RC gh

C � fC 8ghC 2 XC

h

��


gives rise to the operator

Gh : H1=2ðCÞ ! XCh

fC 7! khC:

Notice that this mapping, in contrast with T hR and P h

R, is defined on the whole Sobolev space. MoreoverGhQC = Gh with QC given by (15).

Theorem 5.2. For all hC sufficiently small we have the quasi-optimality estimate

kV�1C QCfC � GhfCk�1=2;C K inf

ghC2XC

h

kV�1C QCfC � gh

Ck�1=2;C 8fC 2 H1=2ðCÞ:

Moreover, there exists u2 with u2(hC)! 0 as hC! 0 such that

kV�1C QCfC � GhfCk�1;C 6 u2ðhCÞ inf

ghC2XC

h

kV�1C QCfC � gh

Ck�1=2;C:

Proof. For d = 3, the convergence in H�1=2n ðCÞ follows from the ellipticity of the single layer operator (see

(31)). For d = 2, we remark that the operator

Kru :¼ � 1

4p

ZC

logðrj � �yjÞuðyÞ dy

is elliptic for r > 0 small enough [10, Theorem 8.16]. The single layer operator for the Stokes system is there-fore a compact perturbation of the diagonal operator constructed with Kr (see the proof of Lemma A.2). Con-vergence follows now from the theory of compact perturbation of elliptic operators, cf. [21], Chapter 13.

Superconvergence in H�1(C) is obtained by applying the Aubin–Nitsche trick. h

By taking fhR as new unknown, the numerical method is seen to be equivalent to the following algorithm:

1. Take cRVh 3 ghR � gR.

2. Solve

fhR 2 cRVh; kh

C 2 XCh s:t:

fhR þ Ih

RSCkhC ¼ gh

R;

GhcCT hRfh

R þ khC ¼ GhgC:

�� ð24Þ

3. Construct

xh :¼ T hRfh

R þSCkhC; qh :¼ P h

RfhR þPCkh

C: ð25Þ

The feasibility of the method relies on existence and uniqueness of solution to (24). On the other hand, andnow from a practical point of view, we must choose an appropriate method to solve this linear system. A clo-ser look shows that one multiplication by the matrix of the system requires to solve simultaneously a finite anda boundary element method. Hence, the use of Krylov methods like GMRES seems appropriate. Moreover,the unknown is not uh but its trace on R. This results in a reduction of the requirements of memory forGMRES. In the final section we will show a numerical experiment dealing with this kind of details.

Eq. (24) leads to consider the operators,

KhCR :¼ GhcCT h

RphR � KCR :¼V�1

C QCcCT R; KhRC :¼ Ih

RcRSC � KRC :¼ cRSC

and

Kh :¼ 0 KhRC

KhCR 0

" #�K:


Therefore, we can write Eq. (24) as

ðIþKhÞfhR

khC

" #¼

ghR

GhgC

� �:

Note that the role of the projection phR, appearing in the definition of Kh

RC, is just for analytical purposes.Specifically, it makes possible to carry out the analysis of IþKh in the continuous frame, i.e., it allows totake right-hand sides not only in the discrete spaces but also in the continuous space H1/2(R) · H1/2(C). How-ever, ph

R is not needed to be computed in practice, since when ghR belongs to the finite element trace space, the

full equation occurs in the discrete frame.Recall r, introduced in (8), is the degree of the continuous piecewise polynomial contained in Vh.

Lemma 5.3. It holds

kðKhRC � KRCÞkCk1=2;R K hrþ1=2

Q kkCk�1;C:

Proof. Note that

KhRC � KRC ¼ ðIR

h � IÞcRSC:

Notice also that SC : H�1ðCÞ ! CmðHÞ is continuous for all m 2 N and for any compact set H having emptyintersection with R. Then the result is consequence of the standard convergence estimates for the interpolationoperator. h

To make the notations simpler, we will denote by h both parameters in such a way that we will write thath! 0 when hQ, hC! 0 simultaneously.

Proposition 5.4. It holds

kK2h �K2k ! 0; as h! 0;

where k Æ k is the operator norm from H1/2(R) · H�1/2(C) into itself.

Proof. By Lemma 5.3 we just have to prove

kðKhCR � KCRÞKRCk; kKRCðKh

CR � KCRÞk ! 0: ð26Þ
k Æ k being the corresponding operator norm in each case.
The first of these terms is the composition of a compact operator, KRC, with another converging strongly tozero. Convergence to zero in operator norm follows from standard approximation results of operatorequations (see [21], Section 10.3, [22]).

The second term in (26) is studied as follows. We first split it into

KRCðKhCR � KCRÞ ¼ KRCðGh �V�1

C QCÞcCT hRph

R þ KRCV�1C QCcCðT h

RphR � T RÞ: ð27Þ

Continuity of KRC : H�1ðCÞ ! H1=2n ðRÞ and Theorem 5.2 proves that kKRCðGh �V�1

C QCÞcCT hRph

Rk ! 0.On the other hand, we recall the well-known inequality for the trace operator (cf. [14])

kcCwk0;R 6 C½ekwk1;Q þ e�1kwk0;Q�

with C independent of w and e > 0. Noting that KRCV�1C QC : H0ðCÞ ! H1=2ðRÞ is continuous (see Remark 4.1),

using Theorem 5.1 and the inequality above with e ¼ffiffiffiffiffiffiffiffiffiffiffiffiu2ðhÞ

p, we obtain convergence to zero of the second

term of (27). h

Theorem 5.5. For all h sufficiently small,

IþKh : H1=2ðRÞ �H�1=2ðCÞ ! H1=2ðRÞ �H�1=2ðCÞ
is uniformly bounded in h with uniformly bounded inverse.


Proof. Let A� :¼ I�K; and A�h :¼ I�Kh. By the special form of A� we have ðfR; kCÞ 2 KerðA�Þ if and

only if ðfR;�kCÞ 2 KerðAþÞ. Therefore and since Aþ is injective, so is A�. Likewise, it follows that A� isonto and therefore invertible.

On the other hand, Proposition 5.4 implies that

Aþh A

�h ¼A�

h Aþh !AþA� ¼A�Aþ; as h! 0:

Hence, Aþh A

�h is invertible for all h small enough with uniformly bounded inverse in h. The result is proven

once we notice that ðAþh Þ�1 ¼A�

h ðAþh A

�h Þ�1. h

Theorem 5.6. Let ðfR; kCÞ 2 H1=2n ðRÞ �H�1=2

n ðCÞ be the solution of (17) and ðfhR; k

hCÞ that of (24). Then, if

(u, p) :¼ (TRfR,PRfR) the following convergence estimate holds

kfR � fhRk1=2;R þ kkC � kh

Ck�1=2;C K kgR � ghRk1=2;R þ hrþ1=2

Q kkCk�1=2;C þ infvh2Vh

ku� vhk1;Q

þ infrh2P h

kp � rhk0;Q þ infghC2XC

h

kkC � ghCk�1=2;C:

Proof. By Theorem 5.5

fR� fhR

kC�khC

" #��

H1=2ðRÞ�H�1=2ðCÞ

K ðIþKhÞfR� fh

R

kC�khC

" #��

H1=2ðRÞ�H�1=2ðCÞ

¼gR�gh

R

ðV�1C QC�GhÞgC

� �þðK�KhÞ

fR

kC

� �� H1=2ðRÞ�H�1=2ðCÞ

:

The first component in H1/2(R) is bounded straightforwardly by Lemma 5.3. For the second component, wenotice that

ðV�1C QC � GhÞgC þ ðKh

CR � KCRÞfR ¼ ðV�1C QC � GhÞðgC � cCT RfRÞ þ GhcCðT h

RphR � T RÞfR

¼ ðV�1C QC � GhÞfC þ GhcCðT h

RphR � T RÞfR:

The proof follows now from Theorems 5.1 and 5.2. h

Finally, we focus on the convergence for the solution of the original problem. Next result shows that theerror estimates for the unknowns, proven in the previous theorem, are actually inherited by the solution of theoriginal Stokes problem (2).

To avoid problems related with non-uniqueness of solution for the pressure, we assume that q is that givenby decomposition (6).

Corollary 5.7. Let (x,q) be the solution of (2), (xh,qh) that given by (25), (26). Then

kx� xhk1;Q þ kq� qhk0;Q K kgR � ghRk1=2;R þ hrþ1=2

Q kkCk�1=2;C þ infvh2Vh

ku� vhk1;Q

þ infrh2P h

kp � rhk0;Q þ infghC2XC

h

kkC � ghCk�1=2;C:

Proof. By Theorem 5.1, it remains to study the term of the solution related with the boundary element prob-lem. For this term, we can apply [16, Section IV, Theorem 6.1], to obtain

kSCðkC � khCÞk1;X þ kPCðkC � kh

CÞk0;X K kVCðkC � khCÞk1=2;C K kkC � kh

Ck�1=2;C: �

6. Numerical experiments

In this section, we illustrate the theoretical results previously shown in this work with a simple numericalexperiment. The domain considered is the one depicted in Fig. 3, with the nine interior obstacles given by ellip-ses of semiaxes 0.1 and 0.06 rotated clockwise 30�.

Fig. 3. Domain of the numerical experiment.


Let us describe briefly the discrete spaces and the numerical procedures we have used. The first stable Tay-lor–Hood pair (continuous piecewise polynomials of degree two for the velocity and linear elements for thepressure [13]) has been taken for the finite element problem. The grids are uniform refinements of an initialunstructured grid. In what follows we denote by NQ the number of triangles of the mesh.

The solution to the corresponding linear system concentrates a great part of the difficulties of the finite ele-ment method. There exists a vast literature on numerical methods designed to solve this kind problems (see thesurvey [23] and references therein). In our numerical experiments we have used the classical Uzawa algorithm.It is well known that this method may require in practice a high number of iterations to achieve the conver-gence. One way to accelerate the convergence is the so-called augmented Lagragian technique. Essentially, thisinvolves working with the continuous equation

�Dv� .rðr � vÞ þ rq ¼ 0 on Q;

r � v ¼ 0;

cRv ¼ fR;

�� ð28Þ

and its corresponding numerical discretization

vh 2 Vh; qh 2 P h s:t:

cRvh ¼ fhR;R

Xrvh : rwh þ .R

Qðr � vhÞðr � whÞ �R

X qhðr � whÞ ¼ 0 8wh 2 Vh \H10ðXÞ;R

X rhðr � vhÞ ¼ 0 8rh 2 P h:

�� ð29Þ

Although (28) is equivalent to the Stokes equation, it does not remain valid for their corresponding discret-izations (29) and (21). However, the analysis exposed in this work can be straightforwardly adapted to coverthis case.

The Uzawa algorithm is now as follows [24]

(i) Solve
ZQru
ðnÞh : rvh þ .

ZQðr � uðnÞh Þðr � vhÞ ¼

ZQ

pðn�1Þh ðr � vhÞ 8vh 2 Vh \H1

0ðQÞ

cRuðnÞh ¼ fh

R:

(ii) Compute
ZQ
sðnÞh rh ¼Z

Qðr � uðnÞh Þrh 8rh 2 P h

and actualize pðnÞh ¼ pðn�1Þh � asðnÞh .

Note that there exists an additional parameter a in the algorithm, specifically when the pressure term isactualized. This parameter must be taken in (0, 2(1 + .)) to guarantee the convergence with the valuea = . + 1 suggested by several authors as the best choice [24]. The couple of parameters (a,.) controls thevelocity of convergence of the algorithm. Hence, higher values . > 0 improve the convergence for the velocity


but the system becomes more and more ill conditioned which results finally in a very poor approximation ofthe pressure.

However, the unknown of system (24) is just cRuh, and so it is possible to use a (relatively) high value of .and to take a smaller value to construct the solution in (25) if we are interested in good approximations for thepressure term.

Regarding the boundary element method, we have used a fully discrete Galerkin scheme with piecewiseconstant functions as trial and test space following the ideas presented in [25–27]. The discrete space is con-structed on an uniform grid of of say NC elements per curve based on a parameterization of them. The result-ing method has order three for right-hand sides smooth enough.

The potentials for both velocity and pressure are computed by means of a quadrature rule of order three.We remark that although the kernels of the single layer potentials are smooth, they suffer from a loss of qualitywhen evaluated close to C since both kernels become singular. This is especially problematic for the pressurebecause the kernel of the potential develops a non-integrable singularity. In contrast, the potential for thevelocity has a logarithmic singularity which is easier to deal with.

One way of circumventing this instability for the pressure is using the identities

PCsC ¼ PCnC ¼ 0; in extC;

where sC denotes the tangent vector. Let x be close to C and take ex 2 C the orthogonal projection of x on C.We take a and b such that

kðexÞ :¼ anCð~xÞ þ bsCð~xÞ:
Then, the numerical approximation for the pressure in x is computed by using
ðPCkCÞðxÞ ¼1

2p

ZC

ðx� yÞ � ðkCðyÞ � anCðyÞ � bsCðyÞÞjx� yj2

dy: ð30Þ

Note that the kernel is now continuous when x! C improving the quality of the numerical solution.Finally, the solution of the full linear system is carried out with GMRES. The good properties of the con-

tinuous equation IþK (recall that K is not only compact but its range is actually the trace of analytic func-tions) are inherited by the numerical method. Hence, the number of iterations required to achieve convergenceis low and independent of the discrete parameters. In our experiments we have checked that 25–30 iterationsare enough to converge to the solution with a tolerance of about 10�6.

We finish by showing the numerical results obtained in our test. We take the data so that the solution isgiven by ðuþSCkC; p þPCkCÞ where

u1ðx; yÞ ¼1

4

x2

x2 þ ðy � 0:6Þ2� x2

x2 þ ðy þ 0:6Þ2� 1

2log

x2 þ ðy � 0:6Þ2

x2 þ ðy þ 0:6Þ2

!" #;

u2ðx; yÞ ¼1

4

xðy � 0:6Þx2 þ ðy � 0:6Þ2

� xðy þ 0:6Þx2 þ ðy þ 0:6Þ2

" #;

and

ðSCkCÞ1ðx; yÞ ¼ðxþ 0:95Þ2

ðxþ 0:95Þ2 þ ðy � 0:25Þ2� ðxþ 0:525Þ2 þ y2

ðxþ 0:525Þ2 þ y2

!� 1

2log

ðxþ 0:95Þ2 þ ðy � 0:25Þ2

ðxþ 0:525Þ2 þ y2

!;

ðSCkCÞ2ðx; yÞ ¼ðxþ 0:95Þðy � 0:25Þðxþ 0:95Þ2 þ ðy � 0:25Þ2

� ðxþ 0:525Þyðxþ 0:525Þ2 þ y2

:

The associated pressure terms, up to an additive constant, are

pðx; yÞ ¼ 1

2

x

x2 þ ðy � 0:6Þ2� x

x2 þ ðy þ 0:6Þ2

" #

ðPCkCÞðx; yÞ ¼ 2xþ 0:95

ðxþ 0:95Þ2 þ ðy � 0:25Þ2� xþ 0:525

ðxþ 0:525Þ2 þ y2

" #:


Tables 1 and 3 show the maximum error in the nodes for the velocity for both the finite and the boundaryelement method. We emphasize that the method works independent of the level of discretization used in theboundary or finite problem. In each row/column the error decreases first and stagnates afterwards, just whenthe error of the other scheme, which are keeping with same discretization level, becomes the main source to thetotal error.

The same behavior is exhibited for the pressure term in Tables 2 and 4. Since PCkC is a higher degree ofmagnitude than ph we have written the relative errors in order to give a better comparison of both approxi-mations. Notice that the boundary element approximation deteriorates when evaluated close to the curve (seealso Fig. 6). Hence we conclude that the strategies developed in (30) mitigates the effects of this instability, butcertainly it seems that it is not enough to obtain good approximation in the vicinity of C when using coarsegrids.

In Tables 5 and 6 we write the maximum of the error |x(x) � xh(x)| and |q(x) � qh(x)| whenx 2 {(0.3,�0.3), (0.3,�0.2), . . . , (0.3,0.3)}. These points are far away from C in such a way that we can profitfrom the best order of the Galerkin scheme no matter in which norm it is attained. We observe that only a few

Table 1ku � uhk1,Q

NQnNC 20 40 80 160

534 1.392E�02 2.497E�03 2.449E�03 2.442E�032136 1.411E�02 2.434E�04 1.790E�04 1.784E�048544 1.416E�02 1.947E�04 2.451E�05 1.579E�05

Table 3kSCkC �SCkh

Ck1;XNQnNC 20 40 80 160

534 1.035E�01 7.020E�03 2.215E�03 2.208E�032136 1.034E�01 7.105E�03 1.963E�04 1.312E�048544 1.544E�01 2.203E�02 4.049E�04 5.467E�05

Table 5Pointwise error for |x � xh|

NQnNC 20 40 80 160

534 1.064E�04 1.424E�04 1.416E�04 1.416E�042136 7.147E�05 1.311E�05 1.287E�05 1.285E�058544 7.157E�05 1.640E�06 1.263E�06 1.309E�06

Table 4kPCkC �PCkh

Ck1;X=kPCkhCk1;X

NQnNC 20 40 80 160

534 1.332E�01 5.326E�03 7.553E�04 7.557E�042136 1.332E�01 5.336E�03 7.159E�04 1.883E�048544 1.588E�01 3.869E�02 1.280E�03 3.362E�04

Table 2kp � phk1,Q

NQnNC 20 40 80 160

534 8.305E�02 6.441E�02 6.456E�02 6.455E�022136 7.980E�02 1.052E�02 1.103E�02 1.104E�028544 7.897E�02 3.019E�03 2.979E�03 2.997E�03

Table 6Pointwise error for |q � qh|

NQnNC 20 40 80 160

534 5.302E�02 5.584E�03 6.021E�03 6.189E�032136 4.735E�02 2.030E�03 1.567E�03 1.579E�038544 4.674E�02 7.085E�04 2.307E�04 2.423E�04


degrees of the boundary element method are needed to obtain an approximation of the same order as thatgiven by the finite element scheme.

Finally, we depict in Figs. 4–6 the numerical solution for NQ = 8544 and NC = 160 as well as the committederror.

Fig. 6. Computed pressure of the FEM (left) and BEM (right) and the committed error (on the bottom).

Fig. 5. Components of the velocity computed by the BEM problem (top row) and the committed error (bottom row).

Fig. 4. uh (top row) and committed error (bottom row).


Acknowledgments

The authors are partially supported by FEDER/MCYT Project MTM2004-01905, DGA (Grupo consolid-ado PDIE), Gobierno de Navarra resolucion 18/2005 and MCYT project VEM2003-269.

Appendix A. Invertibility of the single layer operator

We study here which conditions ensure the invertibility of the single layer operator. We remark that ford = 3, there exists c > 0 depending only on C such that (see [28], Chapter 11)

ðkC;VCkCÞC P ckkCk2�1=2;C 8kC 2 H�1=2

n ðCÞ: ð31Þ

The invertibility of VC : H�1=2n ðRÞ ! H1=2

n ðCÞ is then a simple byproduct.The rest of this section is devoted to studying the bidimensional case, where in a similar way as happens

with the Laplace operator, the invertibility of the single layer operator depends finally on some geometricproperties of the curve.

Consider the exterior problem for the Stokes equation

v 2 H1locðR2Þ; q 2 L2

locðR2Þ; s:t:

cCv ¼ gC;

�Dvþrq ¼ 0 on R2 n C;

r � v ¼ 0 on R2 n C:

�� ð32Þ

The behavior of the velocity at infinity is essential and, in the final, it determines the solution of the problem.Hence, we can demand

limjxj!1

vðxÞ ¼ v1: ð33Þ

but, in contrast with the three dimensional case, we cannot prescribe its value a priori and therefore it turnsout to be a new unknown of the problem. This is the so-called Stokes paradox (cf. [16,1]).

For (v,q) solution of the Stokes equation, we introduce the normal stress on C

tCðv; qÞ :¼ T ½v; q�nC

nC being the outward normal vector at C and

T ½v; q� :¼ �p Id þ ðrvþ ðrvÞ>Þ
the stress tensor. We introduce also the double layer potential [1]
ðDCkCÞðxÞ :¼Z

CDðx; yÞkCðyÞ dy; x 2 R2 n C; ð34Þ

where D is the 2 · 2 matrix of functions whose jth column is

�pjðx� yÞnCðyÞ þ ½ryujðx� yÞ þ ðryujðx� yÞÞ>�nCðyÞ:

Note that each column is the result of applying tC to the corresponding column and component of U and p, thefundamental solution of Stokes system, in the variable y.

The solution of the Stokes system can be written in terms of the single and double layer operator. Hence, if(v,q) is a solution of the Stokes equation on the interior of C

v :¼ SCðtCðv; qÞÞ �DCðcCvÞ: ð35Þ
Analogously, if (v,q) is a solution of the exterior problem satisfying (33), then
v :¼ v1 �SCðtCðv; qÞÞ þDCðcCvÞ: ð36Þ
A similar formula exists for the pressure. For a proof of this result we refer to [1,16].


Notice that the normal stress tensor must satisfy
ZC
tCðv; qÞ ¼ 0

to fulfill the asymptotic behavior at infinity (33). This condition is always satisfied for any solution of theStokes system in the interior of C (obviously with the normal stress taken from the interior), consequence itselfof the second Green identity for the Stokes system as we observed in Section 4. However this argument failsfor solutions in the exterior of bounded domains since the value at infinity of the velocity and pressure must bealso considered.

In the light of this discussion, we introduce the following subspace:

H�1=20;n ðCÞ :¼ gC 2 H�1=2

n ðCÞZ

CgC ¼ 0

�� :

Theorem A.1. The mapping

H�1=20;n ðCÞ � R2 ! H1=2

n ðCÞðkC; cÞ 7!VCkC þ c

is an isomorphism.

Proof. Given gC 2 H1=2n ðCÞ, we denote by (v,q) the corresponding solutions of Stokes equation in R2 n C, with

the exterior solution satisfying (33).We recall the jump relations for the second layer potential

limx!C

DCgðxÞ ¼ 1

2gðxÞ þ

ZC

Dðx; yÞgðyÞ dy; ð37Þ

where the signs correspond to take the limit from the interior and the exterior of C, respectively. As a con-sequence of (35)–(37), we obtain

gC ¼1

2ðcCvþ þ cCv�Þ ¼ 1

2v1 þVCðt�C ðv; qÞ � tþC ðv; qÞÞ;

from where surjectivity follows. Injectivity is proven in a similar way. h

Let G:X! Y be a linear operator between two Hilbert spaces, and denote by R(G) the range of G. It is saidthat G is Fredholm if both the kernel and R(G)?, the orthogonal subspace of R(G), are finite dimensional. Theindex of the operator is then defined as

indG :¼ dim KerðGÞ � dimðRðGÞÞ?:
The index of an operator remains unchanged under compact perturbations. Moreover, if F:X! Y andG:Y! Z are Fredholm, indGF = indF + indG.
Lemma A.2. The mapping VC : H�1=2n ðCÞ ! H1=2

n ðCÞ is Fredholm of index zero.

Proof. Take

KC :¼KC 0

0 KC

� �; KCu :¼ � 1

4p

ZC

log j � �yjuðyÞ dy:

It is well known that KC:H�1/2(C)! H1/2(C) is Fredholm of index zero [10]. Moreover, KC :¼VC � KC is anintegral operator with smooth kernel (recall that C is smooth) and therefore compact. Hence, VC, acting fromH�1/2(C) into H1/2(C) is also Fredholm of index zero.

Finally, consider ıC : H�1=2n ðCÞ ! H�1=2ðCÞ the inclusion and the projection QC : H1=2ðCÞ ! H1=2

n ðCÞ givenby

QCgC :¼ gC �1

jCj

ZC

gC � nC

� �nC: ð38Þ


Obviously, ıC and QC are Fredholm of indices �1 and 1, respectively. Therefore,

indQCVCıC ¼ indQC þ indVC þ ind ıC ¼ 1þ 0� 1 ¼ 0;

and the result is proven. h

From the previous lemma, we just have to prove either the injectivity or the subjectivity to derive the invert-ibility of the single layer operator. In the next result we use the following notation: given C and r > 0,rC :¼ {rx | x 2 C}.

Theorem A.3. Given C a smooth simply connected curve, there exist at most two values of r > 0 such that

VrC : H�1=2n ðrCÞ ! H1=2

n ðrCÞ is not invertible.

Proof. Consider the following problem: given ðfC; dÞ 2 H1=2n ðCÞ � R2 find ðkC; cÞ 2 H�1=2

n ðCÞ � R2 such that

VCkC þ c ¼ fC;

ZC

kC ¼ d: ð39Þ

The operator H�1=2n ðCÞ � R2 ! H1=2

n ðCÞ � R2 defined by the left-hand side of this equation is Fredholm ofindex zero as a simple consequence of Lemma A.2. By Theorem A.1 the same operator is one-to-one, so(39) has always a unique solution.

Let then e1 = (1,0)>, e2 = (0,1)> and consider the solutions to

VCnðjÞC þ cj ¼ 0;

ZC

nðjÞC ¼ ej; j ¼ 1; 2

and the matrix

CC :¼ ½c1jc2�:
It is simple to prove that VC is injective, and therefore invertible, if and only if detCC 5 0.
If we scale the curve to rC (r > 0), it follows readily that

nðiÞrC :¼ 1

rnðiÞC

�r

; CrC ¼ CC �

1

4plogðrÞI2

which proves that VrC is invertible unless �1/4plog(r) is an eigenvalue of CC. h

Remark A.4. From the preceding theorem we conclude that VC is invertible except for some special rescalingsof the curve C. It is possible to prove for instance that if C is a circle, both critical sizes coincide with the radiusffiffiffi

ep

.On the other hand, the solution of the exterior problem given by the single layer-based formulation, that is,

kC 2 H�1=2n ðCÞ; s:t: VCkC ¼ gC; v :¼ SCkC; q :¼ PC

satisfies

vðzÞ ¼ 1

4p

ZC

kC

� �log jzj þ Oðjzj�1Þ; as jzj ! 1:

Therefore, this solution is different from that given in (32), (33) except when the density kC 2 H�1=20;n ðCÞ.

References

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A BEM–FEM overlapping algorithm for the Stokes equation

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