Continuous Analysis of the Additive Schwarz Method: a Stable Decomposition in $H^1$ with Explicit Dependence of the Constants on the Shape Regularity of the Decomposition

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Continuous Analysis of the Additive Schwarz Method: a Stable Decomposition in H ¹ with Explicit Dependence of

the Constants on the Shape Regularity of the Decomposition

Martin Gander, Laurence Halpern, Kévin Santugini-Repiquet

To cite this version:

Martin Gander, Laurence Halpern, Kévin Santugini-Repiquet. Continuous Analysis of the Additive

Schwarz Method: a Stable Decomposition in

with Explicit Dependence of the Constants on the

Shape Regularity of the Decomposition. 2011. �hal-00462006v3�

STABLE DECOMPOSITION IN H¹WITH EXPLICIT DEPENDENCE OF THE CONSTANTS ON THE SHAPE REGULARITY OF THE DECOMPOSITION

MARTIN J. GANDER, LAURENCE HALPERN, AND KÉVIN SANTUGINI

ABSTRACT. The classical convergence result for the additive Schwarz preconditioner with coarse grid is based on a stable decomposition. The result holds for discrete versions of the Schwarz preconditioner, and states that the preconditioned operator has a uniformly bounded condition number that depends only on the number of colors of the domain de- composition, the ratio between the average diameter of the subdomains and the overlap width, and on the shape regularity of the domain decomposition.

The classical Schwarz method was however defined at the continuous level, and simi- larly, the additive Schwarz preconditioner can also be defined at the continuous level. We present in this paper a continuous analysis of the additive Schwarz preconditioned operator with coarse grid in two dimensions. We show that the classical condition number estimate also holds for the continuous formulation, and as in the discrete case, the result is based on a stable decomposition, but now of the Sobolev space H¹. The advantage of such a contin- uous result is that it is independent of the type of fine grid discretization, and thus does the more natural continuous formulation of the Schwarz method justice. The upper bound we provide for the classical condition number is also explicit, which gives us the quantitative dependance of the upper bound on the shape regularity of the domain decomposition.

1. INTRODUCTION

With the generalization of parallelism in today’s computers, parallelizable mathematical algorithms are of increasing importance. Domain decomposition methods make it possible to perform numerical simulations in parallel, see for example the books [32, 30, 34], or the monographs [36, 8], and references therein. Consider a partial differential equation to be solved on a big domainΩ. In domain decomposition methods, an iterative approach intro- duced by Schwarz [31] is to decompose the big domainΩinto several smaller overlapping subdomainsΩi,Ω=^Sn_i=1Ωi, and then to compute approximations u^k_i defined by

(1) Lu^k_i = f inΩi,

u^k_i = u^k_j⁻¹ onΓi j,

whereΓi jdenote the interfaces. In practice, it is more efficient to use the general algorithm (1) as a preconditioner for a Krylov subspace method, like GMRES or conjugate gradients, see for example [18, 19] for a more detailed explanation. The Additive Schwarz operator defines one such preconditioned operator, related to (1). For a domain decomposition with both an overlap and a coarse mesh, Dryja and Widlund [13] proved that the condition number of the discrete Additive Schwarz operator is uniformly bounded, i.e. it does not depend on the number of subdomains. However, it depends on the number of colors of the domain decomposition, on the ratio between the diameter of the subdomain and the thickness of the overlaps, and on the shape regularity of the domain decomposition, see

2010 Mathematics Subject Classification. 65N55.

1

also Toselli and Widlund [34, Chap. 2]. Schwarz preconditioners have then mostly been analyzed at the discrete level, see for example [7, 28, 29, 22] for spectral discretizations, [3]

for the non-selfadjoint case, [4] for parabolic problems, [6] for some non-symmetric and indefinite problems, [5] for multiplicative versions of the algorithm, [10] for discretizations on unstructured meshes, [9] when also the coarse grid is non-matching, [15, 11, 26, 27]

for mixed finite element discretizations, [12] for mortar finite element problems, [16] for discontinuous Galerkin discretizations, and [17] for numerical linear algebra techniques.

For lower bounds on the convergence of Schwarz methods, see [2].

Schwarz domain decomposition methods are however naturally defined and analyzed at the continuous level, like in (1), see for example [23, 24, 25]. Schwarz methods were also invented by Schwarz at the continuous level [31], and the more recent class of optimized Schwarz methods was formulated and analyzed at the continuous level, for an introduction see [18] and references therein. It is however much less clear how to analyze a two level method at the continous level. In a recent review on coarse space components [35], we find the comment:

Early on, coarse spaces were not used and only continuous problems were considered; in fact it is unclear what a coarse problem then might be.

The purpose of our paper is to present an analysis of the two level Additive Schwarz opera- tor in a continuous setting, and to prove that its condition number is bounded independently of the number of subdomains. The proof succeeds by establishing the existence of a stable decomposition of every function in H₀¹(Ω)as a sum of functions belonging to the H₀¹(Ωi) plus a coarse function belonging to the space of continuous piecewise linear functions P₁(T)whereT is our coarse triangular mesh.

Our goal in this paper is to obtain at each step explicit upper bounds, including for the constants. To do so requires an explicit and quantitative definition of the notion of shape regularity. This opens the way for upper bounds of the condition of the Additive Schwarz operator when the underlying domain decomposition is not shape regular.

First, we recall in section 2 the definition of the preconditioned additive Schwarz op- erator, and the abstract results giving an estimate of the condition number of the Additive Schwarz operator as soon as three assumptions hold. The rest of the paper is then devoted to showing that these assumptions hold for a decomposition at the continuous level, the key assumption being the existence of a stable decomposition. After specifying in section 3 the geometric parameters of the domain decomposition, we prove in section 4 the existence of a stable decomposition in the continuous case in the absence of a coarse mesh albeit with a constant that depends on the number of subdomains. Section 5 is dedicated to proving our main theorem, Theorem 5.12, which establishes that, in the presence of a coarse mesh, there exists a uniformly stable decomposition with an explicit upper bound that does not depend on the number of subdomains. Using this result, we prove in section 6 that the con- dition number of the additive Schwarz operator has a uniformly bounded condition number in the continuous case when there is a coarse P₁mesh.

2. THEADDITIVESCHWARZ OPERATOR

In this section, we recall the abstract results in Toselli and Widlund [34, chap. 2]. Let (Vi)0≤i≤N be Hilbert spaces, with V₀being a coarse space. Let V=∑ⁿi=0R^T_iV_i, where the R^T_i are linear extension operators. Let a(·,·)be a symmetric, positive definite bilinear form

on V . We wish to find the unique u in V satisfying

a(u,v) = (f,v)for all v in V.

Letae_i(·,·)be symmetric positive definite bilinear forms on the V_i. We definePe_i: V→V_iby e

a_i(eP_iu,v) =a(u,R^T_i v) for all v in V_i. Let P_i=R^T_iPe_i. The additive Schwarz operator is defined by

(2) P_ad:=

∑

P_i.

This is an a-symmetric a-positive operator. We are interested in bounding the condition number (with respect to the bilinear form a) of the preconditioned operator P_ad.

Definition 2.1. Let a be a symmetric, positive bilinear form on a vector space V . Let P be a continuous linear application from V to V . We call

κ(P) =

max _uuu∈V

a(uuu,uuu)=1a(Pu,u) min _uuu∈V

a(uuu,uuu)=1a(Pu,u) the a-condition number of P.

Assumption 2.2 (Stable decomposition). There exists a constant C₀such that all u in V admit the decomposition

u=

∑

R^T_i u_i, with {u_i∈V_i}, and

∑

e

a_i(ui,ui)≤C²₀a(u,u).

(3)

Assumption 2.3 (Strengthened Cauchy Schwarz inequality). For all i,j≥1, there exist constants 0≤εi j≤1 such that for all u_i∈V_iand u_j∈V_jwe have

(4) |a(R^T_i u_i,R^T_ju_j)| ≤εi ja(R^T_i u_i,R^T_iu_i)¹²a(R^T_ju_j,R^T_ju_j)¹². We denote byρ(E)the spectral radius of the matrixE ={εi j}.

Assumption 2.4 (Local stability). There existsω>0 such that∀i≥0 and∀ui∈range(Pei) we have

(5) a(R^T_iui,R^T_iui)≤ωa_ei(ui,ui).

The following fundamental result can be found in Toselli and Widlund [34], see Theo- rem 2.7.

Theorem 2.5. Let Assumptions 2.2, 2.3 and 2.4, be satisfied. Then the a-condition number κ(P_ad)satisfies

(6) κ(P_ad)≤C₀²ω(ρ(E) +1).

Proof. The proof of Theorem 2.7 in Toselli and Widlund [34] also holds if the V_i have

infinite dimension.

In order to get a more concrete estimate, the strenghtened Cauchy-Schwarz Assump- tion 2.3 is often replaced in the literature by an assumption on the number of colors of the decomposition. The number of colors is defined as follows:

Definition 2.6 (Number of colors). In an abstract domain decomposition into the fine spaces (Vi)1≤i≤N, the number of colors is the smallest integer N_c such that there exists a partition of{1, . . . ,N}into N_csets(Ik)1≤k≤Nc such that R^T_iV_iis a-orthogonal with R^T_jV_j whenever i and j are distinct indices that belong to the same I_k. The fine spaces V_iand V_j are said to have the same color when i and j belong to the same I_k.

Then we can use the number of colors in estimate (6) instead of relying on the spectral radius of the strengthened Cauchy-Schwarz matrix.

Theorem 2.7. Let Assumptions 2.2, and 2.4, be satisfied. Suppose that the fine decompo- sition V_ihas N_ccolors. Then the a-condition numberκ(P_ad)satisfies

(7) κ(P_ad)≤C²₀ω(Nc+1).

Before proving the result we make the following remark:

Remark 2.8. In the literature, three distinct integers are used in estimate (7), and these constants can be defined both in the concrete geometric setting of domain decomposition, and in the abstract setting:

• In the concrete setting of domain decomposition, one can define N_k as the maxi- mum number N_kof neighbors, including itself, a subdomain can have. This integer is the connectivity of the domain decomposition. This number can replaceρ(E) in Theorem 2.5, since we always haveρ(E)≤N_k, see [34, Lemma 2.10] (where N_cis used as the name for this constant). In the abstract setting, one could define N_kas the maximum over i in{1, . . . ,N}of the number of R^T_jV_j, j in{1, . . . ,N}, which are not a-orthogonal to R^T_iV_i.

• The number of colors N_cwe defined in the abstract setting, see Definition 2.6, can also be defined in a transparent way in the concrete geometric setting of domain decomposition, see Definition 3.6. We always have N_c≤N_kin both the concrete and abstract setting, and thus proving a result with the constant N_c implies the result with the constant N_k.

• In the concrete setting of domain decomposition, one can define ˆN as the max- imum number of subdomains a point can belong to. In the abstract setting, one can define ˆN as the largest integer for which there exist ˆN distinct R^T_iV_i whose intersection is not{0}. We always have ˆN≤N_cin both the abstract setting and the concrete setting, so a result with the constant ˆN is the most accurate. In the concrete case, when theaeiare defined as integrals over a subdomain, it is possible to replace N_cwith ˆN in (7), see the original proof of [14, Th. 4.1]. It is unknown to the authors if the result with ˆN can be generalized to an abstract domain decom- position.

In the remainder of this paper, we always work with the number of colors N_c. We now proceed with the proof of Theorem 2.7

Proof. We only need to change part of the proof of Theorem 2.7 in Toselli and Widlund [34]. We already know that a lower bound for the smallest eigenvalue is 1/C₀², see [34, Lemma 2.5]. To get the estimate on the largest eigenvalue, we follow the ideas of [34,

Lemma 2.6] but additionally group the V_iby color. For each color k in{1, . . . ,N_c}, we get:

a(

∑

i∈I_k

P_iu,

∑

j∈I_k

P_ju) =

∑

i∈I_k

∑

j∈I_k

a(Piu,P_ju) =

∑

i∈I_k

a(Piu,P_iu)≤ω

∑

i∈I_k

e

a_i(Pe_iu,Pe_iu)

≤ω

∑

i∈I_k

a(Piu,u)≤ωa(

∑

i∈I_k

Piu,u)≤ωa(

∑

i∈I_k

Piu,

∑

j∈I_k

Pju)¹²a(u,u)¹².

Dividing by a(∑i∈I_kP_iu,∑j∈I_kP_ju)¹², we therefore get a(

∑

i∈I_k

P_iu,

∑

j∈I_k

P_ju)¹² ≤ωa(u,u)¹², and thus can estimate using again the Cauchy-Schwarz inequality

a(

∑

i∈I_k

Piu,u)≤a(

∑

i∈I_k

Piu,

∑

j∈I_k

Pju)¹²a(u,u)¹² ≤ωa(u,u).

We also know that a(P0u,u)≤ωa(u,u), see [34, Lemma 2.6]. Therefore, summing over all colors and P0, we get a(Padu,u)≤(Nc+1)ωa(u,u)for all u in V . While the local stability and the strengthened Cauchy-Schwarz inequality can naturally be extended to the continuous case, the stable decomposition result is traditionaly shown using properties of the fine discretization of the problem, see for example Toselli and Wid- lund [34]. For a continuous formulation, we need to use other techniques, which is the purpose of this paper.

3. GEOMETRY AND DECOMPOSITION INTO SUBDOMAINS

FIGURE1. Domain decomposition with a coarse mesh First, we recall the defintion of a domain:

Definition 3.1. A domain ofR²is an open connected set ofR²whose boundary∂Ωis of null Lebesgue measure¹. We denote by|Ω|the Lebesgue measure of the domainΩ.

We recall the definition of a non overlapping and an overlapping domain decomposition:

1It is possible for a pathological open connected set ofR²to have a boundary with strictly positive measure.

For example(0,1)×(1/4,3/4)∪^S∞j=1S2^j−1−1

k=0 (^2k+1₂j −2^{−4 j},^2k+1₂j +2^{−4 j})×(0,1)is open, connected and dense in(0,1)×(0,1)but has a measure smaller than 9/14.

Definition 3.2 (Non Overlapping Decomposition). LetΩbe a bounded domain ofR². A collection of domains(Ui)1≤i≤N, is a non overlapping domain decomposition ofΩif

(8) Ω=

[N

i=1

U_i, U_i∩U_j=/0 for all i6= j.

Definition 3.3 (Overlapping Decomposition). LetΩbe a bounded domain ofR². A col- lection of domains(Ωi)1≤i≤Nis an overlapping domain decomposition ofΩif

Ω= [N

i=1

Ωi.

In this article, we use the parameter H to represent the average size of subdomains. For the definition of H, we use the concept of diameter, which we recall here:

Definition 3.4 (Domain Diameter). Let U be a bounded subset of R². We define the diameter of U to be

diam(U) =sup

xxx∈U yyy∈U

kxxx−yyyk.

The concept of an overlapping domain decomposition raises the question on how to define the overlap width of the decomposition. We use the following definition:

Definition 3.5 (Overlap of the Decomposition). A domain decomposition(Ωi)1≤i≤N is said to have overlap width²δ >0, if there exists a non overlapping domain decomposition (Ui)1≤i≤NofΩsuch that for all i, 1≤i≤N, U_i⊂Ωiand

{xxx∈Ω|dist(xxx,Ui)<δ} ⊂Ωi.

In practice, it is easier to start with a non overlapping domain decomposition, and then to build from it an overlapping one. If(Ui)1≤i≤Nis a non overlapping domain decomposition ofΩ, then the(Ωi)1≤i≤N defined by

(9) Ωi={xxx∈Ω|dist(xxx,Ui)<δ_}

forms an overlapping domain decomposition ofΩ with overlap widthδ. We denote by ((Ui)1≤i≤N,(Ωi)1≤i≤N)such a decomposition.

Definition 3.6 (Colors of the Decomposition). The number of colors of an overlapping domain decomposition(Ωi)1≤i≤N of domainΩis the smallest integer N_c such that there exists a partition of{1, . . . ,N}into N_csets(Ik)1≤k≤Ncsuch that

Ωi∩Ωj=/0, whenever i6=j and i,j both belong to the same color I_k.

In order to determine the number of colors of the decomposition, it is easiest to consider the nonoverlapping decomposition(Ui)1≤i≤N. Clearly the number of colors can only in- crease withδ. However forδ small enough it remains constant: there existsδ0and N_c>0 that depend only on the(Ui)such that for allδ^{, 0}<δ <δ0, the overlapping domain de- composition(Ωi)1≤i≤Nwith overlap widthδ derived from the(Ui)1≤i≤N has a number of colors equal to N_c.

2Geometrically, the parameterδcorresponds to half the overlap of the subdomains

Remark 3.7. The geometric Definition 3.6 for the number of colors is equivalent to the algebraic Definition 2.6 for the bilinear forms implied by the geometric domain decom- position, like in this paper, where R^T_iV_i contains all functions that are H₀¹ inΩand null outsideΩiand a is an integral overΩ.

4. STABLE DECOMPOSITION WITHOUT A COARSE MESH

To understand the importance of the coarse mesh, we begin by proving the existence of a stable decomposition without a coarse mesh. In that case, the constant C₀in (7) stemming from the stable decomposition depends on the number of subdomains. We consider a bounded domainΩ being decomposed into N overlapping subdomains Ωi with overlap widthδ. We make the following assumptions on the domain decomposition:

Assumption 4.1. The domain decomposition(Ωi)1≤i≤Nis derived from a non overlapping one by formula (9), and we refer to it by((Ui)1≤i≤N,(Ωi)1≤i≤N).

Assumption 4.2. Let H be the smallest diameter among the diameters of the subdomains U_i. We suppose there exist uniform parameters C_d>0, c_a>0 and C_a>0 such that for all i in{1, . . . ,N}

(10) H≤diam(Ui)≤C_dH, c_aH²≤ |U_i| ≤C_aH², where|U_i|is the Lebesgue measure of the subdomain U_i.

To construct the stable decomposition, we use a partition of unity.

Lemma 4.3 (Partition of Unity). LetΩbe an open domain ofR², N>0 be the number of subdomains, and(Ui)1≤i≤N be domains ofR²satisfying (8). Withδ >0 the overlap width, we define

Ωei={xxx∈R²|dist(xxx,Ui)<δ},

and denote by Ncthe number of colors of this domain decomposition³. Then, there exists a universal⁴constantλ2, 0<λ2≤6, and N functions(ψi)1≤i≤N inC^∞(R²)having the following properties:

(1) For all i in{1, . . . ,N},ψivanishes outside ofΩei. (2) For all xxx inR², 0≤ψi(xxx)≤1.

(3) For all xxx inΩ,∑iψi(xxx) =1.

(4) For all xxx inΩ,∑^N_i=1k∇ψi(xxx)k²≤2λ₂^2(Nc_δ⁻2¹⁾².

Proof. The result is classical and well known, see [1, Th. 3.15], we only show how to obtain the explicit constant in the bound given in 4. We start with a functionρ^inC∞(R²) which vanishes outside the unit ball, and satisfies for all xxx inR²that 0≤ρ(xxx)≤1, and the integral^R_R2ρ(xxx)dxxx=1. For allε>0, we then setρε(xxx) =_ε¹₂ρ(^xxx_ε), and we define for all i in{1, . . . ,N}the function

h_i(xxx) =

(1 if dist(xxx,Ui)<^δ₂, 0 otherwise.

We now regularize the functions hiusing a convolution, φi:=ρ_δ/2∗h_i.

3The ˜Ωican extend beyond the domainΩ, in contrast to theΩidefined earlier

4It depends only on the dimension but we have restricted ourselves to two-dimensional domains

The functionsφivanish outside ofΩei, are identically equal to 1 in U_i, and for all xxx inR² we have 0≤φi≤1. Moreover, sincek∇φikL^∞(R²)≤ k∇ρδ

2kL¹(R²)kh_ikL^∞(R²), k∇φikL^∞(R2)≤2k∇ρkL¹(R²)

δ ^.

We then set

ψi=φi i−1 k=1

∏

(1−φk), and the(ψi)1≤i≤Nare then a partition of unity. Moreover

∇ψi=∇φi i−1

∏

(1−φk)−

i−1

∑

φi∇φj i−1 k=1

∏

k6=j

(1−φk).

At a given point xxx, at most N_c−1 terms of the above sum may be non zero, therefore, by the Cauchy-Schwarz inequality, we obtain

∑

k∇ψi(xxx)k²≤(Nc−1)

∑

k∇φi(xxx)k²

i−1 k=1

∏

(1−φk)²+

∑

i−1

∑

|φi|²k∇φj(xxx)k²

i−1 k=1,k

∏

(1−φk)²

≤(Nc−1)

∑

k∇φi(xxx)k²

i−1 k=1

∏

(1−φk)²+

∑

∑

|φj|²k∇φi(xxx)k²

j−1 k=1,k

∏

(1−φk)²

≤(Nc−1)

∑

k∇φi(xxx)k²

i−1 k=1

∏

(1−φk)² 1+

∑

|φj|²

j−1 k=i+1

∏

(1−φk)²

!

≤(Nc−1)

∑

k∇φi(xxx)k²

i−1 k=1

∏

(1−φk)² 1+

∑

φj j−1 k=i+1

∏

(1−φk)

!

≤(Nc−1)

∑

k∇φi(xxx)k²

i−1 k=1

∏

(1−φk)² 2−

∏

(1−φk)

!

≤2(Nc−1)

∑

k∇φi(xxx)k².

Moreover, each term is bounded by max_1≤j≤ik∇φjk²_L∞(Ω), and at no point xxx in Ω, there may be more than N_c−1 nonzero terms in the sum. Hence, for all xxx inΩ

∑

k∇ψi(xxx)k²≤8(Nc−1)²k∇ρk²_L1(R²)

δ^{2 .}

Settingλ2:=2k∇ρk²_L1(R²), the result follows. Note that herek∇ρkL¹(R2)=^RR²(|∂xρ|²+

|∂yρ|²)^1/2dxxx. Using the W^1,1(R²)functionρ(xxx) =1− kxxxk2, we obtain the estimateλ2=

6.

It is easy to build a stable decomposition using a partition of unity, however to get an estimate in H/δ instead of an estimate in H²/δ²we need more assumptions on the regularity of the U_i, specifically we must control the curvature of the boundary of the U_i. Unfortunately, the subdomains of a non overlapping domain decomposition are at best piecewiseC^∞: there will always be corners at cross points. For this reason, we introduce the notions of pseudo normal and pseudo curvature:

Assumption 4.4. Let U be a bounded domain ofR². We suppose there exist an open layer L containing∂U and a vector field XXX continuous on L∩U ,C^∞on L∩U such that:

DXXX(xxx)(XXX(xxx)) =0, kXXX(xxx)k=1

and such that there existsε0>0 such that for all positiveε<ε0and for all ˆxxx in∂^{U :} ˆxxx+ε^XXX(ˆxxx)∈U, ˆxxx−ε^XXX(ˆxxx)∈/U.

The vector field XXX is called an interior pseudo normal.

Setting for all positiveδ

U^δ ={xxx∈U s.t. dist(xxx,∂U)<δ}, V^δ ={ˆxxx+sX(ˆxxx),ˆxxx∈∂U,0<s<δ},

we assume there exist ˆR>0,θXXX, 0<θXXX≤π/2 andδ0, 0<δ0≤R sinˆ θXXXsuch that V^Rˆ⊂L∩U,

U^δ⊂V^δ/sinθXXX for all positiveδ≤δ0.

The parameterθXXX formally represents the smallest angle between the pseudo normal and the tangents. We finally set

R :=˜ 1 kdiv XXXkL^∞(L)

. We call ˜R the XXX -pseudo curvature of U .

When the boundary of the domain U isC¹, XXX is the interior normal. Unfortunately, as the Uiform a non overlapping domain decomposition ofΩ, they cannot be supposed to be C¹. It is perfectly reasonnable to assume the existence of a pseudo normal for Lipshitz domains, see [21, §1.5].

Using these assumptions, we can prove the following lemma:

Lemma 4.5. Let U be an open domain that satisfies assumptions 4.4, then for allδ _≤δ0, we have

kuk²_L2(U^δ)≤2

1+Rˆ R˜

δ^Rˆ

sinθXXXk∇uk²_L²_(U)+2

1+Rˆ R˜

δ

R sinˆ θXXXkuk²_L²_(U). Proof. We have

kukL²(U^δ)≤ kuk_L²_(V^δ/sinθXXX). For all xxx in V^Rˆ, we define

d(xxx) =inf{s,xxx−sXXX(xxx)∈/L∩U}.

The function d is lower semicontinuous. Note that d(xxx+sXXX(xxx)) =d(xxx+sXXX)provided the whole segement[xxx,xxx+sXXX(xxx)]belongs to L∩U . Also note that for allδ<Rˆ

V^δ={xxx∈V^Rˆs.t. d(xxx)<δ}. Define functionψ^by

ψ(xxx) =xxx+ (R sinˆ θXXX

δ ^{−1)d(xxx)XXX(xxx).}

for all xxx in V^δ/sinθXXX. We have d(ψ(xxx) =^{R sinθˆ} _δ ^XXXd(xxx)and:

Z

V^δ/sinθXXX|u(xxx)|²dxxx≤2 Z

V^δ/sinθXXX|u(ψ(xxx))|²dxxx

| {z }

I

+2 Z

V^δ/sinθXXX

d(xxx)(R sinˆ θXXX

δ ⁻¹⁾

Z _d(xxx)(^{R sinˆ θXXX}

δ −1)

0 |∇u(xxx+sXXX(xxx))|²dsdxxx

| {z }

II

. (11)

To further estimate the term I, we need to compute the Jacobian ofψ: let us first suppose that d isC¹. In the orthonormal basis(τττ1,τττ2)whereτττ1=XXX(xxx)andτττ2is orthogonal to τττ1, we have

Jψ(xxx) =

"_{R sinθˆ}

X X

δ X (^{R sinθˆ} _δ ^XXX −1)_∂τττ^∂d

2

0 1+ (^{R sinθˆ} _δ ^XXX −1)d(xxx)div XXX(xxx)

# .

Therefore, sinceψ(V^δ/sinθXXX) =V^Rˆ, we get det(Jψ(xxx)) =R sinˆ θXXX

δ ^{(1+ (} R sinˆ θXXX

δ ⁻1)d(xxx)div XXX(xxx)).

This does not depend on the derivatives of d. Besides, one can prove that for all s inR such that the segment[xxx,xxx+sXXX(xxx)]is included in L∩U :

(1+s div XXX(xxx))(1−s div XXX(xxx+sXXX)) =1.

Therefore, setting yyy=ψ(xxx), we get I=

Z

V^δ/sinθXXX|u(ψ(xxx))|²dxxx

= δ

R sinˆ θXXX Z

V^Rˆ|u(yyy)|²(1−(1− δ R sinˆ θXXX

)d(yyy)div XXX(yyy))dyyy

≤(1+Rˆ R˜) δ

R sinˆ θXXX Z

V^Rˆ|u(yyy)|²dyyy.

This formula holds even when d is notC¹: the idea is to prove by Fubini that the formula holds on open subsets of the form V_xxx={xxx+rτττ2+sXXX(xxx+rτττ2),0<r,s<ε}whereτττ2is orthogonal to XXX(xxx), and then to proceed by way of a partition of unity. Therefore we have

(12) |I| ≤(1+Rˆ

R˜) δ

R sinˆ θXXXkuk²_L²_(L∩U). We now deal with the term II: we compute

II= (R sinˆ θXXX

δ ⁻¹⁾ Z

V^δ/sinθXXX

d(xxx)

Z _d(xxx)(^{R sinˆ} _δ^θXXX₋₁₎

0 |∇u(xxx+sXXX(xxx))|²dsdxxx

= (R sinˆ θXXX

δ ⁻¹⁾

Z Rˆ−sin^δθ_XXX

0

Z

V^Rˆχ _ˆ ^s

R sinθXXX/δ−1 <d(xxx)< δ sinθXXX

d(xxx)|∇u(xxx+sXXX(xxx))|²dxxxds,

and then using the change of variables yyy=xxx+sXXX(xxx)we obtain

II= (R sinˆ θXXX

δ ⁻¹⁾

Z Rˆ−sin^δθ_XXX

0

Z

V^Rˆχ ^{s ˆR sin}θXXX

R sinˆ θXXX−δ ^<d(yyy)<

δ sinθXXX

+s

(d(yyy)−s)|∇u(yyy)|²(1−s div XXX(yyy))dyyyds

= (R sinˆ θXXX

δ ⁻¹⁾ Z

V^Rˆ Z ^d(yyy)_ˆ

R (Rˆ−_sin^δ_θ

XX X) (d(yyy)−δ/sinθXXX)+

(d(yyy)−s)|∇u(yyy)|²(1−s div XXX(yyy))dyyyds

≤(R sinˆ θXXX

δ ⁻¹⁾ Z

V^Rˆ|∇u(yyy)|² Z ^d(yyy)

Rˆ (Rˆ−δ/sinθXXX) (d(yyy)−δ/sinθXXX)+

(d(yyy)−s)(1−s div XXX(yyy))dsdyyy

≤(R sinˆ θXXX

δ ⁻¹⁾⁽¹⁺

Rˆ−δ/sinθXXX

R˜ ) δ² sin²θXXX

Z V^Rˆ

1−d(yyy) Rˆ

|∇u(yyy)|²dyyy

≤(1+Rˆ

R˜)(Rˆ− δ sinθXXX

) δ sinθXXX

Z

V^Rˆ|∇u(yyy)|²dyyy.

We thus obtain the estimate

(13) |II| ≤

1+Rˆ

R˜

Rˆ δ

sinθXXXk∇uk²_L2(V^Rˆ).

Combining inequalities (12) and (13) with inequality (11) concludes the proof.

Theorem 4.6 (Stable Decomposition without Coarse Grid). LetΩbe a bounded domain ofR², and(Ui)1≤i≤Nbe a non overlapping domain decomposition ofΩ. We suppose there exist ˜R, ˆR and 1/sinθXXX such that for each U_i there exists an open layer L_i containing

∂^Ui, a vector field XXX_icontinuous on L_i∩U_i,C^∞on L_i∩U_isuch that DXXX_i(xxx)(XXX_i(xxx)) =0, kXXX_i(xxx)k=1,kdiv XXX_ik∞≤1/R, and˜ ε0>0 such that for all positiveε<ε0and for all ˆxxx in∂^Ui, ˆxxx+ε^XXXi(ˆxxx)∈U and ˆxxx−ε^XXXi(ˆxxx)∈/U . Setting, for all positiveδ^{′, U}_i^δ′ :={xxx∈ U, dist(xxx,∂^Ui)<δ^′}, and V_i^δ′ :={ˆxxx+sXXX_i(ˆxxx),ˆxxx∈∂^Ui,0<s<δ^′}, we assume there exists aδ0, 0<δ0≤R sinˆ θXXX such that V_i^Rˆ⊂L_i∩U_iand U_i^δ′ ⊂V_i^{δ′/sinθXXX} for all positive δ^′≤δ0.

Letδ <δ0be positive. Set Ωi={xxx∈Ω|dist(xxx,∂Ωi)<δ}. The(Ωi)1≤i≤N form an overlapping domain decomposition ofΩ.

Then, if u is in H₀¹(Ω), there exist(ui)1≤i≤Nsuch that for all i, 1≤i≤N, uiis in H₀¹(Ωi) and

u=

∑

u_i, with (14)

∑

ku_ik²_L²_(Ωi)≤ kuk²_L²_(Ω), (15)

∑

k∇uik²_L²_(Ωi)≤2k∇uk²_L²_(Ω)+4λ₂²(Nc−1)²

δ^{2 kuk2L2(Ωδ),} (16)

whereλ2is the universal constant of Lemma 4.3 and whereΩ^δ=^S_i6=jΩi∩Ωj. We further have:

∑

k∇uik²_L²_(Ωi)≤

2+8λ2²(Nc−1)² 1+Rˆ

R˜ Rˆ

δ^sinθXXX

k∇uk²_L²_(Ω) +8λ2²(Nc−1)²

1+Rˆ R˜

1 Rˆδ^sinθXXX

kuk²_L²_(Ω). (17)

Proof. We use Lemma 4.3 and set u_i:=ψiu, which satisfies already (14). We then estimate

∑

Ω|u_i(xxx)|²dxxx= Z

Ω

∑

|ψi(xxx)u(xxx)|²dxxx= Z

Ω|u(xxx)|²

∑

(ψi(xxx))²dxxx≤ Z

Ω|u(xxx)|², since∑^N_i=1(ψi(xxx))²≤1, which shows (15). We finally need to estimate the derivative term.

We have∇ui=ψi∇u+u∇ψi, and therefore (18)

∑

Ω|∇ui(xxx)|²dxxx≤2 Z

Ω|∇u(xxx)|²

∑

|ψi|²dxxx+2 Z

Ω|u(xxx)|²

∑

|∇ψi|²dxxx.

The first term on the right in (18) can be bounded as above. To bound the second term, we use result 4 in Lemma 4.3:

(19)Z Ω|u(xxx)|²

∑

|∇ψi(xxx)|²dxxx≤ k

∑

|∇ψi|²kL^∞(R²) Z

Ω|u(xxx)|²dxxx≤2λ₂²(Nc−1)²

δ^{2 kuk2L2(Ωδ).} Combining these estimates leads to (16). To get (17), one first notice that

kuk²_L2(Ω^δ)≤

∑

kuk²_L2(U_i^δ)

then apply Lemma 4.5 on each U_i^δ.

The lone 1/δ²factor in estimate (16) can further be treated using the Poincaré inequality onΩ, see [1, Th. 6.30], which then explicitly reveals the dependence on the number of subdomains:

Corrollary 4.7. LetΩbe a bounded domain. Let U_i,Ωi and(ui)1≤i≤N be as in Theo- rem 4.6. Then we have

∑

k∇u_ik²_L2(Ωi)≤

2+8λ₂²(Nc−1)² 1+Rˆ

R˜ Rˆ

δ^sinθXXX

+λ₂²CaN(Nc−1)² 1+Rˆ

R˜

(diam(Ω))²

|Ω|

H R sinˆ θXXX

H δ

k∇uk²_L²_(Ω), (20)

where C_ais the constant of Assumption 4.2.

Proof. We start with (16), and use Poincaré’s inequality on H₀¹(Ω), i.e.

kuk²_L²_(Ω)≤Ck∇uk²_L²_(Ω),

but we need an estimate of the constant C. Since Ω is, up to a rotation, a subset of (0,diam(Ω))×R, the constant C is bounded by the Poincaré constant for(0,diam(Ω))×R,