Weighted regularization for composite materials in electromagnetism

DOI:10.1051/m2an/2009041 www.esaim-m2an.org

WEIGHTED REGULARIZATION FOR COMPOSITE MATERIALS IN ELECTROMAGNETISM

Patrick Ciarlet, Jr.

, Franc ¸ois Lef` evre

, St´ ephanie Lohrengel

and Serge Nicaise

Abstract. In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materi- als. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace ofH(curl; Ω) whose fieldsu satisfy w^αdiv(εu) ∈L²(Ω) and have vanishing tangential trace or tangential trace inL²(∂Ω). The weight functionw(x) is equivalent to the distance of xto the geometric singularities and the minimal weight parameter αis given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of typeP1 andP2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed.

Mathematics Subject Classification. 78M10, 65N30, 78A48.

Received September 10, 2008.

Published online November 3, 2009.

1. Introduction

The question of approximability of the solution of Maxwell’s equations by means of nodal finite elements has been widely studied in the last ten years (see e.g. [2,4,5,10,18,31] for perfect conducting boundary con- ditions and homogeneous materials). In a regular domain of class C¹ as well as in a convex polyhedron, the discretization of the time-harmonic Maxwell equations can be performedviastandard Lagrange Finite Elements by solving an equivalent regularized variational formulation similar to the vector Helmholtz equation (see [24]).

In a non-convex polyhedron, however, this approximation fails since the electromagnetic field does in general present singularities near the reentrant edges and corners (see e.g. [7,8,17]) and the discretization space is

Keywords and phrases.Maxwell’s equations, interface problem, singularities of solutions, density results, weighted regularization.

1 Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32 boulevard Victor, 75739 Paris Cedex 15, France. [email protected]

2 Laboratoire de Math´ematiques, FRE 3111, UFR Sciences exactes et naturelles, Universit´e de Reims Champagne-Ardenne, Moulin de la Housse – B.P. 1039, 51687 Reims Cedex 2, France. [email protected];

[email protected]

3 LAMAV, Universit´e de Valenciennes et du Hainaut Cambr´esis, Le Mont Houy, 59313 Valenciennes Cedex 9, France.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2009

no longer dense in the vector space of the variational formulation. The same situation does occur in composite materials where the electric permittivity and the magnetic permeability are piecewise constant functions. The electromagnetic field then presents singularities near the exterior and interior edges and corners of the different subdomains [19].

In order to overcome the lack of density, several possibilities have been studied. The singular complement method [6] and singular field method [25] add explicitly the singularities to the discretization space according to the splitting of the electromagnetic field into a regular part and a singular part deriving from a scalar potential.

Another possibility is the penalization of the perfect conducting boundary condition by an impedance-like condition. From a theoretical point of view, the density result of the FE-space in the variational space holds true for any homogeneous material (see [15,16]) and some composite materials (see [28]). The numerical performances of this method, however, are rather poor. The idea of weighted regularization has been developed in [18] for homogeneous materials. It consists in looking for the solution in the subspace of H(curl; Ω) of fields with divergence in a weighted L²-space, whereas the classical regularized formulation corresponds to the L²-space without weight.

In this paper, we study the method of weighted regularization for composite materials and prove the density of the space of piecewise regular vector fields in the space of the weighted regularization method, for an appropriate choice of the weight parameter. The idea of the proof is similar to the proof in [28] where the case of classical regularization with impedance boundary condition has been addressed. It consists in proving that the orthogonal of the closure of the space of piecewise regular vector fields is reduced to {0}. However, if the density result for classical regularization with impedance boundary condition always holds true in the case of homogeneous materials, it may fail for some composite materials. On the contrary, the method of weighted regularization allows one to choose the weight parameter depending on the singularities of a scalar second-order transmission problem and hence, the density result may be recovered for any composite material.

The paper is organized as follows: the theoretical aspects of the problem are dealt with in Section2. More precisely, in Section2.1, we give the geometric setting and the functional framework including a perfect conduct- ing boundary as well as an impedance boundary condition. We also address equivalence between the weighted regularized formulation and the original Maxwell equations. In Section 2.2, we show that the density problem for vector fields can be reduced to a similar density problem for the associated scalar potentials. The weight function in two dimensions will be defined in Section2.3. The proof of the density result in a two-dimensional domain is developed for a more general family of two-dimensional scalar problems depending on a real param- eter. This turns out to be useful in order to deal with the three-dimensional case where the real parameter represents the (local) edge variable. Section2.4is devoted to the proof of the density result in a polyhedron. In Section3, we state precisely the discretization by means of Lagrange Finite Elements of typeP_kand give a basic convergence proof. Finally, Section4 is devoted to a series of numerical tests performed in two dimensions. In Section4.1, we present the resolution of the static problem with source term for the electric field in an L-shaped domain with three subdomains. Depending on the value of the electric permittivity, the main singularity of the electric field can be arbitrarily strong and thus it is challenging for any numerical method. The numerical results show clearly that the weighted regularization method does converge to the exact singular solution whereas the classical regularization method does not. Further, we provide numerical convergence rates for Finite Elements of typesP₁andP₂on uniform and refined meshes. Next, we study in Section4.2the performance of the weighted regularization method for the eigenvalue problem and we compare our results to a benchmark in the case of an interior singularity in a “checkerboard-like” domain decomposed into four subdomains.

2. Weighted regularization in the case of mixed boundary conditions 2.1. Setting of the problem

In this section we will define precisely the geometric setting, which is the same as the one in [28]. Further, we introduce the variational formulation of the weighted regularization problem as well as the associated functional spaces. Whenever possible, we adopt the notations of [28].

We are concerned with an open bounded set Ω ⊂R^d where d= 2 or 3. We assume that Ω is a Lipschitz polygon (d= 2) or a Lipschitz polyhedron (d= 3) which means that Ω is a Lipschitz domain with piecewise linear (d= 2) or plane (d= 3) boundary ∂Ω. We denote by n the unit outward normal vector to ∂Ω. We further assume that Ω is connected and simply connected and that its boundary∂Ω is connected⁴.

It follows from the Maxwell equations that the electric fieldEis a solution to curl

μ⁻¹curlE

−ω²εE= iωJ, (2.1)

where the time variation is assumed to be in e^−iωt, with ω∈R. In the sequel, we set f= iωJ.

The coefficients ε and μ are, respectively, the permittivity and the permeability of the medium in Ω, and J ∈ L²(Ω)^d is a datum which represents the impressed current density. We assume that J (and thus f) is divergence-free which amounts to saying that the electric charge density vanishes in the whole domain Ω.

In the case of composite materials, the electromagnetic coefficientsε andμare given by piecewise constant functions. This defines a partitionP of Ω into a finite number of subdomains Ω₁, . . . ,Ω_J such that on each Ω_j we haveε(x) =ε_j>0 andμ(x) =μ_j>0.

We assume that each subdomain is itself a polygon (d= 2) or a polyhedron (d= 3) with Lipschitz boundary, and we denote byF_jk the edges or faces of∂Ω_j∩∂Ω_k. We distinguish between the setsF_intandF_ext of interior faces (contained in Ω) and exterior faces (contained in∂Ω). Without loss of generality, we may assume that the subdomains are connected and simply connected and have a connected boundary (see a similar remark in [28]).

In order to deal both with the boundary condition of a perfect conductor and an impedance boundary condition, let{ΓD,Γ_I}denote a partition of∂Ω such that

Γ_D∪Γ_I =∂Ω,

Γ◦_D∩Γ^◦_I=∅. (2.2)

This induces a partition ofFext intoFD={F ∈ Fext|F⊂Γ_D} andFI ={F ∈ Fext|F ⊂Γ_I}. The electric field then satisfies the following mixed boundary condition:

E×n= 0 on Γ_D,

n×(E×n) +λ(n×μ⁻¹curlE) = 0 on Γ_I. (2.3) Above, λis a complex number proportional to the characteristic impedance of the surrounding conductor and satisfying

Re λ≤0.

The variational formulation of problem (2.1)–(2.3) is given on the functional space u∈ H(curl; Ω)div(εu) = 0; (u×n)_|ΓD = 0; (u×n)_|ΓI ∈L²(Γ_I)ⁿ

,

where n = 1 if d = 2 and n = 3 if d = 3, and enters within the framework of the Fredholm alternative.

Hence, (2.1)–(2.3) admits a unique solutionu satisfying div(εu) = 0 if, and only if,ω²∈σ(curl,divε⁰) where σ(curl,divε⁰) is the discrete spectrum of the involved partial differential operator.

4On the one hand, the case where∂Ω consists of a finite number of connected components could be easily included, but would result in more complicated notations. On the other hand, the case of a multiply connected domain is more involved, since one has to deal with cuts, and we refer to [1,30] for a more detailed discussion. However, our results should carry over to this more general setting, since they depend only onlocalgeometry considerations.

As in [18] for the homogeneous case, we consider a weighted regularized formulation of problem (2.1)–(2.3).

To this end, we denote byY a (separable) Hilbert space with scalar product ·,·Y such that

L²(Ω)→Y →H⁻¹(Ω). (2.4)

The variational spaceW[Y] is then given by W[Y] =

u∈ H(curl; Ω)div(εu)∈Y; (u×n)_|ΓD = 0; (u×n)_|ΓI ∈L²(Γ_I)ⁿ

. (2.5)

The spaceW[Y] is equipped with its canonical norm

||u||_W[Y] =

||u||²_0,Ω+||curlu||²_0,Ω+||div(εu)||²_Y +u×n²_0,ΓI1/2 .

The variational formulation corresponding to the spaceY now reads as follows

(P[Y]) Findu∈W[Y] such that

a(u,v)−ω²(εu,v) = (f,v)∀v∈W[Y], where the sesquilinear forma(·,·) is given by

a(u,v) =

Ω

μ⁻¹curlu·curlvdx+sdiv(εu),div(εv)Y

−λ⁻¹

ΓI

(u×n)·(v×n) ds.

(2.6)

Here,s >0 is a real parameter, but it could be defined as a positive piecewise constant function.

Equivalence between problems (P[Y]) and (2.1)–(2.3) involves a scalar transmission operator Δ^Dir_ε [Y] = divεgrad with range inY and Dirichlet boundary condition. The domain of Δ^Dir_ε [Y] is given by

D(Δ^Dir_ε [Y]) :=

ϕ∈H₀¹(Ω)|div(εgradϕ)∈Y

. (2.7)

In the sequel, we note Δ_εϕ= divεgradϕfor anyϕin H₀¹(Ω). SinceY is a subspace ofH⁻¹(Ω), the definition ofD(Δ^Dir_ε [Y]) is natural. Indeed, forϕ∈H₀¹(Ω), we haveq= Δ_εϕ∈H⁻¹(Ω) in the sense of

Ωεgradϕ·gradψdx=− q, ψ_H−1(Ω)−H₀¹(Ω) ∀ψ∈H₀¹(Ω) where ·,·_H⁻¹_(Ω)−H0¹_(Ω) denotes the duality product betweenH⁻¹(Ω) andH₀¹(Ω).

Further, the Riesz representation theorem yields the existence of a bounded operatorK_ε such that K_ε : D(Δ^Dir_ε [Y]) −→ Y

ϕ −→ K_εϕ (2.8)

whereK_εϕis the unique element inY such that

p, K_εϕY = p, ϕ_H⁻¹_(Ω)−H0¹_(Ω) ∀p∈Y.

We are now able to state the following equivalence result:

Theorem 2.1. Let f∈L²(Ω)^d be divergence-free, divf= 0inΩand assume thatω= 0. Letu be a solution to (P[Y]). If the range of the operatorΔ^Dir_ε [Y] + ^ω_s²K_ε is dense in Y, then div(εu) = 0 inΩ anduis a solution to(2.1)–(2.3).

The idea of the proof is the same as in [18] and is omitted here. It is obvious that any solution of (2.1)–(2.3) satisfies (P[Y]). Under the condition of Theorem 2.1, problem (P[Y]) has thus a unique solution whenever ω²∈σ(curl,divε⁰).

Remark 2.2.

(1) The result of Theorem2.1carries over to the caseω= 0, since the range of Δ^Dir_ε [Y] is the whole spaceY provided thatY ⊂H⁻¹(Ω).

(2) If the imbedding ofY in H⁻¹(Ω) is compact, we can prove in a similar way as in [18], thatW[Y] is compactly imbedded in L²(Ω)^d (see also [21] for the caseY =L²(Ω)). The sesquilinear forma(·,·) is thus coercive on the spaceW[Y].

(3) As in [18], the space Y will be defined later on as a weighted L²-space. Therefore, the range of Δ^Dir_ε [Y] +^ω_s²K_εwill be dense inY if and only if ^ω_s² does not belong to the spectrum of a scalar positive self-adjoint operator with compact inverse (see [18] for details). Hence, takingssuch that ^ω_s² is smaller than the smallest eigenvalue of this operator guarantees the equivalence between (P[Y]) and the original problem.

Let us finally introduce the following spaces of piecewise regular functions P H^s(Ω;P) =

ϕ∈L²(Ω)|ϕ_j ∈H^s(Ω_j), j= 1, . . . , J

, (2.9)

whereϕ_j denotes the restriction ofϕto Ω_j. We denote byPH^s(Ω;P) the corresponding spaces of vector fields.

The remainder of this first part is to show that the space W[Y]∩PH¹(Ω;P) is dense in W[Y] for an appropriated choice of the spaceY. As mentioned before, the main application is the possibility to approximate the problem (P[Y]) by means of nodal finite elements.

2.2. Scalar potentials

With regard to the density results that we address here, we prove in this subsection that it is sufficient to deal with the question in terms of scalar potentials only. We introduce the following functional space

H[Y] =

ϕ∈H¹(Ω)Δ_εϕ∈Y; ϕ_|ΓD = 0; ϕ_|ΓI ∈H¹(Γ_I); l(ϕ) = 0

(2.10) wherelis a continuous linear form onH¹(Γ_I) such thatl(1)= 0. The spaceH[Y] is equipped with its canonical norm

||ϕ||_H[Y] =

||ϕ||²_1,Ω+||Δ_εϕ||²_Y +

F∈FI

||ϕ||²_{1,F 1/2}

. (2.11)

It is a space of scalar potentials associated with the space of vector fieldsW[Y] in the sense that gradH[Y]⊂W[Y].

Notice that in general, scalar potentials are uniquely determined up to an additive constant. Here, the linear forml is introduced in the spaceH[Y] in order to fix this constant. In the case where∂Ω is not connected, one linear form for each connected component including a part from Γ_I should be included inH[Y].

The first step will be the decomposition of the elements ofW[Y] into a (piecewise) regular part and a singular part, the singular part deriving from a scalar potential.

Theorem 2.3. Let u ∈ W[Y]. There is a scalar function ϕ ∈ H[Y] and a piecewise regular vector field uR∈W[Y]∩PH¹(Ω;P)such that

u=uR+ gradϕ. (2.12)

Further, there is a constant c >0 independent fromu such that

||uR||_{P H}1(Ω;P)+||ϕ||_H[Y]≤c||u||_W[Y]. (2.13)

Proof. Let u ∈ W[Y]. Since div(εu) ∈ Y ⊂ H⁻¹(Ω), there is a unique function χ ∈ H₀¹(Ω) such that Δ_εχ= div(εu). Thus, the vector fieldv=u−gradχ satisfies

curlv=curlu in Ω div(εv) = 0 in Ω v×n= 0 on Γ_D v×n=u×non Γ_I.

Hence,vbelongs to the standard regularization spaceW[L²(Ω)]. From [28] (Thm. 3.2), we deduce the existence of a regular vector potentialuR∈W[L²(Ω)]∩PH¹(Ω;P) satisfying

curluR=curlvin Ω div(εuR)∈L²(Ω) uR×n= 0 on∂Ω as well as the estimate

||uR||_{P H}1(Ω;P)+||div(εuR)||_0,Ω≤c||curlv||_0,Ω. (2.14) Sincecurl(u−uR) =curl(v−uR) = 0 in Ω, there is a unique scalar potentialϕ∈H¹(Ω) such that

gradϕ=u−uRin Ω and l(ϕ) = 0.

We obviously have Δ_εϕ∈Y. Moreover,ϕ_|ΓI belongs toH¹(Γ_I) since

grad_Tϕ_|F = gradϕ_|F×n=u_|F×n∈L²(F)ⁿ ∀F ⊂Γ_I. This shows that ϕbelongs toH[Y].

We prove in Lemma2.4below that

||ϕ||_H[Y] ≤c

||Δ_εϕ||²_Y +

F∈FI

||grad_Tϕ||²_{0,F 1/2}

∀ϕ∈H[Y].

The estimate of ||Δεϕ||_Y follows from the continuous imbedding of L²(Ω) in the vector space Y and (2.14), taking into account thatcurlv=curlu:

||Δ_εϕ||_Y ≤ ||div(εu)||_Y +c||div(εu_R)||_0,Ω

≤ c(||div(εu)||_Y +||curlu||_0,Ω)

≤ c||u||_W[Y] whereas the second term is equal to

F∈FI

||u×n||²_0,F

according to the definition ofϕ. This proves (2.13).

In the proof of Theorem2.3, we made use of the following equivalence result between norms:

Lemma 2.4. Let Y be such that(2.4)holds. The application

|·|_H[Y] : H[Y] −→ R⁺

ϕ −→

||Δεϕ||²_Y +

F∈FI

||grad_Tϕ||²_{0,F 1/2}

defines a norm onH[Y]equivalent to the canonical norm ||·||_H[Y].

Proof. It is obvious that |·|_H[Y] defines a semi-norm on H[Y]. Now, let ϕ∈ H[Y] be such that |ϕ|_H[Y] = 0.

Hence, Δ_εϕ= 0 on Ω which yieldsϕ= 0 on Ω if Γ_D=∅. If ΓD=∅, we have grad_Tϕ= 0 on all exterior faces.

Hence,ϕ_|∂Ωis a constant and this constant must be 0 sincel(ϕ) = 0 and l(1)= 0.

We next prove equivalence between|·|_H[Y] and the canonical norm. Letϕ∈H[Y]. There is a unique function r∈H¹(Ω) such that

Δ_εr= 0 in Ω r=ϕ on∂Ω.

It follows from classical results in variational theory and the continuous imbeddingH¹(Γ_I)→H^1/2(Γ_I) that

||r||_1,Ω≤cϕ_1/2,∂Ω≤c

F∈FI

||ϕ||²_{1,F 1/2}

. (2.15)

Next, let ˜ϕ=ϕ−r. The function ˜ϕis the variational solution inH₀¹(Ω) to the following Dirichlet problem with data in Y:

Δ_εϕ˜= Δ_εϕ in Ω ϕ= 0 on∂Ω.

We deduce from Poincar´e’s inequality and the definition of the parameterεthat

||ϕ||˜ ²_1,Ω ≤ c

Ωε|grad ˜ϕ|²dx

= −c Δεϕ,ϕ˜ _H−1(Ω)−H₀¹(Ω)

≤ cΔεϕ_−1,Ω||ϕ||˜ _1,Ω, and thus

||ϕ||˜ _1,Ω≤c||Δεϕ||_Y (2.16)

since the imbedding Y →H⁻¹(Ω) is continuous. Finally, we deduce from (2.15) and (2.16) that

||ϕ||_1,Ω≤c

||Δ_εϕ||²_Y +

F∈FI

||ϕ||²_{1,F 1/2}

and the result follows from the equivalence between theH¹-norm and the seminorm

F∈FI| · |1,F in the space w∈H¹(Γ_I)|l(w) = 0

.

Note that the above decomposition (2.12) is not unique. For instance takeψ∈ D(Ωj) for a fixedj ∈ {1, . . . , J} and letu_R=uR+||u||_W[Y]gradψandϕ=ϕ− ||u||_W[Y]ψ. Obviously,

u=u_R+ gradϕ andu_Randϕ satisfy (2.13).

Nevertheless, due to the decomposition (2.12) and estimate (2.13), we are able to define a linear continuous application Φ :W[Y]−→H[Y] which maps any vector fieldu ∈W[Y] on the corresponding scalar potential ϕ∈H[Y] in the sense that

u−grad(Φ(u))∈W[Y]∩PH¹(Ω;P) and (2.17)

Φ(gradϕ) =ϕ∀ϕ∈H[Y]. (2.18)

Since gradH[Y]⊂W[Y], Φ is well defined and onto. Moreover, Φ maps regular vector fields on regular scalar potentials,i.e.

Φ

W[Y]∩PH¹(Ω;P)

⊂H[Y]∩P H²(Ω;P). (2.19)

Indeed, let u∈W[Y]∩PH¹(Ω;P). Due to (2.17), we have grad(Φ(u))∈PH¹(Ω;P) which implies that Φ(u) =ϕ∈P H²(Ω;P).

We are now able to state the main result of this subsection:

Theorem 2.5. The space of vector fieldsW[Y]∩PH¹(Ω;P)is dense inW[Y]if, and only if, the corresponding space of scalar potentials H[Y]∩P H²(Ω;P)is dense in H[Y].

Proof. The proof of Theorem 2.5follows directly from the definition and the properties of the application Φ.

We refer to [28] (Proof of Thm. 3.1) for details.

2.3. Two-dimensional results

In this subsection, we prove the density result in the case of a polygon for an appropriate choice of the space Y. We further state some preliminary results which will be helpful for the edge singularities in three dimensions. In this subsection, Ω is a fixed polygon of the plane with the assumptions of Section2.1.

Let us start with the definition of the spaceY. Forα∈]−1,1[, we denote Y =

g∈H⁻¹(Ω)w^αg∈L²(Ω)

, (2.20)

where the weight function w is assumed to be positive on Ω. There are several possibilities to define the functionw (see [18]). Roughly speaking,wwill be chosen to be equivalent to the distance function to the set of vertices of the subdomains. The space Y is a Hilbert space equipped with the scalar product

f, gY =

Ω

w^2α(x)f(x)g(x) dx.

In order to provide a rigorous definition of the weight functionw, we introduce the following notations. Let S be the set of vertices of at least one Ω_j. The set of exterior vertices will be denoted bySext,

S_ext={S ∈ S |S ∈∂Ω}. This set is split into two subsets, namely,

SD=S_ext∩Γ^◦_D SI =Sext\ SD. The set of interior vertices is given bySint=S \ Sext.

Definition 2.6 (weight function in two dimensions). Let Ω ⊂R² be a polygon. For any vertex S ∈ S, let (r_S, θ_S) denote the local polar coordinates with respect toS. The weight function wis defined by

w(x) =

S∈S0

r_S (2.21)

whereS₀ is a subset ofS.

This definition is similar to the one ofsimplified weightsin [18]. Notice thatw(x) is equivalent to the distance functiond(x) = dist(x,S). Moreover, in a sufficiently small neighbourhoodVS of the vertexS containing no other vertex of Ω, the weight function is equivalent tor_S if the weight is “active”, whereasw(x)≈1 far away from the vertices. Let us now introduce

L²_α(Ω) =

g∈H⁻¹(Ω)

S∈S0

r_{S α}

g∈L²(Ω)

.

The following result shows thatL²_α(Ω) is an admissible choice for the spaceY: Proposition 2.7. Let Y =L²_α(Ω). Then(2.4) does hold for anyα∈[0,1[.

Proof. Sinceα≥0 andwis continuous on ¯Ω, the imbeddingL²(Ω)→L²_α(Ω) is obvious.

On the other hand, we deduce from a classic Hardy inequality (see for instance [32], Lem. 4.1, p. 38) that H¹(Ω)→L²_−α(Ω)

for allα∈[0,1[, if one recalls that the weight functionwis equivalent to the distance function near the vertices and equivalent to 1 anywhere else.

The result of the proposition follows by duality since (L²_−α(Ω))=L²_α(Ω).

From now on, letY be as in Proposition2.7. Forξ∈R, we introduce the space of dual singularitiesNε,ξ[Y] defined as the orthogonal inY of (Δ_ε−εξ²I)(H[Y]∩P H²(Ω;P)) with respect to the scalar product ofY. In other words, an elementg∈Y belongs toNε,ξ[Y] if, and only if,

g,(Δ_ε−εξ²I)ϕY = 0∀ϕ∈H[Y]∩P H²(Ω;P). (2.22) We next recall the space of standard dual singularities N_ε,Dir,ξ defined as follows: g ∈ N_ε,Dir,ξ if, and only if,

g∈L²(Ω) and

Ωg(Δ_ε−εξ²I)ϕdx= 0 ∀ϕ∈ D(Δ^Dir_ε [L²(Ω)])∩P H²(Ω;P) (2.23) whereD(Δ^Dir_ε [L²(Ω)]) is defined in analogy with (2.7). Taking into account the definition of the scalar product ·,·Y, we are now able to state the following link betweenNε,ξ[Y] andN_ε,Dir,ξ:

Proposition 2.8. Let ξ∈R. For anyg∈ Nε,ξ[Y], the functiong_α defined by g_α=w^2αg

belongs to the space of standard dual singularities, N_ε,Dir,ξ. Proof.

(1) Letg∈ Nε,ξ[Y]. Sinceg belongs toY =L²_α(Ω), the function w^αgbelongs to L²(Ω). The definition of the weight functionwthen guarantees thatg_α=w^2αg∈L²(Ω).

Ω Ω Ω

1 2 3

θ=0 θ=σ θ=σ θ=σ

1 2 3

S

Figure 1. Subdomains having a common vertex (J_S = 3).

(2) In order to prove thatg_αsatisfies the orthogonality relation (2.23), letϕ∈ D(Δ^Dir_ε [L²(Ω)])∩P H²(Ω;P).

Then,ϕalso belongs toH[Y]∩P H²(Ω;P), sinceL²(Ω)⊂L²_α(Ω) andϕ_|∂Ω= 0. Hence

Ωg_α(Δ_ε−εξ²I)ϕdx= g,(Δ_ε−εξ²I)ϕY = 0

which proves (2.23).

In view of the forthcoming Theorem 2.10, we need to recall the singularities of the transmission problem involving the operator Δ_εwith domainD(Δ^Dir_ε [L²(Ω)]) (see [27,33,34] for details).

For S ∈ Sext, let Λ_ε,S be the set of positive singular exponents of the operator Δ^Dir_ε [L²(Ω)] that we now describe shortly. Without loss of generality we may assume that the set of subdomains Ω_j having S as vertex is{Ωj}^J_j=1^S , for some positive integerJ_S. For j∈ {1, . . . , JS} letω_j be the interior opening of Ω_j atS and set σ₀ = 0 andσ_j =σ_j−1+ω_j. Then a real number λbelongs to Λ_ε,S if, and only if, there exists a non trivial solutionφ_λ∈H¹(]0, σ_JS[), φ_λ = (φ_λ,j)^J_j=1^S , to

φ_λ,j+λ²φ_λ,j= 0 in ]σ_j−1, σ_j[, j= 1, . . . , J_S, (2.24)

φ_λ,1(0) =φ_λ,JS(σ_JS) = 0, (2.25)

[φ_λ]_σj−1 = [εφ_λ]_σj−1 = 0, j= 1, . . . , J_S−1, (2.26) where (r_S, θ_S) are the local polar coordinates with respect toS, the half-lineθ_S=σ_j containing an edge of Ω_j, forj= 1, . . . , J_S while the half-lineθ_S = 0 contains an edge of Ω₁ (see Fig. 1).

Note that in the homogeneous case, i.e., ε_j =ε, for all j= 1, . . . , J_S, the set Λ_ε,S is equal to{_σ^kπ

JS :k∈N, k = 0} and is independent of ε. In the inhomogeneous case this set is not explicitly known but may be approximated numerically (seee.g. [27,33,34]).

We proceed similarly for S ∈ Sint, replacing the Dirichlet boundary condition (2.25) by the transmission conditions

φ_λ,1(0) =φ_λ,JS(2π), ε₁φ_λ,1(0) =ε_JSφ_λ,JS(2π).

Let us notice that ifS ∈ Sext then λ∈Λ_ε,S is simple (see [34]). In other words, the solutionφ_λ to (2.24)–

(2.25) is unique up to a multiplicative factor. On the other hand, if S ∈ Sint, then λ ∈ Λ_ε,S has a finite multiplicity and in that caseλis repeated in Λ_ε,Saccording to its multiplicity.

The standard singularities of the operator Δ^Dir_ε [L²(Ω)] at the vertexS∈ S are

S_S,λ=η_Sr^λφ_λ, forλ∈Λ_ε,S, (2.27)

where η_S = η_S(r) is an appropriate cut-off function (η_S ≡ 1 in a neighbourhood of S and η_S ≡ 0 in a neighbourhood of the other vertices).

Next, we need to characterize the elements of the space N_ε,Dir,ξ. To this end, we recall Proposition 2.8 of [28] (for any details, see [33] for the case ξ = 0 and [23] for ξ = 0). Let us begin with some classical

Grisvard notations. For = 1,2, letH^−1/2(∂Ω) be the range of the trace mapping, starting from H(Ω); for all facesF ∈ Fext, the restrictions of those sets to F is denoted by H^−1/2(F). Then, define H^−1/2(F) as the set of elements ofH^−1/2(F) whose continuation by zero to∂Ω belongs toH^−1/2(∂Ω). Finally, letH^1/2−(F) denote the dual space of H^−1/2(F). In the same manner, one can introduce similar spaces for the interior faces, starting from interior domains.

Proposition 2.9. g∈ Nε,Dir,ξ if, and only if, g∈L²(Ω)is solution to (Δ−ξ²I)g= 0 in Ω_j ∀j, g= 0 inH^−1/2(F)∀F ∈ F_ext. [g] = 0 in H^−1/2(F)∀F ∈ Fint, [ε∂_ng] = 0 in H^−3/2(F)∀F ∈ Fint.

In order to give an appropriate basis ofNε,Dir,ξ, we set for any vertexS ∈ S and allλ∈Λ_ε,S,

g_S,λ,ξ=η_Se^−|ξ|rr^−λφ_λ−v_S,λ,ξ, (2.28) wherev_S,λ,ξ∈H₀¹(Ω) is the unique variational solution to

(Δ_ε−εξ²I)vS,λ,ξ= (Δ_ε−εξ²I)(ηSe^−|ξ|rr^−λφ_λ), (2.29) i.e., is the unique solution to

Ωε(gradv_S,λ,ξ·gradw+ξ²v_S,λ,ξw) dx=−

Ω(Δ_ε−εξ²I)(ηSe^−|ξ|rr^−λφ_λ)wdx, ∀w∈H₀¹(Ω).

Notice that this problem is well defined since the right hand side of (2.29) belongs to L^q(Ω) withq < _1+λ² (see Lems. 4.4 and 4.5 of [26]).

The functiong_S,λ,ξ belongs toNε,Dir,ξ and satisfies (thanks to Green’s formula, see Prop. 2.5.5 of [23])

Ω

(Δ_ε−εξ²I)(η_Tr^μφ_μ)g_S,λ,ξdx= 2λδ_S,Tδ_λ,μ. (2.30) Furthermore, under the assumption

1∈Λ_ε,S, ∀S∈ S, (2.31)

the set{gS,λ,ξ}_{λ∈Λε,S∩]0,1[,S∈S} is a basis ofN_ε,Dir,ξ.

The following theorem provides an appropriate condition on the weight exponent αsuch that the density result for the scalar potentials (and thus also for the corresponding space of vector fields) holds true:

Theorem 2.10. Let Y be as in Proposition2.7. Let the subsetS0⊂ S satisfy the following inclusion:

{S∈ SD∪ Sint| Λ_ε,S∩]0,1[=∅} ∪ {S∈ SI| Λ_ε,S∩]0,1/2[=∅} ⊂ S0. (2.32) Further, let αbe such that

α >1−min

S∈SD∪Sint

(Λ_ε,S∩]0,1[)∪

S∈SI

(Λ_ε,S∩]0,1/2[)

. (2.33)

Assume that (2.31)holds. ThenH[Y]∩P H²(Ω;P)is dense inH[Y].

Proof. In order to prove the density result, we will characterize the elements of some complementary spaceO[Y] that we define here. Let

H₀[Y] =

ϕ∈H¹(Ω)Δ_εϕ∈Y; ϕ_|ΓD = 0; ϕ_|F ∈H₀¹(F), ∀F ∈ FI

.

As in [28], we prove thatH₀[Y] is continuously imbedded inH[Y] if we choose the linear formlin the definition ofH[Y] as follows:

l(ϕ) =

S∈SI

ϕ(S).

Now, letξ∈Rand letO[Y] be the orthogonal complement of H₀[Y]∩P H²(Ω;P) for the inner product (ϕ, ψ)_ξ,Y = (Δε−εξ²I)ϕ,(Δ_ε−εξ²I)ψY +

F∈FI

(grad_Tϕ,grad_Tψ)_0,F +ξ²(ϕ, ψ)_0,F .

Then

H[Y] =H[Y]∩P H²(Ω;P)⊕ O[Y]

since the complementary space of H₀[Y] in H[Y] is spanned by a finite number of functions that belong to H[Y]∩P H²(Ω;P). Notice that arguments, similar to those of Lemma2.4, allow one to show that the norm ϕξ,Y = (ϕ, ϕ)^1/2_ξ,Y is equivalent to||·||_H[Y] with equivalence constants that depend onξ.

As in [28] (Prop. 4.3), we are able to prove that for anyf ∈ O[Y], there is a uniqueg∈ Nε,ξ[Y] with

(Δ_ε−εξ²I)f =g inY, (2.34)

(Δ_T −ξ²I)f =−ε∂νg_α inH⁻¹(F), ∀F ∈ FI, (2.35) fξ,Y ≤c

||g||_Y +

F∈FI

ε∂νg_α−1,F

(2.36) where g_α=w^2αg is the standard dual singularity inN_ε,Dir,ξ corresponding tog, according to Proposition2.8.

The functiong_αis thus uniquely represented as g_α=

S∈S

λ∈Λε,S∩]0,1[

c_λ,Sg_S,λ,ξ.

As in the proof of Theorem 4.4 in [28], condition (2.35) implies that c_λ,S= 0∀λ∈Λ_ε,S∩[1/2,1[, ∀S∈ S_I since∂_νg_S,λ,ξ≈r^λ−1_S near S.

Now, let λ∈Λ_ε,S ∩]0,1[ for a vertexS ∈ SD∪ Sint or λ∈Λ_ε,S ∩]0,1/2[ for S ∈ SI. Taking into account that w^αg belongs toL²(Ω), we deduce thatw^−αg_S,λ,ξ∈L²(Ω) wheneverc_λ,S = 0. But

w^−αg_S,λ,ξ≈r_S^−(α+λ)Φ_λ,S(θ_S)

nearS, andr^−(α+λ)_S Φ_λ,Sbelongs toL²(Ω) if, and only if,α+λ <1 which is in contradiction with the assumption onα. Thereforec_λ,S= 0 for anyλwhich yieldsg= 0 in Ω.

Finally, we deduce from (2.36) that f = 0 in Ω which completes the proof.

Remark 2.11. One could also considergeneral weights with an exponent that depends on the vertexS of S.

Namely, one could replacew^αby

S∈S

r_S^αS, (2.37)

with (α_S)_S∈S in ]0,1]^|S| such that

α_S >1−min (Λ_ε,S∩]0,1[) ifS ∈ SD∪ Sint;

α_S >1−min (Λ_ε,S∩]0,1/2[) ifS ∈ SI. (2.38)

2.4. Density results in three-dimensional domains

In this subsection we investigate a suitable condition on the weight exponentαin order to obtain the density result in the case of a three dimensional Lipschitz-polyhedron.

In order to define the weight functionw, we introduce the following notations which describe the domain Ω near the geometric singularities.

Let S (resp. E) be the set of vertices (resp. edges) of at least one Ω_j. The subscripts “ext” and “int”

will denote exterior and interior vertices or edges as before, and the setS_ext (resp. E_ext) admits the following splitting, according to the different boundary conditions:

SD=S_ext∩Γ^◦_D, SI =S_ext\ SD, ED=Eext∩Γ^◦_D, EI =Eext\ ED.

For a vertexS∈ S, let ΓSbe the polyhedral cone which coincides with Ω nearSand letG_Sbe the intersection of Γ_S with the unit sphere. We shall use local spherical coordinates (r_S, σ_S) centered at S. To each edge e adjacent to the vertexS, corresponds a corner ofG_S denoted byS_e. A neighbourhood of the pointS_emay thus be mapped on an infinite plane sector which can be written in polar coordinates as

C_S,e={(ϑS,e, ϕ_S,e)|ϑ_S,e>0, 0< ϕ_S,e< ω_S,e}.

Next, let e∈ E be an (exterior or interior) edge with opening angle ω_e ∈ ]0,2π] (ω_e = 2π if, and only if, e∈ Eint). Without loss of generality, we may assume thateis supported by thez-axis and we denote (r_e, θ_e, z) the corresponding cylindrical coordinates. In particular, we have

r_e(x) = dist(x,¯e)∀x∈Ω.

Let us fixR_e>0 andh_e>0 and introduce the two-dimensional domain

Ω_e:={(recosθ_e, r_esinθ_e)|0< r_e< R_e, 0< θ_e< ω_e} such that the dihedral cone

D_e= Ω_e×R (2.39)

coincides with Ω for any z ∈ ]−h_e, h_e[ and does contain no other edge nor any vertex of Ω. To each Ω_j containinge, there corresponds a unique set Ω_e,j ⊂Ω_e. Therefore the partitionP induces a natural partitionPe

of Ω_e(and thusD_e) for whichεandμare piecewise constant and depend only onθ. Namely, we take ε_e,j =ε_j on Ω_e,j×R,

μ_e,j =μ_j on Ω_e,j×R.

We finally denote Γ_e,0 (resp. Γ_e,ω) the edges of Ω_e andF_e,0= Γ_e,0×R(resp. F_e,ω) the corresponding exterior faces ofD_econtaininge.

If we denote byd_S(x) (resp. d_E) the distance function to the setS (resp. E),i.e.

d_S(x) = dist(x,S) andd_E(x) = dist(x,E),

we clearly have

d_S ≈r_S in any sufficiently small neighbourhoodVS of the vertexS, and

d_E ≈r_e in Ω_e×]−he, h_e[ for sufficiently small numbersR_eandh_e.

In order to define the weight function, we need to introduce another distance functionρ_etaking into account the edge/vertex interaction. Lete∈ E be the segment between the two verticesS andS. Then we defineρ_eby

r_e=ρ_er_Sr_S. (2.40)

In a sufficiently small neighbourhood of the vertexS, the functionρ_eis equivalent to the angular distanceϑ_S,e near the edgee, while

ρ_e≈d_E far fromS.

The definition of the weight function then reads as follows (see the definition of global weights in [18]):

Definition 2.12 (weight function in three dimensions). Let Ω⊂R³ be a Lipschitz-polyhedron. The weight functionwis defined by

w(x) =

S∈S0

r_S

e∈E0

ρ_e

(2.41) where S0⊂ S andE0⊂ E satisfy the following compatibility condition: ife∈ E0is an edge with end points S andS, then S∈ S0 andS∈ S0.

It has been proven in [18] that an equivalent definition is w(x) = dist (x,S0∪ E0).

This corresponds to thesimple weights of Costabel-Dauge, where the set S₀∪ E₀ is a so-calledwire basket, in the spirit of [35].

As in two dimensions, we have:

Proposition 2.13. Let Y =L²_α(Ω) with a weight function as in Definition2.12. Then(2.4)does hold for any α∈[0,1[.

Proof. As in two dimensions, the first imbedding L²(Ω) → L²_α(Ω) is obvious. For the second imbedding, we proceed by duality, proving that

H₀¹(Ω)→L²_−α(Ω).

Near an edge, this follows as in two dimensions from the classical Hardy inequality. Near the vertices, we may use Proposition 5.1. in [29] since the definition of weights therein is equivalent to Definition2.12.

We next describe the vertex and edge singularities of the operator Δ_ε with Dirichlet boundary condition, i.e. with domainD(Δ^Dir_ε [L²(Ω)]). The set Λ_ε,Sof positive vertex singular exponents is related to the spectrum of the nonnegative Laplace-Beltrami operatorL_ε,S. More precisely, it is the Friedrichs extension of the triple (H_ε,S, V_S, a_ε,S), whereH_ε,S=L²(G_S) with the inner product

(ψ, φ)_ε=

GS

εψφdσ,

the spaceV_S being equal toH₀¹(G_S) ifS∈ S_ext, andV_S =H¹(G_S) ifS∈ S_int, and finally a_ε,S:V_S×V_S →R: (ψ, φ)→

GS

εgrad_Tψ·grad_Tφdσ.