On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient

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On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one

sign-changing coefficient

Patrick Ciarlet

To cite this version:

Patrick Ciarlet. On the approximation of electromagnetic fields by edge finite elements. Part 4:

analysis of the model with one sign-changing coefficient. 2021. �hal-03273264�

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On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient

Patrick Ciarlet Jr.

This version: June 29, 2021

Abstract In electromagnetism, in the presence of a negative material sur- rounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this set- ting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded setΩofR³, and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale (H^s(Ω))s>0. With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived.

Introduction

We study the numerical approximation by finite elements of electromagnetic fields governed by the time-harmonic Maxwell equations, in the presence of a negative material surrounded by a classical material. A negative material can be a metal at optical frequencies, or a metamaterial, see for instance [46, 1]. So, in this setting, the electric permittivity, and possibly the magnetic permeabil- ity, can exhibit a sign-change at the interface between the two materials. We consider such a model in a bounded set ofR³, supplemented with a vanishing boundary condition on the tangential trace. To the author’s knowledge, the first attempt to address this situation theoretically can be found in [10, 9]; see also [15, 42]. However, little is known regarding the numerical approximation of the model. In the present paper, we provide the numerical analysis for a model with one sign-changing coefficient. In a similar setting, we refer to [34] for the analysis of the two-dimensional time-harmonic Maxwell equations, and to [22]

for the analysis of the three-dimensional, div-curl, or div-curlcurl, problem.

POEMS, CNRS, INRIA, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France (E-mail: patrick.ciarlet@ensta-paris.fr).

For the numerical approximation, we use (low-order) edge finite elements. We use some recent results [21, 31, 22] to interpolate low-regularity solutions that can occur both in a classical setting, that is for a model with fixed-sign, piece- wise smooth coefficients [24, 6, 23], and in the presence of an interface between a classical material and a negative material.

In what follows, we shall assume that the electric permittivity εhas a sign- change, while the magnetic permeability µ has a fixed sign (when the roles of ε and µ are reversed, we refer to section 8). Typically, this corresponds to an interface model between a metal surrounded by a classical material (in some ad hoc frequency range). Classically [2,§8], for solving the time-harmonic Maxwell equations, one can choose first-order formulations in both the electric and magnetic fields, or second-order formulations in the electric field only, or in the magnetic field only. Our choice will be a second-order formulation in the electric field.

The outline is as follows. We begin by introducing some notations, together with a precise definition of the mathematical framework considered hereafter.

Before investigating the solution of this problem, we propose some comments in section 2 to help identify the difficulties to be addressed. For that, we rely on some well-known facts regarding the classical setting (fixed-sign coefficient), that we shall apply to the new model. We introduce the companion scalar problem and tools, such as the T-coercivity to realize the inf-sup condition.

In section 3, we explain how to solve the time-harmonic Maxwell equations.

Next, in section 4, we recall the numerical approximation via edge finite el- ements, and in particular how one can interpolate the electric field, which can (possibly) be of low-regularity. To prove the results regarding convergence of the numerical method, we use some results regarding practical discrete T- coercivity (for the companion scalar problem) which is achieved with the help of T-conform meshes. These are recalled in the appendix A. As a matter of fact, these results allow us to prove the uniform discrete inf-sup condition for the time-harmonic Maxwell equations: this is the object of the next two sec- tions. Then, in section 7, we provide a numerical illustration to check that the expected convergence order is achieved, and how the use of T-conform meshes may impact the convergence rate. In section 8, we outline how one can solve theoretically and numerically the case ofµhaving a sign-change, andεhaving a fixed sign. Finally, we give some concluding remarks in the last section.

1 Setting of the problem

As in [21], we denote constant fields by the symbolcst. Vector-valued (respec- tively tensor-valued) function spaces are written in boldface character (resp.

blackboard bold characters). Unless otherwise specified, we consider spaces of real-valued functions. Given a non-empty open setOofR³, we use the notation (·|·)_0,O (respectivelyk · k_0,O) for theL²(O) and theL²(O) := (L²(O))³ inner products (resp. norms). More generally, (·|·)_s,O andk·k_s,O (respectively|·|_s,O) denote the inner product and the norm (resp. semi-norm) of the Sobolev spaces

H^s(O) andH^s(O) := (H^s(O))³ for s∈ R(resp. for s>0). The index zmv indicates zero-mean-value fields. If moreover the boundary∂Ois Lipschitz,n denotes the unit outward normal vector field to ∂O. It is assumed that the reader is familiar with function spaces related to Maxwell’s equations, such as H(curl;O), H₀(curl;O), H(div;O), H₀(div;O) etc. A priori, H(curl;O) is endowed with the “natural” normv 7→(kvk²_0,O+kcurlvk²_0,O)^1/2, etc. We refer to the monographs of Monk [40] and Assous et al [2] for details.

The symbolCis used to denote a generic positive constant which is indepen- dent of the meshsize, the mesh and the fields of interest ; C may depend on the geometry, or on the coefficients defining the model. We use the notation A.B for the inequalityA≤CB, whereA andB are two scalar fields, and C is a generic constant.

LetΩbe adomaininR³, ie. an open, connected and bounded subset ofR³ with a Lipschitz-continuous boundary∂Ω. The domainΩcan betopologically trivial(tt) ornon-trivial(tnt) [33]. This means that we assume that one of the two conditions below holds:

– (tt) ’for all curl-free vector field v ∈C¹(Ω), there exists p∈C⁰(Ω)such that v=∇pin Ω’;

– (tnt)’there existInon-intersecting, piecewise plane manifolds,(Σj)_j=1,···,I, with boundaries ∂Σ_i ⊂∂Ω, such that, if we letΩ˙ =Ω\SI

i=1Σ_i, for all curl-free vector fieldv, there exists p˙∈C⁰( ˙Ω)such that v=∇p˙ inΩ’.˙ To simplify the computations (without restricting the scope of the study), we assume that the boundary∂Ω isconnected.

We let Ω be surrounded by a perfect conductor. We recall that, for a given pulsationω >0, the time-harmonic Maxwell equations set inΩcan be expressed in terms of the complex-valued electric fieldeonly. They write







Finde∈H₀(curl;Ω)such that curl(µ⁻¹curle)−ω²εe=ıωj in Ω divεe=%in Ω.

(1) Above, the real-valued coefficientεis the electric permittivity tensor and the real-valued coefficient µ is the magnetic permeability tensor. The complex- valued source termsjand%are respectively the current density and the charge density. They are related by the charge conservation equation

−ıω%+ divj= 0 inΩ. (2)

Classically, in (1), the equation divεe = % is implied by the second-order equationcurl(µ⁻¹curle)−ω²εe=ıωj, together with the charge conservation equation (2), so it is omitted from now on. We fix thea priori regularity of the current density to j ∈ L²(Ω), which implies that % ∈ H⁻¹(Ω), with dependencek%k_H⁻¹_(Ω)=ω⁻¹kdivjk_H⁻¹_(Ω)≤ω⁻¹kjk_L2(Ω).

Finally, note that one can split the problem into two parts, where <(e)∈ H0(curl;Ω) is related to−=(j)∈L²(Ω), resp.=(e)∈H0(curl;Ω) is related to <(j) ∈ L²(Ω). So, we carry on with e standing either for <(e) or =(e), resp.f standing for−ω⁻¹=(j) orω⁻¹<(j), that is withreal-valued fields. One can check that the equivalent variational formulation inH₀(curl;Ω) writes

Finde∈H₀(curl;Ω) such that

a_ω(e,v) =ω²(f|v)_0,Ω, ∀v∈H₀(curl;Ω), (3) where

a_ω(u,v) := (µ⁻¹curlu|curlv)_0,Ω−ω²(εu|v)_0,Ω, ∀u,v∈H₀(curl;Ω).

Note that with these notations, one has divεe=−divf inΩ.

Then, the real-valued coefficient ξ ∈ {ε, µ} fulfills one of the two sets of conditions below, which we refer to as theclassical caseand theinterface case hereafter.

Classical case:

ξ is a real-valued, symmetric, measurable tensor field onΩ,

∃ξ₋, ξ+>0,∀z∈R³, ξ₋|z|²≤ξz·z≤ξ+|z|² a.e. inΩ. (4) Interface case:Ωis partitioned into the non-trivial partitionP := (Ωp)_p=+,−, whereΩ_± are domains, andδξfulfills (4), withδ_|Ω+= +1 andδ_|Ω− =−1.

For our studies of the time-harmonic Maxwell equations in the electric field, we assume from now on that

εis as in theinterface case;µis as in theclassical case.

2 Some comments

Observe that if the electric field is curl-free, ie. curle = 0, then it may be written as e=∇pe for some pe∈H₀¹(Ω) (cf. Theorem 3.3.9 in [2], as ∂Ω is connected). Moreover, p_e is such that divε∇pe =−divf in H⁻¹(Ω). So to ensure well-posedness, one must make an assumption on thecompanion scalar problemwith Dirichlet boundary condition:

Find s∈H₀¹(Ω)such that (ε∇s|∇q)_0,Ω=hg, qi_H1

0(Ω), ∀q∈H₀¹(Ω), (5)

namely, that this scalar problem is well-posed. In other words,

∃C?>0, ∀g∈H⁻¹(Ω), ∃!ssolution to (5), withksk_H¹

0(Ω)≤C?kgk−1,Ω. (6) To measure elements ofH₀¹(Ω), we choose the normq7→ kqk_H1

0(Ω):=k∇qk0,Ω. If the permittivityε were to fulfill (4), well-posedness of the scalar problem wouldautomaticallyhold, as an obvious consequence of the fact that (q, q⁰)7→

(ε∇q|∇q⁰)0,Ω defines an inner product on H₀¹(Ω), whose associated norm is equivalent to thek · k_H¹

0(Ω)-norm.

However, in the present setting, since εis as in the interface case, this is an additional assumption, which is addressed with the help of T-coercivity [12, 8]. We recall the abstract framework below, see [20, 18] for details. Let V be a Hilbert space with normk · kV, anda(·,·) asymmetric, continuous bilinear form onV ×V. Then, the well-posedness of the problem

Find u∈V such thata(u, v) =hf, vi_V, ∀v∈V, (7) which reads

∃C >0, ∀f ∈V⁰, ∃!usolution to (7), withkuk_V ≤Ckfk_V⁰, (8) can be addresssed as follows. One has to prove that the formaisT-coercive, cf. Theorem 1 and Remark 2 of [18]:

∃α >0, ∃T ∈ L(V), ∀v∈V, |a(v, T v)| ≥αkvk²_V. (9) In other words, the operatorT realizes the classical inf-sup condition (see eg.

[5])explicitly.

Hence, for the scalar problem (5), and because ε is a symmetric tensor field, well-posedness is equivalent to (q, q⁰) 7→ (ε∇q|∇q⁰)_0,Ω fulfilling an inf- sup condition:

∃γ0>0, ∀q∈H₀¹(Ω), sup

q⁰∈H₀¹(Ω)\{0}

|(ε∇q|∇q⁰)0,Ω| kq⁰k_H1

0(Ω)

≥γ0kqk_H1

0(Ω). (10) Or, as noted above, this is equivalent to

∃α0>0, ∃T0∈ L(H₀¹(Ω)),∀q∈H₀¹(Ω), |(ε∇q|∇(T0q))_0,Ω| ≥α₀k∇qk²_0,Ω. Note that the absolute value can be removed. Indeed, the quadratic mapping q7→(ε∇q|∇(T₀q))_0,Ω is continuous inH₀¹(Ω) and vanishes only forq= 0, so it takes either positive, or negative, values everywhere in H₀¹(Ω). Thus, (10) is also equivalent to

∃α₀>0, ∃T₀∈ L(H₀¹(Ω)),

∀q∈H₀¹(Ω), (ε∇q|∇(T₀q))0,Ω≥α₀k∇qk²_0,Ω. (11) To recapitulate, we assume from now on that

(10)-(11) holds for the companion scalar problem (5).

When we perform the numerical analysis, and in order to obtain explicit convergence rates between the exact and approximate solution to the time- harmonic Maxwell equations, we shall maketwo additional assumptions:

– the coefficientsε, µare piecewise smooth: there exists a partition{Ωp}_p=1,···,P

of Ω, made of disjoint domains (Ωp)p=1,···,P, withΩ=∪^P_p=1Ωp, and such that ε_|Ωp, µ_|Ωp ∈W^1,∞(Ωp) for p= 1,· · ·, P. In relation to the partition and fors≥0, we define

P H^s(Ω) :={v∈L²(Ω) : v_|Ωp ∈H^s(Ω_p), 1≤p≤P}, (12) endowed with the “natural” normkvk_{P H}^s_(Ω):= X

1≤p≤P

kvpk²_s,Ωp1/2

. – the dataf has extra-regularity, in the sense that

divf ∈H^−1+τ0(Ω), withτ₀∈(0,1] given. (13) For further analysis, let us introduce the scalar problem withmodified right- hand side

Find s∈H₀¹(Ω)such that (ε∇s|∇q)0,Ω=hg, qi_H1

0(Ω)+ (εg,∇q)0,Ω, ∀q∈H₀¹(Ω). (14) Ifεwere as in the classical case [24, 43, 29, 38, 35, 6, 27], one could prove a shift theorem for the problem (14) when the data (g,g) has extra-regularitylike

g∈H^−1+τ0(Ω), g∈H¹(Ω), withτ₀∈(0,1] given.

In the interface case, there exist similar results in this direction. We refer to [25, 13, 17, 16, 11] for a piecewise constant coefficientε. So we introduceτDir∈(0,1]

depending only on the geometry and onεsuch that

∀s∈[0, τDir)\ {1/2}, ∀(g,g)∈H^−1+s(Ω)×H¹(Ω), the solutionsto (14) is such thats∈P H^1+s(Ω), and kskP H^1+s(Ω).(kgk−1+s,Ω+kgk1,Ω).

Above, the constant hidden in.may depend ons, but not ong nor ong. By a slight abuse of vocabulary, we call this result theshift theorem, respectively τ_Dirthelimit regularity exponent. We assume from now on that

a shift theorem holds with τDir∈(0,1] for the modified scalar problem (14).

Remark 1 We shall also need a shift theorem for the scalar problem involving the magnetic permeabilityµwith Neumann boundary condition, see (31) be- low. Result can be found in the above-mentioned references, becauseµ is as

in the classical case. ut

So far we focused on curl-free fields. To tackle fields with a non-vanishing curl, we use an ad hoc splitting inH0(curl;Ω). Define

KN(Ω, ε) :={v∈H0(curl;Ω) : divεv = 0}.

An equivalent (variational) definition is

KN(Ω, ε) :={v∈H0(curl;Ω) : (εv|∇q)0,Ω= 0, ∀q∈H₀¹(Ω)}.

Proposition 1 One has the continuous, direct sum

H0(curl;Ω) =∇[H₀¹(Ω)]⊕KN(Ω, ε). (15) Proof Obviously, ∇[H₀¹(Ω)] +K_N(Ω, ε) is a subset of H₀(curl;Ω). Let v ∈ H0(curl;Ω). According to (6), there existspv ∈H₀¹(Ω) such that

(ε∇pv|∇q)0,Ω= (εv|∇q)0,Ω, ∀q∈H₀¹(Ω). (16) Now, letkv =v− ∇pv, one haskv ∈ KN(Ω, ε) by construction. It follows thatH0(curl;Ω) =∇[H₀¹(Ω)] +KN(Ω, ε).

Next, let z ∈ ∇[H₀¹(Ω)]∩KN(Ω, ε) be given. There exists s ∈H₀¹(Ω) such that z =∇s and, by definition of KN(Ω, ε), s is governed by (5) with zero right-hand side. By uniqueness of the solution, one has s= 0 and so z = 0:

the sum is direct.

Finally, by definition (16) ofp_v and according to (11), one hasα₀k∇pvk²_0,Ω≤ (ε∇pv|∇(T0p_v))_0,Ω= (εv|∇(T0p_v))_0,Ω≤ kεvk0,Ωk∇(T0p_v)k0,Ω, so that

k∇p_vk_H(curl;Ω)=k∇p_vk_0,Ω≤α⁻¹₀ kεk_∞,ΩkT₀k_L(H1

0(Ω))kvk_0,Ω, and kkvk_H(curl;Ω)≤(1 +α⁻¹₀ kεk_∞,ΩkT0k_L(H1

0(Ω)))kvk_H(curl;Ω).

So the sum is continuous. ut

In other words, we may introduce the operators of L(H0(curl;Ω), H₀¹(Ω)), resp. ofL(H₀(curl;Ω))

π₁ :

H0(curl;Ω)→H₀¹(Ω) v7→pv

, π₂ :

H0(curl;Ω)→KN(Ω, ε) v7→kv

and write, for allv∈H0(curl;Ω),v =∇(π1v) +π2v. Note that (π2)²=π2. We finally recall an important result on the measure of elements ofKN(Ω, ε).

For its proof, we refer the reader to Corollary 5.2 of [9].

Theorem 1 Elements of K_N(Ω, ε) can be measured with the kcurl·k_0,Ω- norm:

∃CW >0, ∀k∈KN(Ω, ε), kkk0,Ω≤CWkcurlkk0,Ω, (17)

∃C_W⁰ >1, ∀k∈K_N(Ω, ε), kkkH(curl;Ω)≤C_W⁰ kcurlkk0,Ω. (18)

3 Solving the exact problem

Recall thatµis as in the classical case (cf. (4)), resp.εis as in the interface case, and assumption (10)-(11) holds. Using operators π1 and π2, one can provide an equivalent reformulation of the variational formulation (3). Its solution e may be split as

e=e0+∇φ, withe0=π2eandφ=π1e. (19)

By using the (variational) definition ofKN(Ω, ε) (recall thatεis a symmetric tensor field), we notice thate0 andφare respectively governed by

Find e0∈KN(Ω, ε)such that

aω(e0,v) =ω²(f|v)0,Ω, ∀v ∈KN(Ω, ε). (20) Find φ∈H₀¹(Ω)such that

(ε∇φ|∇q)0,Ω=hdivf, qi_H¹

0(Ω), ∀q∈H₀¹(Ω). (21) Actually, there is an equivalence result (the proof is left to the reader).

Proposition 2 A fieldeis a solution to (3) if, and only if,π2e is a solution to (20) and π1e is a solution to (21).

According to the assumption on ε, we already know that problem (21) is well-posed. Hence proving the well-posedness of (3) amounts to proving the well-posedness of (20). We recall Theorem 8.15 of [9].

Theorem 2 The imbedding of K_N(Ω, ε)in L²(Ω)is compact.

As a consequence (cf. Theorem 8.16 of [9]), one has the

Corollary 1 The variational formulation (20) with unknown e₀ enters the Fredholm alternative:

– either the problem (20) is well-posed, ie. it admits a unique solution e0 in H0(curl;Ω), which depends continuously on the dataf:

ke0k_H(curl;Ω).kfk0,Ω;

– or, the problem (20) has solutions if, and only if, f satisfy a finite num- ber of compatibility conditions; in this case, the space of solutions is an affine space of finite dimension, and the component of the solution which is orthogonal (in the sense of theH₀(curl;Ω)inner product) to the corre- sponding linear vector space, depends continuously on the data f.

Finally, each alternative occurs simultaneously for variational formulation (20) and variational formulation (3) with unknowne.

From now on, we assume that variational formulation (3) iswell-posed:

∀f ∈L²(Ω), ∃!e∈H₀(curl;Ω) solⁿto (3) andkekH(curl;Ω).kfk0,Ω. (22)

4 Approximation by N´ed´elec’s finite elements

For the ease of exposition¹, we assume thatΩand{Ωp}_p=1,···,P are Lipschitz polyhedra. We consider a family of simplicial meshes ofΩ, and we choose the N´ed´elec’s first family of edge finite elements [41, 40] to define finite dimensional subspaces (Vh)h of H0(curl;Ω). So Ω is triangulated by a shape regular family of meshes (Th)h, made up of (closed) simplices, generically denoted by K. Each mesh is indexed by h := maxKh_K (the meshsize), where h_K is the diameter ofK. And meshes are conforming with respect to the partition {Ω_p}_p=1,···,P induced by the coefficients ε, µ: namely, for all h and all K ∈ T_h, there exists p ∈ {1,· · ·, P} such that K ⊂ Ω_p. N´ed´elec’s H(curl;Ω)- conforming (first family, first-order) finite element spaces are then defined by

Vh:={vh∈H0(curl;Ω) : v_h|K ∈ R1(K), ∀K∈ Th}, whereR1(K) is the vector space of polynomials onK defined by

R₁(K) :={v ∈P₁(K) : v(x) =a+b×x, a,b∈R³}.

To approximate the curl-free fields, we need to define a suitable approximation of elements ofH₀¹(Ω). So we introduce finite dimensional subspaces (Mh)h of H₀¹(Ω). Lagrange’s first-order finite element spaces are defined by

M_h:={qh∈H₀¹(Ω) : q_h|K∈P₁(K), ∀K∈ Th}.

The discrete companion scalar problems are Find sh∈Mh such that

(ε∇sh|∇qh)0,Ω=hg, qhi_H1

0(Ω), ∀qh∈Mh. (23)

For approximation purposes, one can use the Lagrange interpolation operator Π_h^L, or the Scott-Zhang interpolation operator Π_h^SZ. The latter allows one to interpolate any element of H₀¹(Ω), with values in Mh, at the expense of local interpolation operators that are not localized to each tetraedron, but are localized to the union of the tetrahedron and its neighbouring tetrahedra. We refer to [30] for details. Unless otherwise specified, we chooseΠ_h^grad=Π_h^SZ.

Forhgiven, the discrete variational formulation of the time-harmonic prob- lem (3) is

Find e_h∈V_h such that

a_ω(e_h,v_h) = (f|vh)_0,Ω, ∀vh∈V_h. (24)

1 The results obtained in this paper carry over to curved polyhedra, that is domains with piecewise smooth boundaries (see eg. p. 81 in [2] for a precise definition). When dealing with the discretization by first-order edge finite elements inH0(curl;Ω), one may use [26].

Respectively, when dealing with the discretization by Lagrange’s first-order finite elements inH₀¹(Ω), one may use [32]. In particular, it is proven there that optimal interpolation properties hold, ie. one may recover up toO(h) accuracy, provided the field to be interpolated is sufficiently smooth.

To obtain explicit error estimates for the time-harmonic Maxwell equations, a natural idea is to use the interpolation of its solutione. This requires some additional definitions and a priori analysis of the regularity ofe, and of its curl.

Let Π_h^curl be the classical global Raviart-Thomas-N´ed´elec interpolant in H0(curl;Ω) with values in Vh [41]. We then denote by Π_h^div the classi- cal global Raviart-Thomas-N´ed´elec interpolation operator inH₀(div;Ω) with values inW_h [45, 41], where (W_h)hare designed with the help ofH(div;Ω)- conforming, first-order finite element spaces:

Wh:={wh∈H0(div;Ω) : w_h|K∈ D1(K), ∀K∈ Th}, whereD1(K) is the vector space of polynomials onK defined by

D₁(K) :={v∈P₁(K) : v(x) =a+bx, a∈R³, b∈R}.

Let us recall a few useful properties (see Chapter 5 in [40]). To start with, Proposition 3 For all h, it holds that

∇[M_h]⊂V_h; (25)

∀vh∈Vh, Π_h^curlvh=vh; (26)

curl[Vh]⊂Wh; (27)

∀wh∈Wh, Π_h^divwh=wh; (28)

∀v∈H0(curl;Ω)s.t.Π_h^curlv exists, Π_h^div(curlv) =curl(Π_h^curlv).(29) There are useful additional properties regarding Π_h^curl listed below. Below, when we refer to piecewise-H^s fields, the partition is understood as in (12).

Proposition 4 (discrete exact sequence [41]) Let h be given, and let v ∈H0(curl;Ω) that can be written asv =∇q in Ω, for some q∈H₀¹(Ω).

Then if Π_h^curlv is well-defined, there exists qh∈Mh such that Π_h^curlv=∇qh

inΩ.

Proposition 5 (classical interpolation results)Assume thatv∈P H^s(Ω) andcurlv∈P H^s0(Ω)for somes>1/2,s⁰ >0. Then one can defineΠ_h^curlv and, in addition, one has the approximation result [4]:

kv−Π_h^curlvkH(curl;Ω).h^min(s,s0,1){kvkP H^s(Ω)+kcurlvk_{P H}s0

(Ω)}. (30) Furthermore, if curlv is piecewise constant onT_h, one has the improved ap- proximation result (cf. Theorem 5.41 in [40]):

kv−Π_h^curlvkH(curl;Ω).h^min(s,1){kvkP H^s(Ω)+kcurlvk0,Ω}.

Remark 2 WhenΩ2 is a domain ofR², note that one can define the Raviart- Thomas-N´ed´elec interpolant of a fieldv∈H(curl;Ω2) as soon asv∈P H^s(Ω2) for somes>0 (there is no requirement on the regularity of curlv). This re- sult is proven in [3] for fields in H(div;Ω2), and it obviously carries over to fields in H(curl;Ω₂) by appropriate coordinates transform. Further, one has the approximation result:

kv−Π_h^curlvk_H(curl;Ω2).h^min(s,1){kvk_{P H}^s_(Ω2)+kcurlvk0,Ω₂}. ut Our aim is to applyΠ_h^curl to the electric fieldegoverned by (3).

First, one must havecurle∈P H^s0(Ω) for somes⁰ >0. Sincef ∈L²(Ω), we immediately find thatc=µ⁻¹curlebelongs to

XT(Ω, µ) :={v∈H(curl;Ω) : µv∈H₀(div;Ω)}.

Then, using a shift theorem for the companion scalar problem with Neumann boundary condition

Finds∈H_zmv¹ (Ω)such that (µ∇s|∇q)0,Ω =hg⁰, qi_H1

zmv(Ω), ∀q∈H_zmv¹ (Ω), (31) and a regular plus gradient splitting (see eg. [24, 21]), we introduceτN eu∈(0,1]

depending only on the geometry and onµsuch that XT(Ω, µ)⊂ ∩_s⁰_{∈[0,τN eu})P H^s0(Ω),

with continuous imbedding for alls⁰∈[0, τN eu). Furthermore, using a Weber inequality (cf. Theorem 6.2.5 in [2]), one has that for alls⁰ ∈[0, τN eu),

∀v∈XT(Ω, µ), kvk_{P H}s0

(Ω).kcurlvk_0,Ω+kdivµvk_0,Ω+ X

1≤i≤I

|hµv·n,1i_H1/2(Σi)|. (32) As a consequence, we note that sinceµis piecewise smooth, it also holds that curle∈ ∩_s⁰_{∈[0,τN eu)}P H^s0(Ω). (33) Then, to guarantee thatΠ_h^curlcan be applied to the electric fielde, one must check whethere∈P H^s(Ω) for some s>1/2. To evaluate the exponentsa priori, we use the following decomposition (see Lemma 2.4 of [36]).

Proposition 6 There exist operators

P ∈ L(H0(curl;Ω),H¹(Ω)), Q∈ L(H0(curl;Ω), H₀¹(Ω)), such that

∀v∈H₀(curl;Ω), v=P v+∇(Qv). (34) This yields some useful results for elements ofKN(Ω, ε).

Corollary 2 The a priori regularity of elements ofKN(Ω, ε) is governed by the imbedding:

KN(Ω, ε)⊂ ∩_s∈[0,τDir)P H^s(Ω).

Moreover, for alls∈[0, τDir),

∀k∈KN(Ω, ε), kkk_{P H}^s_(Ω).kcurlkk0,Ω. (35) Proof Let k ∈ K_N(Ω, ε). According to proposition 6, one can write k = k^?+∇sk with k^? ∈H¹(Ω), resp. s_k ∈H₀¹(Ω), and it holds that kk^?k1,Ω+ ks_kk_H1

0(Ω) .kkk_H(curl;Ω). In particular, div(ε∇s_k) =−divεk^? in Ω, so s_k solves the modified scalar problem (14) with data (0,−k^?). Thanks to the shift theorem, we know that, for all s ∈ [0, τDir),sk belongs to P H^1+s(Ω), with the boundkskk_{P H}1+s(Ω).kk^?k1,Ω.Using the triangle inequality, we conclude that

∀s∈[0, τDir), k∈P H^s(Ω), andkkk_{P H}^s_(Ω).kkk_H(curl;Ω).

This proves the first part of the corollary. Using finally theorem 1 on the equivalence of norms inKN(Ω, ε), we conclude that (35) holds. ut Hence, it may happen that the field to be interpolated, eg. the electric field e, does not belong to ∪_s>1/2P H^s(Ω). In the classical case, the occurence of such a situation is explained in Section 7 of [24]. In the interface case, this can be inferred from the results obtained in [8, 11].

On the other hand, to interpolate such a low regularity field, one may still choose the quasi-interpolation operator of [31], or the combined interpolation operator of [21, 22]. We choose the latter. To get a definition for the combined interpolation operator, denoted by Π_h^comb, one needs to be able to split low regularity fields defined onΩ. To that end, we apply proposition 6.

Definition 1 (combined interpolation operator) Letv ∈ H₀(curl;Ω), withcurlv∈H^s0(Ω) for somes⁰ >0. We define

Π_h^combv:=Π_h^curl(P v) +∇(Π_h^grad(Qv)).

Then, the approximation results for the combined interpolation are a straight- forward consequence of the available results forΠ_h^curl andΠ_h^grad.

Proposition 7 (combined interpolation results) Let v ∈ H₀(curl;Ω), with Qv ∈P H^1+s(Ω) and curlv ∈P H^s0(Ω) for some s≥0,s⁰ >0. One has the approximation result:

kv−Π_h^combvk_H(curl;Ω).h^min(s,s0,1){kvk_H(curl;Ω) (36) +kQvkP H^1+s(Ω)+kcurlvk_{P H}s0

(Ω)}.

Furthermore, if curlv is piecewise constant onTh, one has the improved ap- proximation result:

kv−Π_h^combvkH(curl;Ω).h^min(s,1){kvkH(curl;Ω)+kQvkP H^1+s(Ω)}.

Together with this definition of the combined interpolation operator, we have the results below, to be compared with the well-known results (26) and (29) for the classical interpolation operator.

Proposition 8 For all h, it holds that

∀vh∈Vh,∃qh∈Mh, Π_h^combvh=vh+∇qh; (37)

∀v∈H0(curl;Ω) s.t.curlv∈H^s0(Ω) for somes⁰ >0, (38) Π_h^div(curlv) =curl(Π_h^combv).

Proof Letvh∈Vh. We note that becausevh is piecewise smooth onTh, one hasvh,curlvh∈P H^t(Ω) for allt∈[0,1/2). HenceΠ_h^combvhis well-defined according to definition 1. If we write vh= (vh)^?+∇sv_h, with (vh)^? =P vh, resp.sv_h =Qvh, we haveΠ_h^combvh:=Π_h^curl(vh)^?+∇(Π_h^gradsv_h).

On the other hand,∇sv_h =vh−(vh)^?. SinceΠ_h^curl(vh−(vh)^?) is well-defined, so isΠ_h^curl(∇sv_h) and, according to proposition 4, there exists q_h⁰ ∈Mhsuch thatΠ_h^curl(∇svh) =∇q⁰_h. Applying nowΠ_h^curlto (v_h)^?=v_h−∇svh, it follows that

Π_h^curl(vh)^?=Π_h^curlvh− ∇q⁰_h=vh− ∇q⁰_h,

where the second equality now follows from (26). One concludes that Π_h^combvh:=vh+∇(Π_h^gradsv_h−q_h⁰),

which is precisely (37) withqh=Π_h^gradsv_h−q⁰_h∈Mh.

To check (38), letvbe split asv=P v+∇(Qv). Sincecurl(P v)∈H^s0(Ω), ac- cording to proposition 5 one may apply (29) toP v, leading toΠ_h^div(curl(P v)) = curl(Π_h^curl(P v)). On the other hand, because of the definition 1 ofΠ_h^combv= Π_h^curl(P v) +∇(Π_h^grad(Qv)) one has

curl(Π_h^curl(P v)) =curl(Π_h^combv− ∇(Π_h^grad(Qv))) =curl(Π_h^combv).

Using finally the equalitycurlv =curl(P v) leads to the claim. ut We now have all the required results to bound the interpolation error of the electric fielde.

Proposition 9 Letebe the solution to the time-harmonic Maxwell equations.

Let the extra-regularity of the data f be as in (13) with τ0 > 0 given. One can define Π_h^combe, and moreover one has the approximation result, for all s∈[0,min(τ0, τDir, τN eu)),

ke−Π_h^combek_H(curl;Ω).h^s{kdivfk_−1+s,Ω+kfk0,Ω}.

Proof Lets∈[0,min(τ0, τDir, τN eu)) ; becauseτDir≤1, one hass<1.

According to proposition 6, we may writee=e^?+∇sewithe^?∈H¹(Ω),se∈ H₀¹(Ω), andke^?k1,Ω+ksek_H1

0(Ω).kekH(curl;Ω). By construction,sesolves the modified scalar problem (14) with data (divf,−e^?). Buts<min(τ0, τDir) so,

thanks to the shift theorem,se∈P H^1+s(Ω), with the boundksek_{P H}1+s(Ω). kdivfk−1+s,Ω+ke^?k1,Ω. Using (22) for the last inequality below, we find

ksekP H^1+s(Ω).kdivfk−1+s,Ω+kekH(curl;Ω).kdivfk−1+s,Ω+kfk0,Ω. On the other hand, one hascurle∈P H^s(Ω), cf. (33). Then, with the help of the bound (32) on kµ⁻¹curlek_{P H}^s_(Ω), noting that divcurle= 0 inΩ, and hcurle·n,1i_H1/2(Σi)= 0 for 1≤i≤I (see Remark 3.5.2 in [2]), we find

kcurlek_{P H}^s_(Ω).kµ⁻¹curlek_{P H}^s_(Ω).kcurl(µ⁻¹curle)k_0,Ω .kek0,Ω+kfk0,Ω,

where we used the relation curl(µ⁻¹curle) =ω²εe+ω²f in Ω. Therefore, we one can defineΠ_h^combe and, using (36) and (22) once more, one finds now ke−Π_h^combek_H(curl;Ω).h^s{kek_H(curl;Ω)+ksek_{P H}1+s(Ω)+kcurlek_{P H}^s_(Ω)}

.h^s{kdivfk−1+s,Ω+kfk0,Ω},

which is the desired estimate. ut

Remark 3 In particular, we note that even when the electric fieldedoes not belong to∪s>1/2P H^s(Ω), one may use the combined interpolation operator and still obtain “best” interpolation error. On the other hand, it is well-known by using classical interpolation that, whenτ_Dir=τ_{N eu}= 1, and for aregular dataf ∈H(div;Ω), the interpolation error behaves likeO(h). ut From this point on, to obtain the well-posedness result for the discretized problems, and finally convergence to the exact solutione, one needs to prove a uniform discrete inf-sup condition. For that, we mimic in section 5 the two in- gredients that were used to solve the exact variational formulation: uniformly stable discrete decompositions in the spirit of proposition 1 ; uniform equiva- lence of norms in the spirit of theorem 1. The key ingredient is the study of the approximation of the companion scalar problem (5). And, since it was origi- nally solved with the help of T-coercivity, we consider two situations regarding its approximation. We refer to the appendix A for details. Either we have at hand a ”full” T-coercivity involution operator T0 to solve (5), that can also be used to establish to establish the uniform discrete T-coercivity (58)-(59) of the discrete scalar problems (23). Or, we only have at hand a ”weak” explicit T-coercivity involution operatorT, cf. (57). The first situation is addressed in subsection 5.1, whereas the second situation is addressed in subsection 5.2.

5 Uniform estimates

5.1 Case of a ”full” T-coercivity operator

We assume in this section that we have at hand a ”full” T-coercivity involu- tion operatorT0to solve the companion scalar problem (5) (see section A.1),

and that the meshes are T-conform, such that (58)-(59) are fulfilled, with consequences listed in section A.2. Define, for anyh,

Kh(ε) :={vh∈Vh : (εvh|∇qh)0,Ω= 0, ∀qh∈Mh}. (39) Proposition 10 Assume that (58) holds. For allh, one has the direct sum

Vh=∇[Mh]⊕Kh(ε). (40) Proof Let h be given. Thanks to (25), we know that ∇[Mh] + Kh(ε) is a subset ofV_h. Then forv_h∈V_hand because the discrete scalar problem (23) is well-posed, there exists one, and only one,p_vh∈M_h such that

(ε∇pv_h|∇qh)0,Ω= (εvh|∇qh)0,Ω, ∀qh∈Mh. (41) And one has

kv_h =vh− ∇pv_h ∈Kh(ε), (42) soV_h=∇[Mh] +K_h(ε). Using (58), the fact that the sum is direct is derived exactly as in the continuous case (see the proof of proposition 1). ut For all h, we can use the splitting (40) and the explicit definitions (41)-(42) to introduce the operators

π1h :

Vh→Mh

vh7→pv_h

, π2h :

Vh→Kh(ε) vh7→kv_h

. (43)

In other words, one may write, for all h, for all v_h ∈V_h, v_h =∇(π_1hv_h) + π2hvh. Also, one has for all h, (π2h)² =π2h. Below, we prove the uniform stability of the decomposition (40).

Proposition 11 Assume that (58) holds. The continuity moduli of the oper- ators(π1h)h,(π2h)h are bounded independently ofh.

Proof Given handv_h∈V_h, one has according to (58) and (41)

α⁰₀k∇(π1hu_h)k²_0,Ω≤(ε∇(π1hu_h)|∇(T0(π_1hu_h)))_0,Ω= (εu_h|∇(T0(π_1hu_h)))_0,Ω

≤ kεuhk0,Ωk∇(T0(π1huh))k0,Ω

≤ kεu_hk_0,ΩkT₀k_L(H1

0(Ω))k∇(π_1hu_h)k_0,Ω, so that

k∇(π_1hu_h)k_0,Ω≤(α⁰₀)⁻¹kεk_∞,ΩkT₀k_L(H1

0(Ω))ku_hk_0,Ω. And then

kπ_2hu_hk_H(curl;Ω)≤(1 + (α⁰₀)⁻¹kεk_∞,ΩkT₀k_L(H1

0(Ω)))ku_hk_H(curl;Ω),

so the claim follows. ut

Next, one has to check that kh 7→ kcurlkhk0,Ω defines a norm on Kh(ε).

And, if the answer is positive, whether this norm of uniformly equivalent in h(ie. with constants that are independent ofh) to thek · kH(curl;Ω)-norm on Kh(ε).