We study the regularity in terms of the fractional order Sobolev spacesHs,s∈[0,1]

ON THE APPROXIMATION OF ELECTROMAGNETIC FIELDS BY EDGE FINITE ELEMENTS. PART 3: SENSITIVITY TO

COEFFICIENTS^∗

PATRICK CIARLET, JR.^†

Abstract. In bounded domains, the regularity of the solutions to boundary value problems depends on the geometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spacesH^s,s∈[0,1]. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is, to find some regularity exponentτ∈(0,1), such that they both belong toH^s, for alls∈[0, τ). After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N´ed´elec’s first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations.

Key words. Maxwell’s equations, interface problem, edge elements, sensitivity to coefficients, error estimates

AMS subject classifications. 78A48, 35B65, 65N30 DOI. 10.1137/19M1275383

1. Introduction. We study the numerical approximation of electromagnetic fields governed by Maxwell’s equations. More precisely, our goal is to characterize the dependence of the error between the exact and computed fields, with respect to the coefficients that define the model (PDEs, supplemented with boundary condi- tions). This paper is the third part of the series entitled “On the Approximation of Electromagnetic Fields by Edge Finite Elements” [12, 13].

For Maxwell’s equations, the coefficients are the electric permittivity, the magnetic permeability, and the conductivity. Classically, the model is recast as an equivalent variational formulation. The first goal is to determine the value of the constants that appear in the analysis of the variational formulation, which are the continuity modulus of the forms, and the coercivity or inf-sup constants. Then, one performs the numerical analysis of the model. In addition to the abovementioned constants, one has to estimate the order of convergence, which depends on the (extra-)regularity of the fields; this (extra-)regularity depends itself on the behavior of the coefficients and on the geometry of the model. In particular, it is crucial to use ad hoc norms to measure the fields and the data, and particular care is devoted to the definition of those norms.

We observe that if the coefficients belong to a set not reduced to a singleton (e.g., random coefficients), then the (extra-)regularity may vanish in some limit cases.

The outline of the paper is as follows.

In the next section, we introduce the model problem (see, eg., [2]), set in a bounded region of R³, with volume sources. We prescribe some a priori conditions

∗Received by the editors July 16, 2019; accepted for publication (in revised form) April 27, 2020;

published electronically DATE.

https://doi.org/10.1137/19M1275383

†POEMS, ENSTA Paris, Institut Polytechnique de Paris, 828 Bd des Mar´echaux, 91762 Palaiseau Cedex, France (patrick.ciarlet@ensta-paris.fr).

1

on the coefficients and on the source terms; the coefficients are only supposed to be piecewise smooth, hence they may be discontinuous. The variational formulation is introduced. Then, in section 3, we recall the main results regarding the discretization of electromagnetic fields by N´ed´elec’s finite elements [30, 29, 12, 20]. We also define the function spaces that will allow us to perform the analysis of the model. Particular attention is paid to the fractional order Sobolev spaces, which play a crucial role in the analysis. These are defined either using the real interpolation method [28, 33, 9]

or with the help of the Sobolev–Slobodeckij norms [24, 25, 11]: this yields two ways to measure their elements. In section 4, we provide the (well-known) estimates for the continuity modulus and the coercivity constant related to the variational formulation, with respect to the coefficients of the model.

It remains to estimate the so-called regularity exponent and the stability con- stant, which relate the norm of the electromagnetic fields in ad hoc fractional order Sobolev spaces norms to the norm of the source terms. Estimating these last two quantities with respect to the coefficients of the model is less classical. Hence, most novelties in the paper are contained in the approach developed in sections 5 and 6.

In section 5, we recall how one can split the electromagnetic fields into a regular part and a gradient part [4], so the (lack of) regularity of the fields rests on the regularity of the gradients. This is the subject of section 6. We use a perturbation argument `a la Jochmann [27] or Bonito, Guermond, and Luddens [6] to estimate this regularity with respect to the coefficients of the model. We call it theglobal approach. When the co- efficient are piecewise constant, one may also use thelocal approach; see Appendix B.

The main novelties are threefold: the extension of existing results to problems with complex-valued coefficients, set in a nontopologically trivial domain; the use of the two measures for elements of the Sobolev spaces, and their interplay; and the design of estimates for the numerical error that depend only on the coefficients of the model (see Theorem 6.15). To conclude, we illustrate our results by two examples in section 7.

We denote constant fields by the symbolcst. Vector-valued (resp., tensor-valued) function spaces are written in boldface characters (resp., blackboard bold characters).

Given a nonempty open setOofR³, we use the notation (·|·)_0,O (resp.,k·k_0,O) for the L²(O) and the L²(O) := (L²(O))³ hermitian scalar products (resp., norms). More generally, (·|·)_s,O andk · k_s,O (resp.,| · |_s,O) denote the hermitian scalar product and the norm (resp., seminorm) of the Sobolev spaces H^s(O) and H^s(O) := (H^s(O))³ fors∈R(resp., fors >0). The index zmv indicates zero-mean-value fields. If more- over the boundary∂Ois Lipschitz,ndenotes the unit outward normal vector field to

∂O. Finally, 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 normv 7→(kvk²_0,O+kcurlvk²_0,O)^1/2, etc.

We refer to the monographs of Monk [29] and Assous, Ciarlet, and Labrunie [4] for details. We will define more specialized function spaces later on.

2. The model problem. Let Ω be a domain in R³, that is, an open, con- nected, and bounded subset ofR³ with a Lipschitz-continuous boundary∂Ω. We are interested in solving the time-harmonic Maxwell’s equations (with time-dependence exp(−ıωt), for a given pulsationω >0),

ıωd+curlh=jin Ω, (2.1)

−ıωb+curle= 0 in Ω, (2.2)

divd=%in Ω, (2.3)

divb= 0 in Ω.

(2.4)

Above, (e,d,h,b) are the electromagnetic fields. We suppose thatdandbare related toe andhby the constitutive relations

d=εe, b=µhin Ω, (2.5)

where the real-valued coefficient ε is the electric permittivity and the real-valued coefficientµis the magnetic permeability.

The source terms j and % are respectively the current density and the charge density, and they are related by the charge conservation equation

−ıω%+ divj= 0.

(2.6)

We suppose that the current density may be written as j =j_ext+σe in Ω, (2.7)

where j_ext is an externally imposed current, and the real-valued coefficient σ is the conductivity.

2.1. A priori assumptions. Classically, the electromagnetic fields all belong to L²(Ω) and the coefficients ε, µ, and σ have a fixed sign (positive): we make these assumptions from now on. We also assume throughout this work that these coefficients together with their inverses belong to L^∞(Ω), and we use the notation ε_max=kεkL^∞(Ω),ε_min= (kε⁻¹kL^∞(Ω))⁻¹, etc.

We choose the data (j_ext, %) inH(div; Ω)×H⁻¹(Ω). It is also possible to choose j_ext∈L²(Ω) with divj_ext∈H^−t(Ω) for somet∈(0,1), but we assume for simplicity that divj_ext∈L²(Ω). We refer to section 6.5 for the study of the more general case.

Finally, we assume that the medium Ω is surrounded by a perfect conductor, so that the boundary condition below holds:

e×n= 0 on∂Ω.

(2.8)

Equations (2.1)–(2.8) together with the assumptions on the coefficients and on the source terms define our model problem. When we focus on the discretization (see section 3.3 and afterward), we assume that Ω is a Lipschitz polyhedron, that ε, σ are piecewise smooth on Ω, and that µ is constant on Ω. We call this setting the polyhedral model problem. Let us mention that once the field e is known, then all other electromagnetic fields d, b, and hare known too. As a consequence, we focus on the study of the fielde. In particular, we note thatebelongs to the function space H0(curl; Ω).

2.2. Variational formulation. In the spirit of the charge conservation equa- tion, let us introduce %ext = −ı/ωdivj_ext ∈ L²(Ω). Our model problem can be formulated in the electric fieldeonly, namely







Find e∈H0(curl; Ω)such that

−ω²εσe+curl(µ⁻¹curle) =ıωj_ext in Ω, divεσe=%extin Ω.

(2.9)

Above, the complex-valued coefficient εσ is defined byεσ =ε+ıσ/ω. Note that in (2.9), the equation divεσe=%ext is implied by the second-order equation−ω²εσe+ curl(µ⁻¹curle) =ıωj_ext, together with the charge conservation equation (2.6) and the splitting of the current (2.7), so it can be omitted. Moreover, one can check that the equivalent variational formulation inH0(curl; Ω) is written

Find e∈H0(curl; Ω)such that

(µ⁻¹curle|curlv)_0,Ω−ω²(ε_σe|v)0,Ω=ıω(j_ext|v)0,Ω∀v∈H₀(curl; Ω).

(2.10)

Under the assumptions on the coefficients, this variational formulation is well-posed (see, for instance, [4, section 8.3.2]). In other words,

∃C_(ε,µ,σ)>0 such that∀j_ext∈L²(Ω), ∃!e solution to (2.10), and kek_H(curl;Ω)≤C_(ε,µ,σ)kj_extk0,Ω.

(2.11)

3. Discretization of electromagnetic fields. Here, we define finite element approximations of the electric field e. We also recall how one can build an a pri- ori error estimate between e and its approximation. When we study the numerical approximations and for ease of exposition, we assume that Ω is a Lipschitz polyhe- dron (polyhedral model problem). To define finite dimensional subspaces (Vh)h of H0(curl; Ω), we choose the so-called N´ed´elec’s first family of edge finite elements, defined on simplicial meshes of Ω. We follow here [12, section 2.4]. It is sufficient to use first-order finite elements because we focus on electromagnetic fields with low regularity. Ω is triangulated by a shape regular family of meshes (Th)_h, made up of (closed) simplices, generically denoted by K. A mesh is indexed by h:= max_Kh_K (the meshsize), whereh_K is the diameter of K. Denoting byρ_K the diameter of the largest ball inscribed inK, we assume that there exists a shape regularity parameter ς > 0 such that for all h, for all K ∈ T_h, it holds that h_K ≤ ςρ_K. N´ed´elec’s H(curl; Ω)-conforming (first family, first-order) finite element spaces are defined as

Vh:={vh∈H0(curl; Ω) : v_h|K ∈ R1(K)∀K∈ Th}, whereR1(K) is the six-dimensional vector space of polynomials on K

R1(K) :=

v∈P1(K) : v(x) =a+b×x, a,b∈R³ .

According to [30, Theorem 1], any elementv inR₁(K) is uniquely determined by the degrees of freedom in the moment setME(v):

ME(v) :=

Z

e

v·tdl

e∈AK

.

Above,AKis the set of edges ofK, andtis a unit vector along the edgee. The global set of moments onV_h is then obtained by taking one degree of freedom as above per interior edge of Th. We recall that the basic approximability property is written (cf.

[29, Lemma 7.10])

h→0lim

inf

vh∈Vh

kv−vhk_H(curl;Ω)

= 0 ∀v∈H0(curl; Ω).

(3.1)

Assuming for simplicity that the integrals are computed exactly, the discrete electric problem is written

Find eh∈Vh such that

(µ⁻¹curleh|curlvh)0,Ω−ω²(εσeh|vh)0,Ω=ıω(j_ext|vh)0,Ω∀vh∈Vh. (3.2)

Because the exact problem is well-posed (cf. (2.11)), one may apply C´ea’s lemma to find

∃C_(ε,µ,σ)^] >0∀h, ke−ehk_H(curl;Ω)≤C_(ε,µ,σ)^] inf

v_h∈V_hke−vhk_H(curl;Ω). (3.3)

Classically, the constantC_(ε,µ,σ)^] depends on the coercivity constant and on the norm of the sesquilinear form in the left-hand side of (2.10) and (3.2). This constant is investigated in detail in section 4. It follows from (3.1) that

h→0limke−e_hk_H(curl;Ω)= 0.

In order to obtain a result which is more accurate, typically a convergence rate in the order ofh^sfor somes >0, one has to use information on the (extra-)regularity of the electric field. Let us recall how this can be achieved.

3.1. A few reminders about Sobolev spaces. Let O ⊂ Ω be a nonempty connected open subset ofR³ with Lipschitz boundary. To give a precise meaning to the regularity of a scalar or vector field onO, we use the well-known Sobolev scale (H^s(O))s.

(0) Fors∈N, one uses the standard definition:

H^s(O) :={v∈L²(O) s.t. ∀α∈N³,|α| ≤s, ∂_αv∈L²(O)}, equipped with the normkvk_s,O := (P

α∈N³,|α|≤sk∂αvk²_L2(O))^1/2. Obviously,H⁰(O) = L²(O).

(1) To define those spaces fors > 0,s6∈ N, several possibilities exist. Let us begin with the real interpolation method [28] (see also Appendix A), which allows us to define those Hilbert spaces for noninteger indicess=m+σ, m∈N,σ∈(0,1), as

H^s(O) := (H^m(O), H^m+1(O))σ,2.

The corresponding norm is denoted by k · k_s,O. In particular, for all 0≤ s ≤ t, it holds thatH^t(O)⊂H^s(O) with continuous embedding [9, section 14]:

∃C_(s,t)>0 ∀v∈H^t(O), kvk_s,O ≤C_(s,t)kvk_t,O. Given 0< s0≤s1< t <1,s7→C(s,t) is continuous on [s0, s1].

A well-known alternative is to define, forσ∈(0,1), H^σ(O) :=

v∈L²(O) s.t. |v|_Hσ(O)<∞ , where

|v|H^σ(O):=

Z

O

Z

O

|v(x)−v(y)|²

|x−y|^3+2σ dxdy ^1/2

is the Sobolev–Slobodeckij seminorm, andH^σ(O) is endowed with the Sobolev–Slobodeckij norm

kvkH^σ(O):=

kvk²_0,O+|v|²_Hσ(O)

^1/2 . And then, fors=m+σ,m∈N,σ∈(0,1):

H^s(O) :={v∈H^m(O) s.t. ∀α∈N³ with|α|=m, ∂_αv∈H^σ(O)}, endowed with the Sobolev–Slobodeckij norm

kvk_Hs(O):=



kvk²_m,O+ X

α∈N³,|α|=m

|∂αv|²_Hσ(O)





1/2

.

The Sobolev–Slobodeckij seminorm is

|v|H^s(O):=



 X

α∈N³,|α|=m

|∂αv|²_Hσ(O)





1/2

.

For alls∈R⁺\N, it holds thatH^s(O) =H^s(O) algebraically and topologically:

∃m_(s), M_(s)>0 ∀v∈H^s(O), m_(s)kvk_H^s_(O)≤ kvk_s,O ≤M_(s)kvk_H^s_(O). However, in a bounded set O, there are no results on the uniform equivalence of Sobolev–Slobodeckij norms and real interpolation norms whensspans (0,1), i.e., on bounding one norm with the other times a constant that is independent ofs∈(0,1).

We refer to [25, 11] for illuminating discussions on this topic. On the other hand (see [24] or [9, section 14]), if s spans [s0, s1] with 0 < s0 ≤s1 < 1, there is a uniform equivalence of norms: in other words,m, m⁻¹, M, M⁻¹ are continuous on [s0, s1].

(2) For s ≥ 0, H₀^s(O) is the closure of D(O) in H^s(O). For s ∈ [0,¹₂], it holds thatH₀^s(O) =H^s(O) algebraically and topologically (see, for instance, [21, Theorem 1.4.2.4]), while fors > ¹₂, it holds thatH₀^s(O)(H^s(O).

(3) Fors <0,H^s(O) is the topological dual ofH₀^−s(O).

(4) For s ≥ 0, He^s(O) (also denoted in the literature by H₀₀^s(O)) is composed of elements ofH^s(O) such that the continuation by zero outsideObelongs toH^s(R³);

for s /∈ ¹₂ +N, it holds that He^s(O) = H₀^s(O), while for s ∈ ¹₂ +N, it holds that He^s(O)(H₀^s(O). Going back to the real interpolation method, for noninteger indices s=m+σ, m∈N,σ∈(0,1), one has He^s(O) = (H₀^m(O), H₀^m+1O))σ,2.

For the regularity studies, we choose the real interpolation method, while we use the double-integral Sobolev–Slobodeckij norms and seminorms to perform the numerical analysisand derive convergence rates.

3.2. Piecewise smooth fields. The set P := {Ω_j}_j=1,...,J is called a parti- tion of Ω if (Ωj)j=1,...,J are disjoint domains, and it holds that Ω =∪^J_j=1Ωj. When the (Ωj)j=1,...,J are Lipschitz polyhedra, we use the term polyhedral partition. Given a partition, we introduce the corresponding interface Σ := ∪_1≤j6=j⁰_≤J(∂Ωj ∩∂Ωj⁰).

For a fieldv defined on Ω, we denote by vj its restriction to Ωj for allj. In relation to the partitionP and fors≥0, we define

P H^s(Ω) :=

v∈L²(Ω) : vj∈H^s(Ωj), 1≤j≤J , endowed with kvk_{P H}s(Ω):=



 X

1≤j≤J

kvjk²_s,Ωj





1/2

or kvk_{P H}^s_(Ω):=



 X

1≤j≤J

kvjk²_Hs(Ω_j)





1/2

.

To simplify the notation, the reference to P is usually omitted. Let us recall the technical result (Theorem 4.1 of [1], or Lemma 2.1 of [6]).

Proposition 3.1. For alls∈[0,1], it holds that kvk_{P H}s(Ω)≤ kvks,Ω ∀v∈H^s(Ω).

Note that one hasP H^s(Ω) =H^s(Ω) algebraically and topologically for all parti- tions and for alls∈[0,¹₂).

Finally, we introduce

P H^s(curl; Ω) :={v∈P H^s(Ω) : curlv∈P H^s(Ω)} fors >0 ; P W^1,∞(Ω) :={ζ∈L^∞(Ω) : ζj∈W^1,∞(Ωj), 1≤j≤J}.

P H^s(curl; Ω) is endowed with the graph norm. We observe that one has the embed- dingP H^s(curl; Ω)⊂H(curl; Ω), according to the definition of P H^s(Ω).

And we endowP W^1,∞(Ω) with the normkζkP W^1,∞(Ω):=kζkL^∞(Ω)+|ζ|P W^1,∞(Ω)

and the seminorm |ζ|P W^1,∞(Ω) := max_1≤j≤Jk∇ζjkL^∞(Ω_j). For a piecewise constant coefficientζ, it holds thatkζkP W^1,∞(Ω)=kζkL^∞(Ω)= max_1≤j≤J|ζj|.

When the partition is trivial, that is,P={Ω}, we omit theP orP in the name of the function space.

We note that, for the polyhedral model, the assumption on the coefficients is written ε, σ ∈ P W^1,∞(Ω), and the interface Σ can be viewed as the locus of the discontinuities of at least one of the two coefficients. More generally, if ε, σ, µ ∈ P W^1,∞(Ω), Σ is the locus of the discontinuities of at least one of the three coefficients.

3.3. Finite element interpolation or quasi-interpolation operators. In a Lipschitz polyhedron Ω, one can build finite element interpolation, or quasi-interpola- tion, operators that act on piecewise smooth fields, with range inV_h. For a polyhedral partitionP:={Ωj}j=1,...,J, the family of meshes (Th)_his said to be conforming if, for allh, for allK∈ T_h, there existsj₀such thatK⊂Ω_j0. Let us recall briefly the theory of finite element interpolation. Classically, those results are obtained by studying the properties of the mappings to the reference element, using Sobolev–Slobodeckij semi- norms. It holds, for conforming meshes, that

∀s∈(0,1], ∃C^interp_(ς,s) >0, ∀v∈P H^s(curl; Ω), ∀h,

kv−Π^interp_h vkH(curl;Ω)≤C^interp_(ς,s) h^s{kvkP H^s(Ω)+kcurlvkP H^s(Ω)}.

(3.4)

In (3.4), the interpolation operator Π^interp_h is defined in [29, section 5.5] for s > ¹₂, resp., is the so-called combined interpolation operator of [12, section 4.2]

fors≤¹₂. Regarding the theory of finite element quasi-interpolation, a similar result can be derived, namely, that (3.4) holds, where Π^interp_h now stands for the quasi- interpolation operator defined in [20, section 3.5]. The two finite element interpolation and quasi-interpolation bounds are identical, bearing in mind that Π^interp_h is either the interpolation or the quasi-interpolation operator. In addition, we note thatC^interp_(ς,s) is not proven to be independent ofsin the abovementioned papers. On the other hand, one can check that C^interp_(ς,s) depends continuously ons in (0,1) with the help of the tools proposed in those papers. For the derivation of those continuous dependence results, we refer precisely to the proof of Theorem 3.3 in [20] using abstract estimates from [19, section 5] for the quasi-interpolation, resp., [12, section 4.2] using estimates for the Scott–Zhang interpolation, for the combined interpolation. Both proofs rely on [9, section 14.3].

With the help of the results on the equivalence of norms of section 3.1(1), we conclude that one can write (3.4) with the real interpolation norms

∀s∈(0,1], ∃C_(ς,s)^interp>0, ∀v∈P H^s(curl; Ω), ∀h, kv−Π^interp_h vkH(curl;Ω)≤C_(ς,s)^interph^skvkP H^s(curl;Ω). (3.5)

Furthermore, one can choose s 7→ C_(ς,s)^interp that is continuous on [s0, s1] for all 0 <

s0≤s1<1.

3.4. Extra-regularity of the electric field and convergence rate. Since j_ext∈H(div; Ω), one may prove that the electric field that solves (2.9)–(2.10) enjoys extra smoothness. More precisely, the aim is to prove that

∃τ_(ε,µ,σ)>0, ∀t∈(0, τ_(ε,µ,σ)), ∃C_(ε,µ,σ,t)^? , ∀j_ext∈H(div; Ω), e∈P H^t(curl; Ω) andkek_{P H}t(curl;Ω)≤C_(ε,µ,σ,t)^? kj_extk_H(div;Ω). (3.6)

Above, τ(ε,µ,σ) plays the role of aregularity exponent, whileC_(ε,µ,σ,s)^? can be seen as astability constant.

Let Θ be a set of coefficients (ε, µ, σ) whose elements are all piecewise smooth on the same partition, and assume thatτ = inf_{(ε,µ,σ)∈Θ}τ_(ε,µ,σ)>0, whereτ_(ε,µ,σ)is defined in (3.6), and letj_ext the data be given. Regrouping all the previous results, one concludes first that for all (ε, µ, σ) ∈ Θ, the solution e to (2.9)–(2.10) is in T

s∈[0,τ)P H^s(curl; Ω), and second that one has the error estimates

∀s∈(0, τ),

ke−ehk_H(curl;Ω)≤C_(ε,µ,σ)^] C_(ς,s)^interpC_(ε,µ,σ,s)^? h^skj_extk_H(div;Ω). (3.7)

In the error estimates (3.7), only C_(ε,µ,σ)^] and C_(ε,µ,σ,s)^? depend on the coefficients (ε, µ, σ). Also, for a given ∈(0, τ), since s 7→C_(ς,s)^interp is continuous for s >0, one may replaceC_(ς,s)^interpby thes-independent max_s∈[,τ)C_(ς,s)^interp for alls∈[, τ).

Our purpose is now to estimate more precisely the constants that appear in (3.3), (3.5), (3.6), and (3.7). The dependency of C_(ε,µ,σ)^] on (ε, µ, σ) is addressed in section 4. For τ_(ε,µ,σ) and C_(ε,µ,σ,s)^? , this dependency can be studied via the global approach, which relies on a decomposition of the electric field, and of its curl, into a regular part and a gradient part. To obtain this splitting, we adapt [4, Chapter 6] to the case of complex-valued coefficients, and in the process we generalize the results of [15] to the case of a nontopologically trivial domain: this is the subject of section 5. In section 6, one studies the regularity of the gradient part, where the scalar potential is governed by a second-order elliptic PDE complemented either with Dirichlet boundary conditions (for the electric field) or with Neumann boundary con- ditions (for its curl). The global approach, in the spirit of [27, 6], uses a perturbation argument, where the regularity of the gradient part, i.e., of its scalar potential, is derived in comparison to the regularity of the solution to the Laplace equation with the same boundary condition. Indeed, in a domain Ω and forL²(Ω) volume data, it is known from [26] that the gradient of the solution to the Laplace equation belongs to H¹²(Ω). Using interpolation theory, one can find a regularity exponentτ_(ε,µ,σ) in (3.6) for our problem. To our knowledge, this analysis has only been carried out for PDEs with real-valued coefficients. Here, we check in particular that the analysis proposed in [6] can be extended to the case of complex-valued coefficients. The main results are the derivation of a regularity exponentτ_(ε,µ,σ)that depends polynomially on the coefficients, and the computation of an upper bound for the stability constant C_(ε,µ,σ,s)^? when s spans (0, τ_(ε,µ,σ)). Finally, in section 7 we illustrate the theory in two examples for which the singular behavior can be determined explicitly.

Remark 3.2. For piecewise constant coefficients, there exists an alternative, which focuses on the singular behavior of the gradient part of the solution by finding directly the “best” regularity exponent τ_(ε,µ,σ)^opt attached to this part of the solution. We call it thelocal approach; see Appendix B. On the other hand, providing an upper bound with the local approach forC_(ε,µ,σ,s)^? whensspans (0, τ_(ε,µ,σ)^opt ) is an open question.

4. Estimating the constant C_(ε,µ,σ)^] . Let V := H0(curl; Ω) be endowed withkvkV := (kvk²_0,Ω+kcurlvk²_0,Ω)^1/2, and leta(·,·) be the sesquilinear form onV defined by

(v,w)7→(µ⁻¹curlv|curlw)_0,Ω−ω²(ε_σv|w)_0,Ω.

Then, letC_(ε,µ,σ)^cont be the best continuity constant, or continuity modulus, ofa(·,·), C_(ε,µ,σ)^cont = sup

v,w∈V\{0}

|a(v,w)|

kvk_Vkwk_V,

resp.,C_(ε,µ,σ)^coer be the best coercivity constant ofa(·,·), C_(ε,µ,σ)^coer = inf

v∈V\{0}

|a(v,v)|

kvk²_V .

Proposition 4.1. Let the coefficientsε,µ, andσbe as in section 2.1. Then the sesquilinear forma(·,·)is continuous, withC_(ε,µ,σ)^cont ≤max(ω(ω²ε²_max+σ_max² )^1/2, µ⁻¹), and it is coercive withC_(ε,µ,σ)^coer ≥¹₂σ_minmin(ωε_minε⁻¹_max, µ⁻¹(ω²ε²_max+¹₂σ²_min)^−1/2).

Proof. (We omit the subscript_0,Ω for the L²(Ω)-scalar product and norm). Re- garding continuity, givenv,w∈V, one finds

|a(v,w)| ≤ω²kεσk_L^∞_(Ω)kvk kwk+µ⁻¹kcurlvk kcurlwk

≤max ω²kεσk_L∞(Ω), µ⁻¹

(kvk kwk+kcurlvk kcurlwk)

≤max ω²kε_σk_L∞(Ω), µ⁻¹

kvk_V kwk_V

≤max

ω ω²ε²_max+σ²_max1/2

, µ⁻¹

kvkV kwkV

becausekεσk_L^∞_(Ω)≤(ε²_max+ω⁻²σ_max² )^1/2.

Regarding coercivity, givenv∈V, we letc=curlv, and one finds

|a(v,v)|²= −ω²(εv|v) +µ⁻¹kck²²

+ω²(σv|v)²

=ω⁴(εv|v)²+µ⁻²kck⁴−2ω²µ⁻¹(εv|v)kck²+ω²(σv|v)²

≥ ω⁴−ω²η

(εv|v)²+µ⁻²(1−ω²η⁻¹)kck⁴+ω²(σv|v)² for allη >0 (Young’s inequality). Then,

|a(v,v)|²≥ ω⁴−ω²η

(εv|v)²+µ⁻² 1−ω²η⁻¹

kck⁴+ω²σ_min² kvk⁴

≥ω²ε²_min ω²+σ²_minε⁻²_max−η

kvk⁴+µ⁻² 1−ω²η⁻¹ kck⁴.

As a consequence, choosing η ∈ (ω², ω²+σ²_minε⁻²_max), one derives coercivity. For instance, letη=ω²+¹₂σ²_minε⁻²_max. It follows that

|a(v,v)|²≥ 1

2ω²σ_min² ε²_min

ε²_maxkvk⁴+ σ_min²

2µ² ω²ε²_max+¹₂σ²_minkck⁴

≥ σ_min²

2 min ω²ε²_min

ε²_max, 1

µ² ω²ε²_max+¹₂σ_min²

!

kvk⁴+kck⁴

≥ σ_min²

4 min ω²ε²_min

ε²_max, 1

µ² ω²ε²_max+¹₂σ_min²

! kvk⁴_V.

Hence,|a(v,v)| ≥ ¹₂σminmin(ωεminε⁻¹_max, µ⁻¹(ω²ε²_max+¹₂σ²_min)^−1/2)kvk²_V.

Corollary 4.2. Let the coefficients ε, µ, and σ be as in section2.1. Then the error estimate (3.3)holds with

C_(ε,µ,σ)^] ≤

2 max

ω ω²ε²_max+σ²_max^1/2 , µ⁻¹ σ_minmin

ωε_minε⁻¹max, µ⁻¹ ω²ε²_max+¹₂σ²_min−1/2. Proof. This is an obvious consequence of the fact that one can choose

C_(ε,µ,σ)^] = C_(ε,µ,σ)^cont C_(ε,µ,σ)^coer

in the error estimate (3.3).

5. Splitting into a regular part and a gradient part. Below, we recall some results of [4], and we adapt them to the case of complex-valued coefficients if necessary. Letting ξbe a coefficient defined on Ω, we assume in the current section thatξ fulfills

ξis a complex-valued measurable scalar field on Ω, ξ, ξ⁻¹∈L^∞(Ω),

∃ξ−>0, θ^?∈[0,2π), <(exp(−ıθ^?)ξ)≥ξ− a.e. in Ω.

(5.1)

Lemma 5.1. Let the coefficientsεandσbe as in section2.1. Thenξ=εσ fulfills (5.1), whereθ^? can be any element of [0, π/2].

Remark 5.2. In other words,ξ=εσbelongs to a subclass of those coefficients that are defined by (5.1). In the case where σ ≥ 0 (in particular, in the nonconducting case, that is, when it holds that σ = 0 on some region of Ω), the above result still holds for allθ^?∈[0, π/2). On the other hand, a real-valued, sign-changing coefficient ξdoes not fulfill (5.1). We refer to [8, 7, 18, 10] for those more “exotic” configurations of Maxwell’s equations, in whichεand/orµare real-valued and exhibit a sign change.

Proof. One has ε_σ, ε⁻¹_σ ∈ L^∞(Ω). The result follows from <(exp(−ıθ^?)ε_σ) ≥ cosθ^?εmin+ sinθ^?σmin/ω >0 a.e. in Ω.

Define

XDir(Ω, ξ) :={v∈H0(curl; Ω) : ξv∈H(div; Ω)}, X_{N eu}(Ω, ξ) :={v ∈H(curl; Ω) : ξv∈H₀(div; Ω)}.

The function spaces XDir(Ω, ξ) and XN eu(Ω, ξ) are endowed with the graph norm v7→(kvk²_H(curl;Ω)+kξvk²_H(div;Ω))^1/2. In the particular case whereξis equal to 1, one writesXB(Ω) instead ofXB(Ω,1) forB∈ {Dir, N eu}. We also define the subspaces ofregular fields, resp., thenull subspaces:

HB(Ω) :=XB(Ω)∩H¹(Ω), B∈ {Dir, N eu},

Z_B(Ω) :={v∈X_B(Ω) : curlv= 0, divv= 0 in Ω}, B∈ {Dir, N eu}.

In our case, both εσ and µ fulfill (5.1) and moreover, since j_ext ∈ H(div; Ω), we note that the solution e to (2.9) is such that e ∈ XDir(Ω, εσ), and µ⁻¹curle ∈ XN eu(Ω, µ).

5.1. Geometric framework. The domain Ω can be topologically nontrivial or with a nonconnected boundary. Regarding the first item, we assume that

• either (Top)I=0: “given any curl-free vector field v ∈ C¹(Ω), there exists p∈C⁰(Ω) such that v=∇pinΩ”;

• or (Top)I>0: “there exist I nonintersecting manifolds, Σ1, . . . ,ΣI, with boundaries∂Σi⊂∂Ω, such that, if we letΩ = Ω\˙ SI

i=1Σi, given any curl-free vector fieldv, there existsp˙∈C⁰( ˙Ω)such that v=∇p˙ inΩ.”˙

WhenI = 0, ˙Ω = Ω. For short, we write (Top)_I to cover both instances. One can build cuts that are piecewise plane, see [23, Chapter 6]. Finally, we assume that Ω is a connected set. For the polyhedral model problem, we assume that˙ (Top)_I is fulfilled.

The domain Ω is said to be topologically trivial whenI= 0. WhenI >0, the set Ω has pseudo-Lipschitz boundary in the sense of [3].˙

The a priori regularity of elements ofX_Dir(Ω) and X_{N eu}(Ω) is described in [3, Remark 2.16 and Proposition 3.7]. Below,⊂refers to an algebraical and topological embedding.

Proposition 5.3. LetΩbe a Lipschitz polyhedron: there existsσ_Dir∈(¹₂,1]such that it holds thatX_Dir(Ω)⊂H^σDir(Ω). Assume in addition that (Top)_I is fulfilled:

there existsσN eu∈(¹₂,1]such that it holds thatXN eu(Ω)⊂H^{σN eu}(Ω).

Let Ωbe a domain: the embeddings hold withσDir=σN eu= ¹₂.

Corollary 5.4. With the same assumptions as in Proposition5.3, it holds that ZDir(Ω)⊂H^σDir(Ω) andZN eu(Ω)⊂H^{σN eu}(Ω).

Finally, one can prove that the null spacesZDir(Ω) and ZN eu(Ω) are finite di- mensional vector spaces.

5.2. Splittings of fields. We now provide splittings into a regular part and a gradient part of elements of X_Dir(Ω, ξ) (“electric case”), resp., of elements of X_{N eu}(Ω, ξ) (“magnetic case”), calledregular/gradient splittings. The proofs can be found in sections 6.1.6 and 6.2.6 of [4]. We provide some comments on these splittings below.

Theorem 5.5. LetΩbe a domain such that(Top)_I is fulfilled, and assume thatξ fulfills(5.1). Then, there exists a continuous splitting operator acting fromX_Dir(Ω, ξ) toH_Dir(Ω)×Z_Dir(Ω)×H₀¹(Ω).

More precisely, givenv ∈XDir(Ω, ξ),

∃(vreg,z, p0)∈HDir(Ω)×ZDir(Ω)×H₀¹(Ω), v=vreg+z+∇p0 inΩ.

(5.2) One has

kvregk1,Ω+kvregk_XDir(Ω)+kzkσ_Dir,Ω+kvreg+zk_1/2,Ω≤C_X^Dirkvk_H(curl;Ω). (5.3)

The scalar field p0 is governed by the variational formulation:

Findp0∈H₀¹(Ω) such that

(ξ∇p0|∇ψ)0,Ω=−(ξz|∇ψ)0,Ω−(ξvreg|∇ψ)0,Ω−(divξv|ψ)0,Ω ∀ψ∈H₀¹(Ω).

(5.4)

Theorem 5.6. LetΩbe a domain such that(Top)I is fulfilled, and assume thatξ fulfills(5.1). Then, there exists a continuous splitting operator acting fromXN eu(Ω, ξ) toH¹_zmv(Ω)×ZN eu(Ω)×H_zmv¹ (Ω).

More precisely, givenv ∈XN eu(Ω, ξ),

∃(w_reg,z, q₀)∈H¹_zmv(Ω)×Z_{N eu}(Ω)×H_zmv¹ (Ω), v=w_reg+z+∇q₀ inΩ.

(5.5) One has

kwregk1,Ω+kwregk_{XN eu(Ω)}+kzkσ_{N eu},Ω+kwreg+zk_1/2,Ω≤C_X^{N eu}kvk_H(curl;Ω). (5.6)

The scalar field q0 is governed by the variational formulation:

Findq₀∈H_zmv¹ (Ω) such that

(ξ∇q₀|∇ψ)_0,Ω=−(ξz|∇ψ)_0,Ω−(ξw_reg|∇ψ)_0,Ω−(divξv|ψ)_0,Ω∀ψ∈H_zmv¹ (Ω).

(5.7)

In the splitting (5.2) of v ∈ XDir(Ω, ξ), all three terms vreg,z,∇p0 have vanishing tangential components on the boundary ∂Ω, whereas in the splitting (5.5) of v ∈ X_{N eu}(Ω, ξ),w_regdoes not verify a homogeneous boundary condition in general. Since ξ fulfills (5.1), both variational formulations (5.4) and (5.7) are well-posed. Finally, we note that regarding the a priori regularity in (5.2), one hasv_reg∈H¹(Ω) andz ∈ H^σDir(Ω). Likewise, regarding the a priori regularity in (5.5), one hasw_reg∈H¹(Ω) andz∈H^{σN eu}(Ω).

5.3. Comments. One may easily generalize the splitting theory to the case where ξ is a complex-valued, measurable, tensor field. As a matter of fact, it is straightforward to check that ifξfulfills

ξis a complex-valued measurable tensor field on Ω, ξ, ξ⁻¹∈L^∞(Ω),

∃ξ−>0, θ^?∈[0,2π), ∀z∈C³, <(exp(−ıθ^?)ξz·z)≥ξ−|z|² a.e. in Ω, then the conclusions of Theorems 5.5 and 5.6 still apply. Obviously, (5.4) and (5.7) are well-posed.

In the special case where ξ is a normal tensor field (ξ^∗ξ = ξξ^∗ a.e. in Ω), or equivalently there exists a unitary tensor field Uand a diagonal tensor field D such thatξ=U⁻¹DUa.e. in Ω, one can reformulate the second line of the above condition as

∃ξ−>0, θ^? ∈[0,2π), min

k=1,2,3<(exp(−ıθ^?)Dkk)≥ξ− a.e. in Ω.

(5.8)

6. The global approach for finding a regularity exponentτ_(ε,µ,σ) and a stability constant C_(ε,µ,σ,s)^? . To estimate the regularity exponent, we adapt some results of [6] to the case of complex-valued coefficients. Letξbe a coefficient defined on Ω; we assume in the current section thatξfulfills (5.1). This assumption prescribes that

ξ∈ {z=ρexp(ıθ), ρ∈[ξ₋, ξmax], θ∈[θmin, θmax]}a.e. in Ω, where ξmax:=kξkL^∞(Ω), and 0≤θmax−θmin≤2 arccos _ξ

−

ξmax

(6.1) .

In other words, since arccos(ξ₋/ξ_max)< π/2 the coefficientξtakes its values in some open, half plane in C. If the coefficients εand σ are as in section 2.1, then ξ =ε_σ takes its values in some open,quarter planein C.

We recall that e∈XDir(Ω, εσ) andµ⁻¹curle∈XN eu(Ω, µ). Hence, according to Theorems 5.5 and 5.6, we may write

e=ereg+ze+∇p0 in Ω, ereg ∈H¹(Ω), ze∈H^σDir(Ω) ; (6.2)

µ⁻¹curle=c_reg+z_c+∇q0 in Ω,c_reg ∈H¹(Ω), z_c∈H^{σN eu}(Ω).

(6.3)

In addition, it holds that

keregk1,Ω+keregkX_Dir(Ω)+kzekσ_Dir,Ω+kereg+zek1/2,Ω≤CX^DirkekH(curl;Ω),

kcregk1,Ω+kcregkX_{N eu}(Ω)+kzckσ_{N eu},Ω+kcreg+zck1/2,Ω≤CX^{N eu}kµ⁻¹curlekH(curl;Ω). It now remains to evaluate the regularity and norm of the gradient parts ∇p0 and

∇q0. Note that p0 is governed by the second-order scalar PDE (5.4) with Dirichlet boundary condition, whileq0 is governed by the second-order scalar PDE (5.7) with Neumann boundary condition. We will use this vocabulary in the following to address both cases.

6.1. Preliminary results. To start with, givenO ⊂Ω a nonempty connected open subset of R³ with Lipschitz boundary, let H⁰(O) be equal to L²(O) in the Dirichlet case, resp.,L²_zmv(O) in the Neumann case, and H¹(O) be equal to H₀¹(O) in the Dirichlet case, resp., H_zmv¹ (O) in the Neumann case. We equip H¹(O) with the normkvk_H1(O):=k∇vk_0,O.

Then, for s ∈ (0,1), we introduce H^s(O), the Sobolev space obtained by the real interpolation method between H¹(O) and H⁰(O): if needed, we distinguish the two cases by writing H_Dir^s (O), resp., H^s_{N eu}(O). In particular, by definition (cf.

section 3.1), it holds that H^s_Dir(O) = He^s(O) for all s ∈ [0,1], and we recall that H^s(O) =He^s(O) for alls∈[0,¹₂).

We denote byH^−s(O) the dual space ofH^s(O) fors∈[0,1]. Finally, fors∈[0,1], we define H^1+s(O) := {v ∈ H¹(O) s.t. ∇v ∈ H^s(O)}, equipped with the norm kvk_H1+s(O):=k∇vk_s,O.

Lemma 6.1. Given s∈[0,1], there existsC_(s)^P >0such that

∀v∈ H^1+s(O), kvk_H1+s(O)≤ kvk_1+s,O≤C_(s)^P kvk_H1+s(O).

Proof. The result is obvious fors∈ {0,1}, according to the Poincar´e inequality.

We let nows∈(0,1).

For the left inequality, notice that

∀v∈H¹(O), k∇vk0,Ω≤ kvk1,Ω; ∀v∈H²(O), k∇vk1,Ω≤ kvk2,Ω. As a consequence, the left inequality follows. This is the so-called exact sequence property. Following Appendix A, if we letv∈ H^1+s(O),

kvk_H1+s(O):=k∇vk_s,O

= kt^−s inf

∇v=v0+v1

v0∈L²(O),v1∈H¹(O)

kv0k²_0,O+t²kv1k²_1,O1/2

k_L2(0,∞;^dt_t)

≤ kt^−s inf

v=v0+v1

v0∈H¹(O), v1∈H²(O)

k∇v0k²_0,O+t²k∇v1k²_1,O1/2

k_L2(0,∞;^dt_t)

(cf. above) ≤ kt^−s inf

v=v0+v1

v₀∈H¹(O), v1∈H²(O)

kv0k²_1,O+t²kv1k²_2,O^1/2

k_L2(0,∞;^dt_t)

=:kvk1+s,O.

For the right inequality, let us introduce the Poincar´e constant:

C:= sup

v∈H¹(O))\{0}

kvk0,O

k∇vk_0,O.

Using the equivalence of norms, the definition ofk · k_H1+s(O), the Poincar´e inequality, and finally the equivalence of norms again, we find

kvk1+s,O≤M_(1+s)kvk_H^1+s_(O):=M_(1+s)

kvk²_1,O+|∇v|²_Hs(O)

^1/2

≤M_(1+s)

1 +C²

k∇vk²_0,O+|∇v|²_Hs(O)

^1/2

≤M_(1+s) 1 +C^21/2

k∇vk_Hs(O)

≤M(1+s) 1 +C^21/2

m⁻¹_(s)k∇vks,O=:M(1+s) 1 +C^21/2

m⁻¹_(s)kvk_H1+s(O).

Hence, one may chooseC_(s)^P =M_(1+s)(1 +C²)^1/2m⁻¹_(s).

If we lets∈[0,1], we want to find the a priori regularity of the solution to Find u∈ H¹(Ω) such that

(ξ∇u|∇v)_0,Ω=hf, vi_H1(Ω) ∀v∈ H¹(Ω), (6.4)

andf is some data inH^−s(Ω).

Ifξis constant on Ω, that is, if one considers the Laplace operator with Dirichlet boundary condition, or with Neumann boundary condition, then one may apply the classical results of [26] or [32] (see [6, p. 504]). See also Proposition 6.7 below. In the statement of the next theorem, the constant c^Lap(s) depends on Ω. For the sake of conciseness, we omit this dependence.

Theorem 6.2. Let ξ6= 0 be constant on Ω. Then, for alls∈[0,¹₂), there exists c(s) := c^Lap(s)>0 such that for all f ∈ H^s−1(Ω), the solution u∈ H¹(Ω) to (6.4) belongs toH^s+1(Ω), and

kuk_Hs+1(Ω)≤ c(s)

ξ_maxkfk_Hs−1(Ω).

Definition 6.3. Let the coefficient ξfulfill (5.1). We say thatξfulfills the coef- ficient assumption if there exists a partitionP ofΩsuch that ξ∈P W^1,∞(Ω).

Ifξfulfills the coefficient assumption on a partition, thenξ⁻¹fulfills the coefficient assumption on the same partition.

From now on in the current section, we consider the case whereξ6= 0 is a scalar, nonconstant, complex-valued coefficient that fulfills the coefficient assumption on a partitionP :={Ωj}j=1,...,J. In [6], the authors study the case of a symmetric-tensor, real-valued coefficientξ. There are similarities between the two cases, and also some differences, that are highlighted below. We refer to section 6.5 for a generalization to the case of a normal-tensor, complex-valued coefficientξ. Let

Λ_ξ :=|ξ|_{P W}^1,∞_(Ω) ξ_max .

By definition, it holds thatkξk_{P W}^1,∞_(Ω) = ξmax(1 + Λξ). For a piecewise constant coefficientξ, one has Λξ= 0. Otherwise, Λξ >0.

Fors∈[0,¹₂), choosingO ∈ {Ωj, 1≤j≤J}, we denote byD_j^sthe norm of the natural embedding ofH^s(Ωj) intoHf^s(Ωj):

D^s_j := sup

vj∈H^s(Ωj)\{0}

kvjk

fH^s(Ωj)

kvjks,Ωj

, 1≤j≤J; Ds:= max 1,max1≤j≤JD^s_j

≥1.

(6.5)

Remark 6.4. It holds that lim_s→1

2D^s_j = +∞, because constant, nonvanishing fields defined on Ωj belong toH¹²(Ωj) but not tofH¹²(Ωj).

Also, we denote the Poincar´e constants by C_j:= sup

v_j∈H¹₀(Ω_j)\{0}

kvjk0,Ωj

k∇vjk0,Ω_j

, 1≤j≤J; C:= max

1≤j≤JC_j>0.

(6.6)

We note that, obviously, the constants Λ_ξ, (D^s_j)_j,D_s, (C_j)_j, and C all depend on Ω and on the partitionP. These dependences are omitted.

Then, we define the multiplicative operatormξ ∈ L(L²(Ω),L²(Ω)) bymξv(x) = ξ(x)v(x), for allv∈L²(Ω), a.e. x∈Ω. One may now adapt the proof of Proposition 2.1 of [6] to the complex-valued case, to find the following.

Proposition 6.5. Letξfulfill the coefficient assumption. Then, for alls∈[0,¹₂), it holds that m_ξ∈ L(H^s(Ω),H^s(Ω)) and in addition,

kmξk_L(Hs(Ω),fH^s(Ω))≤ξmaxN_ξ^s, whereN_ξ^s:=Ds 2 1 +C²Λξ2^s/2 . (6.7)

Furthermore, for allr∈[0,¹₂), it holds that

kmξk_L(Hs(Ω),fH^s(Ω))≤ξmax(N_ξ^r)^s/r ∀s∈[0, r].

(6.8)

We then recall the technical Lemmas 3.1 and 3.2 of [6], which are independent of the coefficientξ. Introduce the operatorD∈ L(L²(Ω),H⁻¹(Ω)) defined by

hDv, qi_H1(Ω)= (v|∇q)_0,Ω∀v∈L²(Ω),∀q∈ H¹(Ω).

Proposition 6.6. For alls∈[0,1], one has

D∈ L(fH^s(Ω),H^s−1(Ω))and kDk_L(fHs(Ω),H^s−1(Ω))≤1.

(6.9)

Introduce the operatorL∈ L(H⁻¹(Ω),H¹(Ω)) defined by

(∇(Lv)|∇q)0,Ω=hv, qi_H1(Ω) ∀v∈ H⁻¹(Ω),∀q∈ H¹(Ω).

One may rephrase Theorem 6.2 (withξ= 1) as follows.

Proposition 6.7. For all r ∈ [0,¹₂), one has L ∈ L(H^r−1(Ω),H^r+1(Ω)) and there existsKr≥1 such that it holds that

kLk_L(Hr−1(Ω),H^r+1(Ω))≤Kr; (6.10)

for alls∈[0, r], kLk_L(H^s−1_(Ω),Hs+1(Ω))≤(K_r)^s/r. (6.11)

Obviously,Krdepends on Ω. This dependence is omitted.