Augmented formulations for solving Maxwell equations

Augmented formulations for solving Maxwell equations

Patrick Ciarlet Jr.

ENSTA & CNRS UMR 2706 32, Boulevard Victor, 75739 Paris Cedex 15, France Received 18 May 2004; accepted 18 May 2004

Abstract

We consider augmented variational formulations for solving the static or time-harmonic Maxwell equations. For that, a term is added to the usualH(curl) conforming formulations. It consists of a (weighted) L² scalar product between the divergence of the EM and the divergence of test fields. In this respect, the methods we present areH(curl, div) conforming. We also build mixed, augmented variational formulations, with either one or two Lagrange multipli- ers, to dualize the equation on the divergence and, when applicable, the relation on the tangential or normal trace of the field. It is proven that one can derive formulations, which are equivalent to the original static or time-harmonic Max- well equations. In the latter case, spurious modes are automatically excluded. Numerical analysis and experiments will be presented in the forthcoming paper [Augmented formulations for solving Maxwell equations: numerical analysis and experiments, in preparation].

2004 Elsevier B.V. All rights reserved.

Keywords:Maxwell equations; Augmented variational formulations; Lagrange multipliers

0. Introduction

In recent years, much attention has been devoted to the computation of electromagnetic fields in boundedsingulardomains, that is with a non-smooth and non-convex boundary. This is in particular true with numerical methods based on finite element techniques. Although the numerical computation is fairly standard in a 3d domain with either a smooth or a convex boundary, i.e. aregulardomain, a number of problems need to be addressed in the more general case. In particular, it is common knowledge that in a singular domain, there usually exist intense electromagnetic fields near thegeometrical singularities, that is reentrant corners and/or edges of the boundary.

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doi:10.1016/j.cma.2004.05.021

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For instance, with edge finite element methods[28,29], in order to reach an acceptable approximation of the solution, one way is to use suitable mesh refinement techniques near the corners and edges[30]. It is also possible to adapt the mesh and the degree of the local Finite Element, byhptechniques [20,31,32]. If one computes instead a continuous approximation of the field based on theP1Lagrange FEM (see[7]and Refs.

therein), then one can prove that this method, which works well in a regular domain, fails to capture the electromagnetic field even in a 2d singular domain[6,10,5,24,2,3]. It is well-known that this corresponds to a density problem, i.e. that of the subspace ofH¹-smooth fields in the space of EM fields (see for instance [10]). In this case, it is advised in the above mentioned Refs to introduce additional basis functions to the usual FE basis functions: the resulting method is called the singular function method, or the singular com- plement method.

In terms of functional spaces, edge FEM are H(curl)-conforming, whereas the P1Lagrange FEM is H(curl, div)-conforming. In this paper, we study H(curl, div)-conforming methods for solving Maxwell equations. We investigate three different approaches, from a theoretical point of view:

• With boundary conditions treated asessential boundary conditions. This means that they are explicitly included in the definition of the functional spaces. This approach corresponds more or less to the clas- sical one[7].

• With boundary conditions treated asnatural boundary conditions, in order to overcome the density prob- lem mentioned above. This amounts to solving Maxwell equations in larger functional spaces, in which the subspace ofH¹-smooth fields is dense[14,16].

• In weighted Sobolev spaces. By introducing suitableweights, which depend on the distance to the singu- lar edges, one can prove a second density result in another class of larger functional spaces[17].

Therefore, from a numerical point of view, the last two approaches do not require a singular comple- ment, which leads to a more straightforward implementation.

We are interested in solving static-like Maxwell equations, and the time-harmonicMaxwell equations.

Also, we consideraugmented, andaugmented and mixedVariational Formulations to solve those equations theoretically. The (augmented and) mixed variational formulations, which are introduced here, yield an effi- cient framework to handle thetime-dependentMaxwell equations. Numerical analysis, implementation is- sues and numerical examples will be dealt with in a forthcoming paper [15]. We usually provide detailed proofs for the electric field. For the magnetic field, we only give the proofs, or parts of proof, which can- not be easily inferred from those corresponding to the electric field, or when they lead to different conclusions.

The outline of this paper is as follows. In the next section, we derive the static and time-harmonic mod- els, and we detail the mathematical framework. Then, in Section 2, we validate augmented, and augmented and mixed variational formulations with the boundary conditions treated as essential. This section is split into two subsections: we consider first the solution of the static model, and then the solution of the time- harmonic model. In Section 3, we follow the same framework, with boundary conditions handled as natural boundary conditions. Then, in Section 4, we solve those models with the electric field in a weighted Sobolev space. Finally, we give a few concluding remarks.

Note that, as far asH(curl)-conforming methods are concerned, mixed formulations have already been introduced to solve the static and time-harmonic models (see for instance[25,1]).

1. Derivation of the models and mathematical framework

If we letc,e0andl0be respectively the light velocity, the dielectric permittivity and the magnetic per- meability (e0l0c²= 1), Maxwell equations in vacuum read

e0

oE

ot curlH¼ J; ð1Þ

l₀oH

ot þcurlE¼0; ð2Þ

divðe0EÞ ¼q; ð3Þ

divðl₀HÞ ¼0; ð4Þ

whereEandHare the electric and magnetic fields,qandJthe charge and current densities. These quan- tities depend on the space variablex and on the time variablet.

For the moment, we assume that we consider equations(1)–(4)outside a bounded, open perfect conduc- torO, with a Lipschitz polyhedral boundaryoO. These equations are then supplemented with the boundary condition

En_O ¼0 on oO ð5Þ

withn_O a unit outward normal tooO. Note that(2) and (5)imply l₀o

otðHn_OÞ ¼0 on oO: ð6Þ

The charge conservation equation is a consequence of equations(1) and (3) oq

ot þdivJ¼0: ð7Þ

Last, initial conditions are provided (for instance at timet= 0)

Eð;0Þ ¼E0; ð8Þ

Hð;0Þ ¼H0; ð9Þ

where the coupleðE0;H0Þdepends only on the variablex. It is convenient to assume thatl₀H0n_OjoO¼0, so that(6)is equivalent to

l₀Hn_O ¼0 onoO: ð10Þ

Then, wetruncatethe domain, to define aboundedcomputational domain. We thus introduce abounded, opensubset ofR³nO, calledX, with aLipschitz polyhedralboundaryoX. For convenience, we further as- sume that the domainXissimply connected, and that its boundaryoXisconnected. The goal is to solve Maxwell equations in this domainX.

The boundaryoXis made up of two parts:CCandCA, withCC¼oX\oOtheperfect conductor bound- ary, andCAanartificial boundary. OnCC, conditions(5) and (10)hold, withnO replaced byn, the unit out- ward normal tooX. We further split the artificial boundaryCAintoCⁱ_AandC^a_A. OnCⁱ_A, we model incoming plane waves, whereas we impose onC^a_Aan absorbing boundary condition. Both conditions can be modelled [7]as aSilver–Mu¨llerboundary condition on CA, which reads

E ffiffiffiffiffi l₀ e0

r

Hn

n¼~e^Hn on CA: ð11Þ

By definition, one has~e^H_jCa

A¼0, whereas~e^H_jCi

Ais linked to the incoming plane waves. Where~e^H¼0, condition (11)is actually a first order absorbing boundary condition.Without loss of generality, it is possible

• to choose the artificial boundaryCAsuch that it does not intersect any of thegeometrical singularitiesof oO, i.e. there exists a neighborhoodVof the reentrant corners and/or edges such thatV\CA ¼ ;;

• to splitCAinto twosmoothsubsetsCⁱ_A andC^a_A;

• to assume that the incoming wave is smooth.

A study of the existence of the time-dependent EM field, with perfect conductor and absorbing boundary conditions (Cⁱ_A ¼ ;), has been carried out in[8]. It can be generalized to our case (~e^H6¼0,CA\CC6¼ ;) with no difficulty.

Remark 1. Note that¹, within the H(curl, div) framework, the EM field belongs to H(curl)\H(div), outside ofO, and it satisfies boundary conditions(5) and (10)onoO. Under the above assumption thatC_A contains no geometrical singularities, there exists a neighborhood WA of CA, such that, with W¼WA\ ðR³nOÞ, the field actually belongs to H¹ðWÞ H¹ðWÞ(cf.[1, Theorems 2.9 and 2.12]).

The above equations, considered inXand on its boundaryoX, will be referred to as thetime-dependent Maxwell equations.

As mentioned in the Introduction, the static modelis build in order to provide a sound mathematical framework to thetime-dependent Maxwell equations. So, let us detail the steps, which lead to its definition.

We give below a formal presentation forE, which can be justified mathematically, see[9,8]. It is well-known that one can replace Ampe`res law (1) by a second-order Maxwell equation and an additional initial condition, which read

o²E

ot² þc²curl curlE¼ 1 e0

oJ

ot ; ð12Þ

oE ot ð0Þ ¼ 1

e0

ðcurlH0Jð0ÞÞ: ð13Þ

Now, let us consider a test field in

TE:¼ fv2Hðcurl;XÞ:vn_joX2L²ðoXÞ; vn_jCC¼0g:

There holds, by integration by parts d²

dt²ðE;vÞ₀þc²ðcurlE;curl vÞ₀c²ðcurlEn;v_TÞ_0;oX¼ 1 e0

d

dtðJ;vÞ₀;

wherev_Tstands for the tangential trace components, i.e.v_T=n·(v·n)_joX. But, according to Faradays law (2) and the boundary condition(11)on CA, one finds

c²curlEn_jCA¼co

ot~e^H_T co otET; so that one gets

findE2TE

s:t: d²

dt²ðE;vÞ₀þc²ðcurlE;curl vÞ₀þcd

dtðET;v_TÞ_0;CA

¼ 1 e0

d

dtðJ;vÞ₀þcd

dtð~e^H_T;v_TÞ_0;CA 8v2TE: ð14Þ

1 Sobolev spaces of vector-valued fields are written in boldface.

This suggests strongly that one defines the followingstatic modelfor the electric field: look for a static- like fieldE, solution to

curlE¼f_E in X; ð15Þ

divE¼g_E in X; ð16Þ

En¼~en on oX: ð17Þ

Its a priori regularity is (without forgetting theH(div)-conforming requirement),

E2TE; divE2L²ðXÞ ð18Þ

with the corresponding assumptions on the data

f_E2L²ðXÞ; divf_E¼0; g_E2L²ðXÞ; ~e2H¹⁼²ðoXÞ: ð19Þ NB. The extra regularity on~eis a consequence of Remark 1.

And, finally,fand~eneed to fufill a compatibility condition, which reads, according to[11, p. 23]

fEn_joX¼divCð~enÞ: ð20Þ

One can proceed similarly forH: replace Faradays law by asecond-orderMaxwell equation, and addi- tional boundary and initial conditions, which read

o²H

ot² þc²curlðcurlHJÞ ¼0; ð21Þ

1 e0

ðcurlHJÞ n_jCC ¼0; ð22Þ

oH

ot ð0Þ ¼ 1

l₀curlE0: ð23Þ

Now, let us consider a test field in

TH:¼ fv2Hðcurl;XÞ:vn_joX2L²ðoXÞg:

There holds, by integration by parts d²

dt²ðH;vÞ₀þc²ðcurlHJ;curl vÞ₀c²ððcurlHJÞ n;v_TÞ_0;oX¼0:

According to Ampe`res law and the boundary condition(11)onCA, one finds c²ðcurlHJÞ n_jCA¼ 1

l₀ o

otð~e^HnÞ co otHT; so that one gets

findH2TH

s:t: d²

dt²ðH;vÞ₀þc²ðcurlH;curl vÞ₀þcd

dtðHT;v_TÞ_0;CA

¼c²ðJ;curl vÞ₀þ 1 l₀

d

dtð~e^Hn;v_TÞ_0;CA 8v2TH: ð24Þ

This suggests that one solves amixedproblem, imposingHn onCCand Hn onCA. This is actually what one has to do to solve the time-dependent Maxwell equations. However, we are mainly interested here

in resolving problems related to geometrical singularities and the corresponding (numerical) approximation of intense fields. Thanks to Remark 1, the magnetic field isH¹-smooth in a neighborhood of CA, which means it can only be intense in a neighborhood of, and on,CC. This is the reason why we impose asingle boundary condition on the whole ofoX, the one onCC. We thus obtain the followingstatic modelfor the magnetic field: look for a static-like fieldH, solution to

curlH¼f_H inX; ð25Þ

divH¼g_H in X; ð26Þ

Hn¼h on oX: ð27Þ

Its a priori regularity is

H2TH; divH2L²ðXÞ ð28Þ

with the corresponding assumptions on the data

fH2L²ðXÞ; divfH¼0; g_H2L²ðXÞ; h2L²ðoXÞ ð29Þ

and a compatibility condition, which reads

ðg_H;1Þ₀¼ ðh;1Þ_0;oX: ð30Þ

Remark 2. Notice that bothstatic modelsinclude the usual electrostatic and magnetostatic equations.

To introduce the time-harmonic Maxwell equations, let us consider the case of a resonator cavity X, bounded by a perfect conductor. As before,Xis a bounded and simply connected open subset ofR³, with a connected Lipschitz polyhedral boundary oX. The goal is to solve a source problem or to model eigen- modes of electromagnetic oscillations. In these cases, the solutions to and data of system (1)–(4)are har- monic functions of time. Given x2R, one has relations such as Eðx;tÞ ¼ReðeðxÞexpıxtÞ and Hðx;tÞ ¼ReðhðxÞexpıxtÞ, where the fieldseandhbelong toC³. The same holds for the current density ðj2C³Þand the charge densityðr2CÞ. The equations are

ıxe0ecurl h¼ j inX; ð31Þ

ıxl₀hþcurl e¼0 inX; ð32Þ

divðe0eÞ ¼r in X; ð33Þ

divðl0hÞ ¼0 inX; ð34Þ

en¼0 on oX; ð35Þ

ðl₀hÞ n¼0 on oX: ð36Þ

The charge conservation reads

ıxrþdivj¼0: ð37Þ

It is fairly straightforward to replace(31) and (32)by

x²e0ecurlðl¹₀ ðcurl eÞÞ ¼ıxj inX; ð38Þ

x²l₀hcurlðe¹₀ ðcurl hjÞÞ ¼0 in X; ð39Þ

e¹₀ ðcurl hjÞ n¼0 on oX: ð40Þ

NB. The problems ineandharecoupled, through the data j.

In addition, if one solves find/_e 2H¹₀ðXÞ

s:t: divðe0r/_eÞ ¼r; ð41Þ

one can replacejbyjıx e0$/e,rby 0, andebye+$/e.

To summarize, in the time-harmonic case, we solve a source problem ine⁰

x²e0e⁰curlðl¹₀ ðcurl e⁰ÞÞ ¼j⁰ inX; ð42Þ

divðe0e⁰Þ ¼0 in X; ð43Þ

e⁰n¼0 onoX; ð44Þ

where we setj⁰= ıxj+x²e0$/eande⁰=e+$/e. We can also solve a source problem inh

x²l₀hcurlðe¹₀ ðcurl hjÞÞ ¼0 in X; ð45Þ

divðl₀hÞ ¼0 in X; ð46Þ

ðl₀hÞ n¼0 on oX; ð47Þ

e¹₀ ðcurl hjÞ n¼0 on oX: ð48Þ

We have some a priori regularities on the solution and on the data

e⁰2Hðcurl;XÞ; h2Hðcurl;XÞ; ð49Þ

j⁰2L²ðXÞ; divj⁰¼0; j2L²ðXÞ; divj¼0: ð50Þ

Following for instance[25], we note that Eqs.(31)–(34) and (37)naturally split into decoupled problems (real and imaginary parts): one inðRðeÞ;IðhÞ;IðjÞ;RðrÞÞ, and the other inðIðeÞ;RðhÞ;RðjÞ;IðrÞÞ. Further- more, the boundary conditions(35) and (36)do not yield any coupling between those two quadruples. For these reasons, we shall consider from now onrealvalued data and electromagnetic fields.

In the following Sections, we sete0,l0(andc²) to one.

We now introduce the mathematical framework. First, let us define five Sobolev spaces:

L²_tðoXÞ ¼ fz:z2L²ðoXÞ; zn¼0 a:e:g;

XE ¼ fv:v2Hðcurl;XÞ \Hðdiv;XÞ; vn_joX2L²_tðoXÞg;

X⁰_E ¼ fv:v2XE; vn_joX¼0g;

XH¼ fv:v2Hðcurl;XÞ \Hðdiv;XÞ; vn_joX2L²ðoXÞg;

X⁰_H¼ fv:v2XH; vn_joX¼0g:

Thanks to our assumptions on the data(19) and (29), the static-like and time-harmonic electromagnetic fieldsa priori regularities are

ðE;HÞ 2XEXH; ðe;hÞ 2X⁰_EX⁰_H: ð51Þ

Recall that the domainXis a bounded and simply connected open subset ofR³, with a connected Lips- chitz polyhedral boundaryoX. Then, according to[33,21,1], the following results hold

Theorem 3. InX⁰_EandX⁰_H,

kvk_X⁰¼ fkcurl vk²₀þ kdivvk²₀g¹⁼²

is a norm, which is equivalent to the fullH(curl, div,X) norm.

InXE,

kvk_XE¼ fkcurl vk²₀þ kdivvk²₀þ kvnk²_0;oXg¹⁼²

is a norm, which is equivalent to the fullH(curl, div,X) norm, plusL²(oX) norm of the tangential trace.

InXH,

kvk_XH ¼ fkcurl vk²₀þ kdivvk²₀þ kvnk²_0;oXg¹⁼²

is a norm, which is equivalent to the fullH(curl, div,X) norm, plus L²(oX) norm of the normal trace.

Note thatkv·nk0,oXcan be replaced bykvTk0,oXinkvkX_E.

We denote by (Æ,Æ)X⁰, (Æ,Æ)X_E, and (Æ,Æ)X_H the associated scalar products.

Remark 4. If the boundary is not connected, there appears an additional term (see for instance[19]) in the norm in X⁰_E: from an electrostatic point of view, it corresponds to the constant value of the electrostatic potential, at the surface of each perfect conductor. When the domain is not simply connected, there appears an additional term in the norm inX⁰_H(see[1,22]). For the additional terms in the norms ofXEandXH, we refer the reader to[21].

2. Variational formulations with essential boundary conditions

In this section, we focus on Variational Formulations, with spaces including the boundary condition.

More precisely, the field is explicitly split into two parts: the first one corresponds to the incoming part (cf. the comment after formula(11)), so it is known, and the second one belongs toX⁰_E;H. In the PDE vocab- ulary, this means that the boundary conditions are treated asessential boundary conditions. This framework was introduced in[7].

2.1. The static equations

Since~ebelongs toH^1/2(oX), there exists~ein H¹(X)³such that~e_joX¼~e. We then definef⁰_E¼f_Ecurl~e andg⁰_E¼g_Ediv~e. We thus look forE⁰¼E~e, which satisfies

E⁰2X⁰_E; curlE⁰¼f⁰_E; divE⁰¼g⁰_E: ð52Þ

In order to illustrate our framework, based onvariational formulations, we introduce our first problem (P⁰_E)

findE⁰2X⁰_E

s:t: ðE⁰;vÞ_X⁰¼ ðf⁰_E;curl vÞ₀þ ðg⁰_E;divvÞ₀ 8v2X⁰_E: ð53Þ With respect to(14), the variational formulation (53) is called an augmentedvariational formulation (or AVF later on), since the (divÆ, divÆ)0product appears in addition to (curlÆ,curlÆ)0. According to the Riesz

theorem, it is clear that there exists one, and only one, solution to problem (P⁰_E), which is continuous with respect to the dataðf⁰_E;g⁰_EÞinL²(X)·L²(X).

There holds

Theorem 5. The fieldE⁰ satisfies(52)iff it is a solution to problem (P⁰_E).

Proof. It is clear that ifE⁰is a solution to (52), it also solves(53).

Let us consider the reciprocal assertion: letE⁰be the solution to(53). Giveng2L²(X),9!/2H¹₀ðXÞs.t.

D/=g. Asv¼ r/2X⁰_E, there holdsðdivE⁰;gÞ₀¼ ðg⁰_E;gÞ₀; divE⁰¼g⁰_Efollows.

By construction, divf⁰_E¼0, and, moreover, thanks to(20) f⁰_En_joX¼fEn_joXcurl~en_joX¼fEn_joXdivCð~enÞ ¼0:

According to[23, Theorem 3.6, p. 48],9!w⁰2X⁰_Es.t. divw⁰= 0, andcurl w⁰¼f⁰_E. Asv¼E⁰w⁰belongs to X⁰_E,(53)yieldskcurlðE⁰w⁰Þk²₀¼0, orcurlE⁰¼f⁰_E. h

This allows to prove

Theorem 6. There exists one, and only one, solutionEto (15)–(17)inXE. Proof. Taking E¼E⁰þ~eyields existence.

Uniqueness follows from the fact that ifEandE⁰are two solutions to(15)–(17), their difference satisfies (53)with vanishing data, i.e. it also vanishes. h

Evidently, the solutionEdepends continuously on the data (fE,gE,e) inL²(X)·L²(X)·H^1/2(oX).

For the problem(25)–(27)in H, one uses the same procedure. Indeed, one solves findw2H¹ðXÞ \L²₀ðXÞ

s:t: Dw¼ 1

jXjðg_H;1Þ₀; ow onjoX¼h:

NB. It is a Neumann problem, and the data fulfills the usual compatibility condition, thanks to(30).

Then,~h¼gradwis an admissible lifting, together withf⁰_H¼f_H and g⁰_H¼g_H ðg_H;1Þ₀=jXj. We now look forH⁰¼H~h, solution to

H⁰2X⁰_H; curlH⁰¼f⁰_H; divH⁰¼g⁰_H: ð54Þ We then define the problem (P⁰_H)

findH⁰2X⁰_H

s:t: ðH⁰;vÞ_X⁰¼ ðf⁰_H;curl vÞ₀þ ðg⁰_H;divvÞ₀ 8v2X⁰_H; ð55Þ our second so-called AVF (augmented w.r.t.(24)this time). According to the Riesz theorem, this formu- lation admits one, and only one, solution, continuous w.r.t.f⁰_Handg⁰_H. There follows:

Theorem 7. The fieldH⁰ satisfies(54)iff it is a solution to problem (P⁰_H).

Proof. IfH⁰ is a solution to(54), it also solves (55).

Now, letH⁰be the solution to(55). Giveng2L²₀ðXÞ; 9!/2H¹ðXÞ \L²₀ðXÞs.t.D/=gandon/_joX= 0.

As v¼ r/2X⁰_H, there holdsðdivH⁰;gÞ₀¼ ðg⁰_H;gÞ₀. Since divH⁰g⁰_H belongs to L²₀ðXÞ, divH⁰¼g⁰_H follows.

By construction, divf⁰_H¼0, so, according to[23, Theorem 3.5 p. 47],9w⁰2X⁰_Hs.t.curl w⁰¼f⁰_H. Putting v¼H⁰w⁰ in(53)yieldskcurlH⁰f⁰_Hk²₀¼0. h

And, with a proof similar to the one given in the electric case,

Theorem 8. There exists one, and only one, solution Hto (25)–(27)inXH.

Now that existence, uniqueness (and continuous dependence w.r.t. the data) is proven for the static-like fields, let us investigate a dualization of the divergence of the fields. In other words, what happens if one considers(16) and (26)asconstraints? Note that, in order to solve the time-dependent Maxwell equations, it is important to enforce those relationships, so that one avoids a drift as one iterates in time, if for instance the charge conservation equation is not enforced numerically. This is the approach originally investigated by Assous et al. in [7].

So, let us define a new problem (Q⁰_E) findðE⁰;pÞ 2X⁰_EL²ðXÞ

s:t: ðE⁰;vÞ_X0þ ðp;divvÞ₀¼ ðf⁰_E;curl vÞ₀þ ðg⁰_E;divvÞ₀ 8v2X⁰_E; ð56Þ ðdivE⁰;qÞ₀¼ ðg⁰_E;qÞ₀ 8q2L²ðXÞ: ð57Þ This is our firstmixed,augmentedvariational formulation (or MAVF later on). According to Proposi- tion 3.5 of[4], there holds divX⁰_E¼L²ðXÞ. So, anecessarycondition for (Q⁰_E) to yield the expectedE⁰(i.e.

the solution to (P⁰_E) and(52)), is thatp vanishes.

Proposition 9. In (Q⁰_E), one has p = 0.

Proof. 9!/H2H¹₀ðXÞs.t.D/H=p. TakingvH=$/Hin (56)yields ðdivE⁰;pÞ₀þ kpk²₀¼ ðg⁰_E;pÞ₀:

SinceðdivE⁰;pÞ₀¼ ðg⁰_E;pÞ₀according to(57), one hasp= 0. h Quoting[20], one calls sometimes pthedummy variable.

Remark 10. Notice that one reaches a similar conclusion (in the continuous case), provided the charge conservation equation is true, when one solves the time-dependent Maxwell equations with an MAVF. The well-posedness of such MAVFs is alluded to in Section 5 and AppendixA.

Theorem 11. Problem (Q⁰_E) admits one, and only one, solutionðE⁰;pÞ. Moreover,E⁰is the solution to problem (P⁰_E).

Proof. The existence and uniqueness of the solution to problem (Q⁰_E) stem from theBabuska–Brezzi theory (cf. for instance[23]). The so calledV-ellipticity condition is automatic, since the bilinear form involved is the scalar product of X⁰_E. The inf–sup condition (cf. Appendix A) is proved as follows: "q2L²(X), 9!/2H¹₀ðXÞ s.t. D/=q. As v=$/ belongs to X⁰_E with kvkX⁰=kqk0, there holds ðdivv;qÞ₀=kvk_X0¼ kqk₀, so theinf–supcondition follows with a unit constantb. This proves the first point.

To conclude, it is enough to note that problem (Q⁰_E) reduces to problem (P⁰_E), since p= 0. h NB. As emphasized in the proof, (Q⁰_E) satisfies aninf–supcondition.

Similarly, we introduce an MAVF forH⁰, i.e. a problem (Q⁰_H), findðH⁰;pÞ 2X⁰_HL²₀ðXÞ

s:t: ðH⁰;vÞ_X⁰þ ðp;divvÞ₀¼ ðf⁰_H;curl vÞ₀þ ðg⁰_H;divvÞ₀ 8v2X⁰_H; ð58Þ ðdivH⁰;qÞ₀¼ ðg⁰_H;qÞ₀ 8q2L²₀ðXÞ: ð59Þ

We know from [23, Corollary 2.4, p. 24] that divH¹₀ðXÞ ¼L²₀ðXÞ. Since H¹₀ðXÞ X⁰_H and since (divv, 1)0= 0 for allvin X⁰_H, we derive divX⁰_H¼L²₀ðXÞ. Again, anecessary condition for (Q⁰_H) to provide the same solution as (P⁰_H) and(54)is thatpvanishes. Following the proof of Proposition 9 (with a Neu- mann problem in/H).

Proposition 12. In (Q⁰_H), one has p = 0.

Finally, following the proof of Theorem 11, one easily proves

Theorem 13. Problem (Q⁰_H) admits one, and only one, solution ðH⁰;pÞ. Moreover, H⁰ is the solution to problem (P⁰_H).

NB. Problem (Q⁰_H) satisfies aninf–supcondition.

So far, we defined one augmented variational formulation, and one mixed augmented variational formu- lation, for each of the static-like fieldsEandH. More precisely, for the part which belongs toX⁰_EorX⁰_H, i.e.E⁰andH⁰. We shall see, in the next subsection, that one can try successfully the same mathematical approach, in order to solve the time-harmonic Maxwell equations.

2.2. The time-harmonic equations

In this subsection, we concentrate first on solving the system of equations(42)–(44) ine⁰2H(curl,X), givenj⁰ and x. Then, we derive equivalent variational formulations, set in X⁰_E for the electric field. Let us introduce

V⁰_E:¼ fv2H0ðcurl;XÞ:divv¼0g:

(The setV⁰_E is either a subspace ofH₀(curl,X) or ofX⁰_E.)

One can prove easily that the two formulations below are equivalent to the problem(42)–(44). The obvi- ous one,

finde⁰2H₀ðcurl;XÞ

s:t: ðcurl e⁰;curl vÞ₀¼x²ðe⁰;vÞ₀ ðj⁰;vÞ₀ 8v2H0ðcurl;XÞ; ð60Þ

dive⁰¼0: ð61Þ

One can also choose to solve an AVF, such as finde⁰2V⁰_E

s:t: ðe⁰;vÞ_X0¼x²ðe⁰;vÞ₀ ðj⁰;vÞ₀ 8v2V⁰_E: ð62Þ The reason why a solutione⁰2V⁰_E to(62)solves(60)is simple (why it solves(61)is obvious). Indeed, givenvinH₀(curl,X), one considers first/2H¹₀ðXÞsuch thatD/= divv. Thenv$/belongs toV⁰_E, and used as a test function in(62), it yields (60)forv, since

ðx²e⁰j⁰;r/Þ₀¼ ðx²dive⁰divj⁰;/Þ₀¼0:

The difference between the two formulations is thatX⁰_E––and so V⁰_E as a subset ofX⁰_E––is compactly imbedded intoL²(X)[33], whereas H₀(curl,X) is not. It is therefore possible to use the Fredholm theory to solve problem (62). Indeed, let us defineg_E:L²ðXÞ !V⁰_E by (gEf,v)X⁰= (f,v)0 for allv2V⁰_E, and iE

the compact imbedding ofV⁰_EintoL²(X). By construction,TE¼g_Ei_Eis a (symmetric) compact operator ofV⁰_E, and(62)now amounts to

ðIx²TEÞe⁰¼ gEj⁰ inV⁰_E:

There follows:

Theorem 14. Assume 1/x²is not an eigenvalue ofTE: problem (42)–(44)admits one, and only one, solution.

Assume 1/x²is an eigenvalue of TE: letv₁,. . .,v_pdenote a basis ofkerðIx²TEÞ, then

• either$l2{1,. . ., p} st (j,v_l)050: problem(42)–(44) admits no solution;

• or "l2{1,. . ., p}, (j,v_l)0= 0: problem (42)–(44) admits an affine space of solutions, in the form e⁰¼e⁰₀þP

lb_lv_l,ðb_lÞ_l2R^p.

Remark 15. The orthogonality condition of the Fredholm theory says that (gEj⁰,v_l)X⁰= 0. Or, according to the definition of operatorgE, (j⁰,v_l)0= 0. Now, since v2V⁰_E and/_e2H¹₀ðXÞ, there follows automatically ($/e,v_l)0= 0 (see (41)), so that (j⁰,v_l)0= 0 corresponds to (j,v_l)0= 0 (it is indeed equivalent, since the orthogonality condition is relevant only when x50).

Now that problem(42)–(44)has been solved, we turn to the derivation of equivalent formulations. Let us define our starting point. As a matter of fact, one can consider, in-between (60)–(62), the equivalent formulation, called problemðp⁰_EÞ

finde⁰2X⁰_E

s:t: ðe⁰;vÞ_X⁰¼x²ðe⁰;vÞ₀ ðj⁰;vÞ₀; 8v2X⁰_E; ð63Þ

dive⁰¼0: ð64Þ

Then, we turn to an MAVF, with(64)handled as a constraint, as in the static case: problemðq⁰_EÞ, finde⁰;p2X⁰_EL²ðXÞ

s:t: ðe⁰;vÞ_X⁰þ ðp;divvÞ₀¼x²ðe⁰;vÞ₀ ðj⁰;vÞ₀ 8v2X⁰_E; ð65Þ

ðdive⁰;qÞ₀¼0 8q2L²ðXÞ: ð66Þ

NB. A similar, well-known, mixed VF, set inH0ðcurl;XÞ L²ðXÞ R(withj⁰= 0), had already been con- sidered to solve the related eigenvalue problem by Kikuchi[25].

There holds

Proposition 16. In (q⁰_E), one has p = 0.

Proof. 9!/H2H¹₀ðXÞs.t.D/H=p. TakingvH=$/H in(65)yields ðdive⁰;pÞ₀þ kpk²₀¼0:

Since (dive⁰,p)0= 0 according to(66), one hasp= 0. h

According to Theorem 55, thanks to theinf–supcondition proved for problemðQ⁰_EÞ, one gets finally the Theorem 17. (e⁰, 0) is a solution to problem (q⁰_E) iffe⁰is a solution to (42)–(44).

Second, let us consider the system(45)–(48)in h2H(curl,X), givenjandx. We introduce V⁰_H:¼ fv2Hðcurl;XÞ:divv¼0; vn_joX¼0g:

As in the electric case, it is easily inferred that the two formulations below are equivalent to the original problem (45)–(48)

findh2Hðcurl;XÞ

s:t: ðcurl h;curl vÞ₀¼x²ðh;vÞ₀þ ðj;curl vÞ₀ 8v2Hðcurl;XÞ; ð67Þ

divh¼0: ð68Þ

Or, one can choose to solve equivalently findh2V⁰_H

s:t: ðh;vÞ_X⁰¼x²ðh;vÞ₀þ ðj;curl vÞ₀ 8v2V⁰_H: ð69Þ Since the imbedding ofX⁰_H intoL²(X) is compact[33], one can utilize the Fredholm theory again. Let TH¼g_Hi_H, with obvious notations, be the (symmetric) compact operator of V⁰_H into itself. With j_H2V⁰_Hsuch that (jH,v)X⁰= (j,curl v)0for allv2V⁰_H,(69)is equivalent to

ðIx²THÞh¼j_H in V⁰_H: There follows:

Theorem 18. Assume 1/x²is not an eigenvalue ofTH: problem(45)–(48)admits one, and only one, solution.

Assume 1/x²is an eigenvalue ofTH: letw₁,. . .,w_pdenote a basis ofkerðIx²THÞ, then

• either $l2{1,. . ., p}s.t. (j,curl wl)050: problem(45)–(48)admits no solution;

• or "l2{1,. . ., p}, (j,curl wl)0= 0: problem (45)–(48) admits an affine space of solutions, in the form h¼h0þP

lb_lwl,ðb_lÞ_l2R^p.

Remark 19. It is well known that the operatorsTE andTHhave the same eigenvalues (with the same mul- tiplicity). Indeed, one can actually prove thatv_lis an eigenvector ofTE with eigenvalue kif, and only if, w_l=curlv_l is an eigenvector of TH with eigenvalue k. Finally, the orthogonality conditions are identical for both the electric and the magnetic problems, so that the Fredholm classifications of the solutions in eandh are compatible, as expected.

As far as equivalent formulations of(45)–(48)are concerned, let us begin by the following formulation, called problemðp⁰_HÞ,

findh2X⁰_H

s:t: ðh;vÞ_X⁰¼x²ðh;vÞ₀þ ðj;curl vÞ₀ 8v2X⁰_H; ð70Þ

divh¼0: ð71Þ

With(71)handled as a constraint, we define further the MAVFðq⁰_HÞ, findðh;pÞ 2X⁰_HL²₀ðXÞ

s:t: ðh;vÞ_X⁰þ ðp;divvÞ₀¼x²ðh;vÞ₀þ ðj;curl vÞ₀ 8v2X⁰_H; ð72Þ

ðdivh;qÞ₀¼0 8q2L²₀ðXÞ: ð73Þ

There holds

Proposition 20. In (q⁰_H), one has p = 0.

According to Theorem 55 (inf–supvalid for problemðQ⁰_HÞ), one gets the Theorem 21. (h, 0) is a solution to problem (q⁰_H) iffh is a solution to(45)–(48).

3. Variational formulations with natural boundary conditions

In this section, we focus on Variational Formulations, with boundary conditions treated as natural boundary conditions, i.e. which stem from the formulation. The obvious advantage is computational, since,

according to the density of smooth fields inXE andXH[14,16], one can (theoretically) use the continuous approximation of the field based on theP1Lagrange FEM. Still, one has to validate those formulations, i.e.

prove that they are well-posed! This is the topic we address in this section.

3.1. The static equations

As a starting point, we consider the AVFs and MAVFs of Section 2.1. The first difference with these Variational Formulations is that we now treat the boundary conditions as natural. This requires that one includes those in a variational form. Consequently, and this is the second difference, we look directly for thetotalfieldEorH, i.e. we solve the variational problems inXE andXH. Let us proceed forE: we introduce the linear form

l_EðvÞ ¼ ðfE;curl vÞ₀þ ðg_E;divvÞ₀þ ð~en;vnÞ_0;oX and define the problem (PE)

findE2XE

s:t: ðE;vÞ_XE ¼l_EðvÞ 8v2XE: ð74Þ

With respect to(14), this falls in the AVF category. We infer from the Riesz theorem, that there exists one, and only one, solution to(74), which is continuous w.r.t. the data (cf.(19)). Moreover, one can prove Theorem 22. Eis a solution to(15)–(17)iff it solves (PE).

Proof. It is clear that ifEis a solution to(15)–(17), it also solves(74).

Let us consider the reciprocal assertion: letEbe the solution to(74). Following the proof of Theorem 5, we find divE¼g_E, i.e.(16). Furthermore, according to the same proof, one can write

fE¼curl w withw¼w⁰þ~e:

As v¼Ew belongs to XE, with vn_joX¼En_joX~en, (74) yields kcurlEf_Ek²₀þ kE n_joX~enk²_0;oX¼0: both(15) and (17)hold. h

By construction, Edepends continuously on the dataðfE;g_E;~eÞinL²ðXÞ L²ðXÞ L¹⁼²_t ðoXÞ.

Remark 23. From a numerical point of view, we recall that the difference between (PE) and (P⁰_E) is that H¹(X) is alwaysdenseinXE, whereasH¹ðXÞ \X⁰_Eisnot denseinX⁰_E, whenXis not convex[14,16]. There is no need for a Singular Complement, when one uses a continuousP1FE approximation to solve (PE) (see [15]for some numerical illustrations).

For the problem inH(25)–(27), we introduce the linear form l_HðvÞ ¼ ðfH;curl vÞ₀þ ðg_H;divvÞ₀þ ðh;vnÞ_0;oX;

and consider the AVF (PH) findH2XH

s:t: ðH;vÞ_XH¼l_HðvÞ 8v2XH: ð75Þ

This is again a formulation, which admits one, and only one, solution, continuous w.r.t. the data (cf.(29)), thanks to the often cited Riesz theorem. In addition, the following theorem can be proven.

Theorem 24. His a solution to(25)–(27)iff it solves (PH).

Proof. IfH is a solution to(25)–(27), it also solves (75).

Now, letH be the solution to(75). We prove in a single shot that both divH¼g_H andHn_joX¼h hold, provided the compatibility condition(30)is enforced.

We first deal withconstants:9!/_c2H¹ðXÞ \L²₀ðXÞs.t.D/c=joXjandon/_cjoX=jXj. Sincer/_c2XH, one finds

joXj ðdivH;1Þ₀þ jXj ðHn;1Þ_0;oX¼joXj ðg_H;1Þ₀þ jXj ðh;1Þ_0;oX:

According to the identityðdivH;1Þ₀¼ ðHn;1Þ_0;oXand to the compatibility condition(30), this yields

ðdivH;1Þ₀¼ ðg_H;1Þ₀: ð76Þ

Now, we proceed with the general case. Given g2L²(X) and l2L²(oX), let ccg;l2R such that (g+c, 1)0= (l, 1)0,oX. Then, 9!/2H¹ðXÞ \L²₀ðXÞ s.t. D/= (g+c) and on/_joX=l. As r/2XH, there holds, according to(76)

ðdivH;gÞ₀þ ðHn;lÞ_0;oX¼ ðg_H;gÞ₀þ ðh;lÞ_0;oX:

Takingg¼divHg_Handl¼Hn_joXhyieldskdivHg_Hk²₀þ kHn_joXhk²_0;oX¼0, so(26) and (27)hold.

The formulation (PH) reduces toðcurlH;curl vÞ₀¼ ðfH;curl vÞ₀, for allvinXH. One concludes that(25) is true as in the proof of Theorem 7. h

One notices a difference between the two proofs given for problems (PE) and (PH). For the first one, inE, one deals first with the divergence condition, and then simultaneously with the curl and boundary condi- tions. For the second one, inH, one deals first with the divergence and boundary conditions with a stone, and then with the curl condition.

As in Section 2.1, it is possible to consider an MAVF, with the divergence condition on the field treated as a constraint. But it is not the only possibility, since one can also investigate another MAVF, withtwo constraints. ForE, one on the divergence, and one on thetangential trace. ForH, one on the divergence, and one on thenormal trace.

Remark 25. For the time-dependent Maxwell equations, the MAVF with a single constraint is not relevant. As a matter of fact, it seems unlikely that the boundary condition is resolved, when only the divergence condition is treated as a constraint. On the other hand, the MAVF with two constraints is a good candidate to resolve the time-dependent Maxwell equations. This is actually very similar in the case of the time-harmonic equations, as we shall see in Section 3.2.

We begin by the MAVF with a single constraint, on the electric fieldE. We propose to solve the problem (QE)

findðE;pÞ 2XEL²ðXÞ

s:t: ðE;vÞ_XEþ ðp;divvÞ₀¼l_EðvÞ 8v2XE; ð77Þ

ðdivE;qÞ₀¼ ðg_E;qÞ₀ 8q2L²ðXÞ: ð78Þ

Once more, as divXE ¼L²ðXÞ, anecessary condition for (QE) to yield the desired solution is that p= 0.

Actually, with proofs identical to that of Proposition 9 and Theorem 11, one finds Proposition 26. In (Q_E), one has p = 0.

Theorem 27. Problem (QE) admits one, and only one, solutionðE;pÞ. Moreover,Eis the solution to problem (PE).

Accordingly, we propose a single constraint MAVF forH, problem (QH) findðH;pÞ 2XHL²₀ðXÞ

s:t: ðH;vÞ_XHþ ðp;divvÞ₀¼lHðvÞ 8v2XH; ð79Þ

ðdivH;qÞ₀¼ ðg_H;qÞ₀ 8q2L²₀ðXÞ: ð80Þ

Since divXH¼L²ðXÞ, the usualnecessary condition isp= 0.

Proposition 28. In (QH), one has p = 0.

Theorem 29. Problem (QH) admits one, and only one, solutionðH;pÞ. Moreover,His the solution to prob- lem (PH).

NB. Both problems (QE) and (QH) satisfy aninf–supcondition.

The (M)AVFs (PE,H) and (QE,H) allow to solve the static-like problems(15)–(17)and(25)–(27). In order to deal with the time-dependent Maxwell equations, one has to consider the boundary conditions as con- straints, as mentioned before. The MAVF in E, called (RE), reads

findðE;p;~kEÞ 2XEL²ðXÞ L²_tðoXÞ

s:t: ðE;vÞ_XEþ ðp;divvÞ₀þ ð~kE;vTÞ_0;oX¼lEðvÞ 8v2XE; ð81Þ

ðdivE;qÞ₀¼ ðgE;qÞ₀ 8q2L²ðXÞ; ð82Þ

ðET; ~lÞ_0;oX¼ ð~e_T;~lÞ_0;oX 8~l2L²_tðoXÞ: ð83Þ To go back to(15)–(17), the method of proof is different than the ones we utilized up to now. Let us begin by a preliminary result

Lemma 30. The space spanned by the tangential trace of elements ofXE is K¼ f~l2L²_tðoXÞ:curlC~l2H¹⁼²ðoXÞg:

Proof. Let v2XE. Then, one has v_T 2L²_tðoXÞby definition. Also, since vbelongs toH(curl,X), one has curlCv_T2H^1/2(oX), according to[11, Theorem 3.10]. Sov_T2K.

Conversely, let~l2K. Thanks to [12, Theorem 5.4], there exists v2H(curl,X) s.t. v_T¼~l. Now, divv belongs toH¹(X), so9!/2H¹₀ðXÞs.t.D/= divv. Then, w=v$/is such that

w2Hðcurl;XÞ; divw¼0; w_T¼~l:

Since~lis an element ofL²_tðoXÞ, we conclude thatwis inXE. h In terms of potentials, note that (see[12, Theorem 3.4])

L²_tðoXÞ ¼ rCH¹ðoXÞ ^?curlCH¹ðoXÞ:

The additional condition on the tangential curl of elements ofKmeans that K¼ rCH¹ðoXÞ ^?curl_CHðoXÞ

withHðoXÞ ¼ fq2H¹ðoXÞ:DCq2H¹⁼²ðoXÞg. ThusK6¼L²_tðoXÞ. As a consequence, one has the follow- ing ‘‘negative result’’.

Corollary 31. Problem (RE) does not satisfy the inf–sup condition.

Proof. It is based on the Corollary 4.1 of Chapter I of[23]: this corollary is concerned with mixed prob- lems, for which the V-ellipticity condition is true. Applied to our case, it says that problem (RE) is well- posed, i.e. that

XEL²ðXÞ L²_tðoXÞ !X⁰_EL²ðXÞ L²_tðoXÞ;

ðE;p;~kEÞ7!ðlE;g_E;~eTÞ (

is an isomorphism, iff theinf–supcondition holds. Now, if one takes~e_T2L²_tðoXÞ nK, it is clear that prob- lem (RE) admits no solution: otherwise, (83) would imply ET¼~e_T in L²_tðoXÞ, with E2XE, a contradiction. h

However, one can still prove a density result, as below.

Proposition 32. The spaceKis dense in L²_tðoXÞ.

Proof. Lete> 0 and~l2L²_tðoXÞbe given. By definition,L²_tðoXÞis a subset ofL²(oX), andH^1/2(oX) is dense inL²(oX). So,9~l^e2H¹⁼²ðoXÞs.t.k~l^e~lk_0;oX6e. Using theH^1/2(oX)H¹(X) lifting operator, one can choosev^e2H¹(X) s.t.v^e_joX¼~l^e. Now, v^e is an element ofXE, so~l^e_T ¼v^e_T is an element ofK, and

k~l^e_T~lk_L²_tðoXÞ6k~l^e~lk_0;oX6e;

which yields the result. h This allows to prove

Theorem 33. Problem (RE) admits one, and only one, solutionðE;p;~k_EÞ. In addition,ðp;~k_EÞ ¼ ð0;0Þ, so that Eis the solution to(15)–(17).

Proof. Since the usual Babuska–Brezzi theory is not usable here, we revert to simpler means to prove the existence and uniqueness.

Existence of the solution: let E be the solution to problem (15)–(17), then ðE;0;0Þ is a solution to problem (RE).

Uniqueness of the solution: let ðE;p;~kEÞ be a solution to problem (RE) with ðlE;g_E;~eÞ ¼ ð0;0;0Þ. In particular,(83)yieldsET¼0, soE2X⁰_E. Then, with test fieldsvinX⁰_Eonly,(81) and (82)lead to(56) and (57), with zero right-hand sides:ðE;pÞsatisfies the homogeneous problem (Q⁰_E), and it is therefore equal to (0, 0). Finally,(81)now reduces to

ð~kE;v_TÞ_0;oX¼0 8v2XE: ð84Þ

This gives~kE¼0 according to the density result of Proposition 32.

The obvious by-product of the existence and uniqueness results is thatðp;~k_EÞalways vanishes. h The MAVF onH, called (RH), reads

findðH;p;kHÞ 2XHL²₀ðXÞ L²ðoXÞ

s:t: ðH;vÞ_XHþ ðp;divvÞ₀þ ðkH;vnÞ_0;oX¼l_HðvÞ 8v2XH; ð85Þ

ðdivH;qÞ₀¼ ðg_H;qÞ₀ 8q2L²₀ðXÞ; ð86Þ

ðHn;lÞ_0;oX¼ ðh;lÞ_0;oX 8l2L²ðoXÞ: ð87Þ

We fall back to the often used pattern of proof,²with theBabuska–Brezzi theory.

Theorem 34. Problem (RH) admits one, and only one, solutionðH;p;k_HÞ. In addition, (p,kH) = (0, 0), so that His the solution to(25)–(27).

2 It is possible to choose the formulation with ðH;p;kHÞ 2XHL²₀ðXÞ L²₀ðoXÞ, i.e. with two zero mean-value Lagrange multipliers.