Lecture Notes in Computational Science 59 and Engineering Editors TimothyJ.Barth MichaelGriebel DavidE.Keyes RistoM.Nieminen DirkRoose TamarSchlick Habib Ammari (Ed.) Modeling and Computations in Electromagnetics A Volume Dedicated to Jean-Claude Nédélec With74Figuresand5Tables ABC Editor HabibAmmari ÉcolePolytechnique CNRSUMR7641 CentredeMathématiquesAppliquées 91128PalaiseauCedex France email:[email protected] LibraryofCongressControlNumber:2007933491 Mathematics Subject Classification (2000): 78M05, 78M15, 83C50, 35Q60, 35R30, 65R20,65R32 ISBN978-3-540-73777-3SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2008 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheeditorsandSPiusingaSpringerLATEXmacropackage Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper SPIN:10885101 46/SPi 543210 To Jean-Claude N´ed´elec Preface Modeling and computations in electromagnetics is quite novel and growing discipline, expanding as a result of the steadily increasing demand for de- signingelectricaldevices,modelingelectromagneticmaterials,andsimulating electromagnetic fields in nanoscale structures. The aim of this volume is to bring together prominent worldwide experts to review state-of-the-art developments and future trends of modeling and computations in electromagnetics. This volume is devoted to merging the ex- pertise of scientists working on this active discipline, and to raise interest for challengingissues.Themostsignificantadvancesincomputationaltechniques developedintheverylastyearsandseveralchallengingtechnologicalapplica- tions such as those related to communications, to biomedical devices, and to magnetic storage design are presented in this volume. It covers the following topics: fast algorithms for time-dependent electromagnetic waves, high-order methods for high-frequency electromagnetic scattering, non-reflecting bound- aryconditionsfortime-dependentelectromagneticwaves,multi-scaleanalysis for Maxwell’s equations, time-reversal for electromagnetic waves, and inverse electromagnetic scattering. ThisvolumeisdedicatedtoJean-ClaudeN´ed´elecincelebrationofhis65th birthday and retirement from Ecole Polytechnique in January 2008. Jean- Claude has been one of the most distinguished scientist in the area of com- putational electromagnetics. His outstanding contributions on finite-elements and fast algorithms for Maxwell’s equations have significantly advanced the field. Paris, Habib Ammari May 2007 Contents 1 Stabilized FEM–BEM Coupling for Maxwell Transmission Problems R. Hiptmair and P. Meury ........................................ 1 2 A Posteriori Error Analysis and Adaptive Finite Element Methods for Electromagnetic and Acoustic Problems Z. Chen ........................................................ 39 3 Time Domain Adaptive Integral Method for Surface Integral Equations H. Ba˘gcı, A.E. Yılmaz, J.-M. Jin, and E. Michielssen ................ 65 4 Local and Nonlocal Nonreflecting Boundary Conditions for Electromagnetic Scattering M.J. Grote......................................................105 5 High-Order Methods for High-Frequency Scattering Applications O.P. Bruno and F. Reitich........................................129 6 Recent Studies on Inverse Medium Scattering Problems G. Bao, S. Hou, and P. Li ........................................165 7 Time Reversal of Electromagnetic Waves J. de Rosny, G. Lerosey, A. Tourin, and M. Fink ....................187 8 Addition Theorem B. He and W.C. Chew............................................203 Index..........................................................227 1 Stabilized FEM–BEM Coupling for Maxwell Transmission Problems R. Hiptmair and P. Meury 1.1 Introduction We consider the electromagnetic scattering of monochromatic incident waves from a penetrable, three-dimensional bounded object Ω R3, the scat- ⊂ terer.Inapplicationsoneusuallyencountersscattererswithpiecewisesmooth, Lipschitz continuous boundaries. Thus it is natural to assume the scatterer to be a curvilinear Lipschitz-polyhedron in the parlance of [28, Sect.1]. For the sake of simplicity, we assume that its surface Γ := ∂Ω is connected. However, with slight changes all theorems can be extended to more general situations. The material parameters ε and µ may display some spatial vari- r r ation inside Ω but assume the constant values ε > 0 and µ > 0 in the air 0 0 region Ω+ :=R3 Ω. \ Let Es denote the complex amplitude of the scattered electric field in the air region and E the total electric field inside the scatterer Ω, which emerge as solutions to the Maxwell transmission problem (cf. [49, Sect.5.6.3]) curlµ (x)−1curlE κ2ε (x)E = F(x) in Ω, r r − curlcurlEs κ2Es = 0 in Ω+, − (1.1) γ+Es γ−E = g on Γ, γ+Es µ−1γ−E = g on Γ , t − t D N − r N N lim curlE x iκxE = 0. |x|→∞ × − | | Here, κ := ω√µ0ε0L (with ω > 0 the fixed angular frequency of the exci- tation, L the characteristic length of the scatterer) denotes the normalized wave number and should be considered as a real positive parameter. Further- more, we write γ E for the tangential components of E on Γ and γ E for t N the “magnetic components” curlE n on Γ. The exterior unit normal vec- tor field n on Γ belongs to L∞(Γ)×and is directed from Ω into Ω+. In the 2 R. Hiptmair and P. Meury case of excitation by plane electric waves, whose complex amplitude will be denoted by Einc, the generic jump data g and g evaluate to the following D N traces g := γ Einc, g := γ Einc. D − t N − N Finally, we designate by [γU] :=U U and γU := 1 U +U Γ |Ω+ − |Ω { }Γ 2 |Ω+ |Ω the jump, respectively the average, of some generic t(cid:1)race γ of a fu(cid:2)nction U across the boundary Γ. Using Rellich’s lemma and unique continuation techniques, the following result can be established (cf. [36, Theorem 3.1]). Theorem 1.1. Provided that the relative material parameters µ and ε >0 r r are piecewise smooth and bounded away from zero everywhere in Ω, the prob- lem (1.1) has a unique solution. Boundary element methods (BEM) offer the most flexible way to deal with the homogeneous problem in the unbounded exterior domain Ω+. They are based on boundary integral operators on the interface Γ. Due to poten- tially nonconstant material parameters, the field problem inside Ω may not be amenable to a treatment by means of boundary element methods. Hence, finite element schemes (FEM) have to be used here. Thus the topic of this chapter comes into focus, namely how to derive and discretize stable cou- pled variational formulations, and how to analyze the resulting FEM–BEM formulation. The coupling entails expressing the Dirichlet-to-Neumann (DtN) map of the exterior problem by means of boundary integral operators linking the Cauchy data γ E and γ E for the electric field. There exists a huge vari- t N ety of integral formulations for the exterior electromagnetic boundary value problem. A comprehensive survey is given in N´ed´elec’s monograph [49]. In principle all these methods furnish Dirichlet-to-Neumann maps. However, in many cases, in particular with so-called indirect formulations, the resulting operator lacks structural properties of the Dirichlet-to-Neumann map, for in- stancesymmetry.Thisisobviouslythecaseforsecond-orderellipticproblems. If the structure of the DtN map is not preserved, then the linear systems of equations obtained by a Ritz–Galerkin boundary element discretization are adversely affected. For second-order elliptic problems Costabel [27] discovered that the so- calleddirectboundaryintegralequationmethodsprovidearemedy.Themain ideaistoemploytheCaldero´nprojector,whichactsontheCauchydataofthe problem. For details and theoretical considerations we refer to [21, Sect.4.5] and[30].Inshort,theCaldero´nprojectoryieldstwosetsofboundaryintegral equations. Judiciously combining them yields a version of the Dirichlet-to- Neumann map, which is perfectly suited for a Ritz–Galerkin discretization. 1 Stabilized FEM–BEM Coupling for Maxwell Transmission Problems 3 Costabel’s idea of coupling finite elements with boundary elements is usually referred to as “the symmetric coupling approach.” It has been applied to a wide range of strongly elliptic problems; see, among others [19, 38, 44]. For references to the engineering literature see [53, 55] and the references therein. Unsurprisingly, the Caldero´n projector for the Maxwell system has been thoroughly studied, cf. [20, Sect.1.3.2], [32], [49, Sect.5.5], and [43, Sect.3]. Theideaofsymmetriccouplingforthetransmissionproblemwastheoretically probed in [1, 2, 4], and in [7] for a related problem involving impedance boundary conditions. All these results employ compactness arguments and theFredholmalternative. Tothisend,most authorshavestudied theintegral operators intrinsically on Γ. They have been successful on smooth interface boundaries, but all efforts to adjust the approach to nonsmooth boundaries have been in vain. The fundamental new insights about the traces of electromagnetic fields, presented in [9, 12, 13, 15], paved the way to further progress. That progress couldfinallybeachievedbyrememberingahighlyeffectivepolicyinthemod- ern treatment of boundary integral equations: The guideline is to stay off the boundary as far as possible by studying variational problems instead of the boundary integral operators directly. This policy has demonstrated its efficacy in the work of Costabel [27]. The recent textbook [46] discusses all nuances of this approach for strongly elliptic systems. Moving off the bound- ary helps steer clear of its awkward geometric features. Thus, the foundation for a theory of electromagnetic boundary integral operators could be laid in [16, 18]. Inaddition,inordertoharnesscompactnessarguments,wehavetoemploy decompositions of the surface vector fields on Γ. The classical composition is theso-calledHodgedecomposition[32],whichremainsaveryeffectivetoolon piecewise smooth boundaries, cf. [14], [18], and, in particular [41]. Its coun- terpart on domains istheHelmholtz decomposition. It isimportant to realize that there is some leeway in choosing the decomposition, because the exact orthogonality featured by Hodge or Helmholtz decompositions is of minor importance. Instead, we prefer to use related, but simpler, splittings. Almost all boundary integral equations for the exterior Dirichlet problem in electromagnetic and acoustic scattering are haunted by the presence of “spuriousfrequencies”[15,18,22],forwhichtheequationsfailtohaveunique solutions. Those agree with interior Dirichlet eigenvalues. The symmetrically coupledvariationalformulationpresentedin[39]exhibitsthesamedrawback. In this chapter, we propose a stabilized method for FEM–BEM coupling based on (mixed) Robin-type boundary conditions to ensure unique solvabil- ityofthecorrespondinginteriorboundaryvalueproblem.Theuseofcomplex combinations of boundary integral operators has been an invaluable tool for deriving resonance-free combined field integral equations (CFIE) for electro- magnetic scattering from a perfect conductor, cf. [35]. Furthermore, our ap- proach also features regularizing operators, already used to stabilize Maxwell scatteringproblemsin[17],toensureaG˚ardinginequalityforthesesquilinear 4 R. Hiptmair and P. Meury form underlying the variational formulation. In our case, both problems are tackled by introducing modified trace operators. Basedonthegeneralizedtraces,stabilizedversionsofCalder´onprojectors canbedefinedforthecouplingofdomainbasedvariationalformulationswith boundary integral equations. Thus, we can derive new coupled variational formulations, which feature existence, uniqueness, and stability of solutions for all wave numbersκ>0. Asimilar approach totheonepresentedherecan be found in [54]. To discretize the symmetric and the stabilized coupled variational formu- lations,werelyondiscretedifferentialforms(edgeelements,faceelements)on triangulations of both Ω and Γ. The Ritz–Galerkin approach is straightfor- ward,andyet,inthediscretesettinganotherchallengearises.The Helmholtz and Hodge-type decompositions do not directly carry over to the discrete spaces. For pure indirect boundary element formulations (Rumsey’s princi- ple) remedies have been explored in [41] and [22]. Direct boundary integral equations were tackled in [18]. All these approaches exploit the fact that ap- propriate discrete splittings can approximate their continuous counterparts reasonably well. In this chapter we adapt the ideas in [18] and [39] to the symmetrically coupled FEM–BEM problem. We will use variants of these re- sults that do not require sophisticated elliptic regularity theory. The outline of this chapter is as follows: In the following section we will review the theory of Sobolev spaces and tangential traces. In Sect.1.3 we will introduce the potentials, which form the building blocks of the Stratton–Chu representation formula and the boundary integral operators for the electric field equation. In Sect.1.4 we construct decompositions of the electric field in Ω. The theoretical results for the symmetrically coupled variational formula- tion are reviewed in Sect.1.5. In Sect.1.6 the stabilized coupling strategy is presented.Sofar,allsectionshavebeenmerelyconcernedwiththeanalysisof thecontinuousvariationalproblems.Then,inSect.1.7,weintroducethefinite elementandboundaryelementspaces,whichareusedforaRitz–Galerkindis- cretizationofthecoupledproblem.InSect.1.8wederivediscretecounterparts to the decompositions on the continuous level and establish discrete inf–sup estimates for the underlying sesquilinear forms. Finally, in Sect.1.10 we will establish a priori convergence estimates for the stabilized coupling approach. 1.2 Traces and Spaces The main purpose of this section is to define suitable Sobolev spaces, which can be used to derive weak formulations of (1.1), and review some of their most important properties. The notation and notions we introduce closely follow the ones of [39, Sect.2]. The natural Hilbert space for an analysis of the Maxwell transmission problem (1.1) is the space Hloc(curl, D):= V L2loc(D); curlV L2loc(D) . ∈ ∈ (cid:3) (cid:4)