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Applied Mathematical Sciences Franck Assous Patrick Ciarlet Simon Labrunie Mathematical Foundations of Computational Electromagnetism Applied Mathematical Sciences Volume 198 Editors S.S.Antman,InstituteforPhysicalScienceandTechnology,UniversityofMaryland,CollegePark, MD,USA [email protected] LeslieGreengard,CourantInstituteofMathematicalSciences,NewYorkUniversity,NewYork,NY, USA [email protected] P.J.Holmes,DepartmentofMechanicalandAerospaceEngineering,PrincetonUniversity,Princeton, NJ,USA [email protected] Advisors J.Bell,LawrenceBerkeleyNationalLab,CenterforComputationalSciencesandEngineering, Berkeley,CA,USA P.Constantin,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA R.Durrett,DepartmentofMathematics,DukeUniversity,Durham,NC,USA R.Kohn,CourantInstituteofMathematicalSciences,NewYorkUniversity,NewYork,NY,USA R.Pego,DepartmentofMathematicalSciences,CarnegieMellonUniversity,Pittsburgh,PA,USA L.Ryzhik,DepartmentofMathematics,StanfordUniversity,Stanford,CA,USA A.Singer,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA A.Stevens,DepartmentofAppliedMathematics,UniversityofMünster,Münster,Germany S.Wright,ComputerSciencesDepartment,UniversityofWisconsin,Madison,WI,USA FoundingEditors FritzJohn,JosephP.LaSalle,LawrenceSirovich Moreinformationaboutthisseriesathttp://www.springer.com/series/34 Franck Assous (cid:129) Patrick Ciarlet (cid:129) Simon Labrunie Mathematical Foundations of Computational Electromagnetism 123 FranckAssous PatrickCiarlet DepartmentofMathematics ParisTech ArielUniversity ENSTA Ariel,Israel Palaiseau,France SimonLabrunie UniversitédeLorraine InstitutÉlieCartandeLorraine Vandœuvre-le`s-Nancy,France ISSN0066-5452 ISSN2196-968X (electronic) AppliedMathematicalSciences ISBN978-3-319-70841-6 ISBN978-3-319-70842-3 (eBook) https://doi.org/10.1007/978-3-319-70842-3 LibraryofCongressControlNumber:2018937591 MathematicsSubjectClassification(2010):M13120,P21070,T11030,M12066,P19005,P24040 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword Our interest in the study and computationof electromagneticfields started during the1990s.ForFranckAssous,itoriginatedfromtheneedtocomputepreciselythe motionofchargedparticlesforplasmaphysicsapplications.ForPatrickCiarlet,it beganwith the study of the relationsbetween the electromagneticfields and their potentials from a mathematical point of view. From both the numerical and the theoretical points of view, it soon appeared that one had to be especially careful whendealingwithsingularconfigurations.Atypicalexampleoccurswhenonehas to solve a seemingly elementary problem, namely the computation of the fields in vacuum, around a perfectly conducting body, or inside a perfectly conducting cavity or waveguide.Togetherwith SimonLabrunie,we started to investigatethis problemforaclassofsuchbodiesthatareinvariantbyrotation.Sincethen,wehave collaboratedregularlyonthistopicandmanyothers. Going back to the example, when the interface between the body and vacuum is piecewise smooth and when the computational domain is locally non-convex near this interface, intense electromagneticfields may occur. Pointwise valuesare unbounded,and mathematically,the smoothnessof the fields deteriorates. It turns out that this common situation induces challenging problems, which we address here. Though the contents of this monograph chiefly deal with theoretical issues, most results are derived in order to solve problems numerically,using discretized variationalformulations(wedonotaddresstheissueofdiscretizationinthisbook). The focus of this monographis clearly an applied mathematicalone; however, we begin by discussing the physical framework of electromagnetism and related models.Oneofthemainpointsofthebookistheintroductionofmathematicaltools to characterize electromagnetic fields precisely and, among others, the traces of thosefieldsonsubmanifoldsofR3.Thisissueisespeciallyimportantonnonsmooth submanifolds.Anotherimportantissueisthemathematicalmeasureofthosefields, which can take several forms. Interestingly, this leads to very differentcategories of discretized problems.A third main issue is the introductionand justification of approximate models in a broad sense, such as, for instance static, quasi-static or time-harmonic,andalsoofreducedmodels,namely2Dand21Dmodels.Thelast 2 v vi Foreword important issue deals with the introduction and study of models that govern the motionofchargedparticlesinteractingwithelectromagneticfields. The textis entirelyself-contained:we onlyassumefromthereadera bachelor- level background in analysis, and we give all the necessary basic definitions. Nevertheless, this monographincludes some originalapproachesand novelappli- cationsnotcovered,toourknowledge,inpreviousbooks.Itischieflyintendedfor researchers in applied mathematics who work on Maxwell’s equations and their approximateorcoupledmodels.Muchofitsmaterialmayalsoserveasabasisfor master’s-ordoctorate-levelcoursesonmathematicalelectromagnetism. We are indebted to a number of people who contributed, to various extents, to the topics we address in this monograph. Let all of them be thanked: Régine Barthelmé, Anne-Sophie Bonnet-BenDhia,Annalisa Buffa, Lucas Chesnel, Pierre Degond,EmmanuelleGarcia,ErellJamelot,Pierre-ArnaudRaviart,JacquesSegré, EricSonnendrücker,JunZouandCarloMariaZwölf. Finally,wegratefullyacknowledgethehelpofthefollowingreadersofprelim- inaryversionsofthemanuscript:LucasChesnel,LipengDai,BenjaminGoursaud andClaireScheid. Ariel,Israel FranckAssous Palaiseau,France PatrickCiarlet Vandœuvre-lès-Nancy,France SimonLabrunie Contents 1 PhysicalFrameworkandModels......................................... 1 2 BasicAppliedFunctionalAnalysis ....................................... 73 3 ComplementsofAppliedFunctionalAnalysis .......................... 107 4 AbstractMathematicalFramework...................................... 147 5 AnalysesofExactProblems:First-OrderModels ...................... 191 6 AnalysesofApproximateModels......................................... 223 7 AnalysesofExactProblems:Second-OrderModels ................... 267 8 AnalysesofTime-HarmonicProblems................................... 313 9 DimensionallyReducedModels:DerivationandAnalyses ............ 347 10 AnalysesofCoupledModels .............................................. 393 A IndexofFunctionSpaces.................................................. 429 References......................................................................... 443 Index............................................................................... 453 vii List of Figures Fig.1.1 “Pipe”domain.......................................................... 53 Fig.1.2 Adjustmentofasamplediffractionproblem ......................... 56 Fig.1.3 Adjustmentofasampleinteriorproblem ............................ 56 Fig.1.4 BasicgeometricalstepsfortheconstructionofPMLs ............... 59 Fig.2.1 The“twosugarcubes” ................................................. 81 Fig.9.1 ExampleofanaxisymmetricdomainΩ anditsmeridian sectionω................................................................ 349 Fig.9.2 TopologicalconditionsforanaxisymmetricdomainΩ and itsmeridiansectionω.................................................. 380 ix Chapter 1 Physical Framework and Models The aim of this first chapter is to present the physics framework of electromagnetism, in relation to the main sets of equations, that is, Maxwell’s equationsand some related approximations. In that sense, it is neither a purely physical nor a purely mathematical point of view. The term model might be more appropriate: sometimes,itwillbenecessarytorefertospecificapplicationsinordertoclarifyour purpose,presentedinaselectiveandbiasedway,asitleansontheauthors’personal view.Thisbeingstated,thischapterremainsafairlygeneralintroduction,including theforemostmodelsinelectromagnetics.Althoughthechoiceofsuchapplications isguidedbyourownexperience,thepresentationfollowsanaturalstructure. Consequently,inthefirstsection,weintroducetheelectromagneticfieldsandthe setofequationsthatgovernsthem,namelyMaxwell’sequations.Amongothers,we presenttheirintegralanddifferentialforms.Next,wedefineaclassofconstitutive relations,whichprovideadditionalrelationsbetweenelectromagneticfieldsandare needed to close Maxwell’s equations. Then, we briefly review the solvability of Maxwell’sequations,thatis,theexistenceofelectromagneticfields,inthepresence of source terms. We then investigate how they can be reformulated as potential problems.Finally,werelatesomenotionsonconductingmedia. InSect.1.2,weaddressthespecialcaseofstationaryequations,whichhavetime- periodic solutions, the so-called time-harmonic fields. The useful notion of plane wavesisalsointroduced,asaparticularcaseofthetime-harmonicsolutions. Maxwell’s equations are related to electrically charged particles. Hence, there exists a strong correlationbetween Maxwell’s equationsand models that describe the motion of particles. This correlation is at the core of most models in which Maxwell’s equations are coupled with other sets of equations: two of them—the Vlasov–Maxwell model and an example of a magnetohydrodynamics model (or MHD)—willbedetailedinSect.1.3. ©SpringerInternationalPublishingAG,partofSpringerNature2018 1 F.Assousetal.,MathematicalFoundationsofComputational Electromagnetism,AppliedMathematicalSciences198, https://doi.org/10.1007/978-3-319-70842-3_1