A finite element method for approximating the time

Numer.Math.63,243261(1992)NumerischeMathemaUk9Springer-Verlag1992Afiniteelementmethodforapproximatingthetime-harmonicMaxwellequations*PeterMonkDepartmentofMathematicalSciences.UniversilyofDelaware,Newark,DE19713,USAReceivedAprilI7,199ISummary.InthispaperwestudytheuseofN6d6lec'scurlconformingfiniteelementstoapproximatethetime-harmonicMaxwellequationsonaboundeddomain.Theanalysisiscomplicatedbythefactthatthebilinearformisnotcoercive,andtheprincipleparthasaverylargenull-space.ThisdifficultyiscircumventedbyusingadiscreteHelmholtzdecompositionoftheerrorvector.Numericalresultsarepresentedthatcomparetwodifferentlinearelements.MathematicsSubjectClassification(1991):65M601IntroductionLetf2beaboundedpolygonaldomaininI1t3withboundarysurfaceFandunitoutwardnormaln.ThepropagationofanelectromagneticfieldthroughOisgovernedbyMaxwell'sequations[7].Wesupposethatthematerialinf2isinhomogeneous,linearandisotropic,andthuscanbecharacterizedbyadielectricconstante,apermeability/~,andanelectricconductivitya.Ingeneraleand/~arestrictlypositivefunctionsand~isanon-negativefunctiononf2.Ifo~(x,t)andJ~(x,t)denotetheelectricandmagneticfieldsrespectivelyatpositionxeQandtimetthenMaxwell'sequations[15]statethat(++)(1)~+~g-VxJ4rinr8(2)g~H+VxS=0inO,whereJ(x,t)isaknownfunctionspecifyingtheappliedcurrent.Inaddition,initialdataford~and~mustbespecified,togetherwithanappropriateboundary*ResearchsupportedinpartbygrantsfromAFOSRandNSF244P.Monkcondition.Inthispaperweshallassumethesimplestessentialboundarycondition(3)nx~=0onF,whichdescribesaperfectlyconductingboundary[!5].Frequently,theelectromagneticfieldisassumedtobetime-harmonicinwhichcasewemaywrite(4)~(x,t)=e-iE(x)(5)~(x,t)=e-i~H(x).Using(4)and(5)in(1)and(2),weobtainthetimeharmonicMaxwellequations[15]:(6)-ieJqE-VxH=Jin~2(7)-ico/~H+VxE=0inf2whereq=e+ia/coand(x,t)=e-io'tJ(x).Proceedingformally,wemayelimin-atethemagneticfieldHbytakingthecurlof(7)andsubstitutinginto(6)toobtain(8)V�(/t-1V�E)-coZqE=-icoJin(2.Ofcourse(3)alsoholds.Tosimplifynotation,wedefineF=-icoJ.Toobtainaweakformulationfor(8)weproceedasin[15],andmultiply(8)byanarbitraryfunction$~Ho(curl;f2)where(9)Ho(curl;Q)={uE(L2(Q))3IVxu~(LZ(Q))3,nxtt=0onF}.Thenweintegrateoverf2andintegratethecurltermbypartstoobtaintheweakproblemoffindingE~Ho(curl;f2)suchthat(10)(It-lV�215V~9eHo(curl;f2),where(.,.)isthe(L2(Q))3innerproductonf2.ExistenceanduniquenessforthisproblemarestudiedbyLeisin[15].Therearetwodistinctcases.Ifa0inf2,then(I0)possessesauniquesolutionforanypositivecoandFe(L2(Q))3.Inthiscasethelefthandsideof(10)definesabilinearformthatiscoerciveonHo(curl;f2).Hencethefiniteelementapproximationof(10)maybeanalyzedviaCea'slemmainthestandardway.Theproblemismoreinterestingifcr=0onf2.Then(10)mayfailtohaveasolutionatacountablyinfinitediscretesetofvaluesofcowhicharetermedtheinteriorMaxwelleigenvalues.Thusevenifco0,thebilinearformin(10)isnotcoercive.Furthermore,theprinciplepartof(8)hasalargenull-spacesinceVx(#-1Vx(VqS))=0foranythreetimesdifferentiablescalarfunctionqS.Thesetwopropertiesofthebilinearformin(10)complicatetheerroranalysisweshallpresent.Todiscretize(10)weuseN6d61ec'scurlconformingelements,whichwediscussindetailinSect.2.ThusweconstructafamilyoffinitedimensionalsubspacesV~cHo(curl;f2).ThediscreteproblemistofindEheV~suchthat(11)(]2-1VxEh,VxOh)--fo2(r]Eh,~th)=(F,~lh)[e-mail].3weshallprovethatprovidedcoisnotaMaxwelleigenvalue,andprovidedhissmallenoughthen(1I)hasauniquesolutionandE-Ehisofoptimalorder.FiniteelementmethodforapproximatingtheMaxwellequations245OurchoiceofN6d~lec'sspacestodiscretizeHo(curl;t2)ismotivatedbythefollowingthreeobservations.FirstifV.J=0int2,thenV-(r/E)=0andwecanshow(seeSect.3)thatifN6d61ec'sspacesareused,t/Enisdiscretedivergencefree.ThusN6d61ec'selementsallowprecisecontrolofthedivergenceoftheelectricfield.AsecondreasonforchoosingN+d61ec'selementsisthatifeisdiscontinuousacrossasurfaceinIR3,itisknownthatthetangentialcomponentofEiscontinuousacrossthesurfacewhereasthenormalcomponentisdiscontinuous.IfthesurfaceofdiscontinuitycoincideswithaunionoffacesofelementsinO,thenthediscontinu-ityofthenormalcomponentishandledbyN6d~lec'selementswithoutmodifica-tion.Bycomparison,standardcontinuouselementsrequireelaboratemodificationacrossthesurfaceofdiscontinuity(cf.[14]).Finally,wenotethattheboundarycondition(3)iseasytoimplementusingN6d61ec'selements.However,thisboundaryconditionisbynomeanstheonlyinterestingboundaryconditionforMaxwell'sequations.OuranalysisismotivatedbytheworkofSchatz[22]whoanalyzedthefiniteelementapproximationoftheHelmholtzequation.WealsoemploythetechniquesofGirault[8]inapplyingadiscreteHelmholtzdecompositiontoanalyzetheerror.Unfortunately,whena--0wecanonlyproveconvergencewhen~tand~/areconstant(althoughthemethodisapplicableinmoregenerality).ForageneralreferenceonstandardfiniteelementmethodsappliedtoMaxwell'sequationsseeforexample[23]andforerrorestimates[13].Kikuchi[12]hasanalyzedtheuseofN6d61ec'selementsforaneigenvalueproblemasso-ciatedwith(8)intwodimensions,andLevillain[16]hasanalyzedthethreedimensionalcase(seealso[2]).Foradiscu