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Uniform a posteriori error estimation for the heteregeneous Maxwell equations
Sarah Cochez and Serge Nicaise
UniversitedeValenciennesetduHainautCambresis LAMAV
Institut des Sciences et Techniques de Valenciennes F-59313 - Valenciennes Cedex 9 France Sarah.Cochez,Serge.Nicaise@univ-valenciennes.fr
February 14, 2006
We consider residual based a posteriori error estimators for the heteregeneous Maxwellequationswithdiscontinuouscoecientsinaboundedtwodimensional domain. The continuous problem is approximated using conforming approximated spaces. The main goal is to express the dependence of the constants in the lower and upper bounds with respect to chosen norm and to the variation of the coecien ts. Forthatpurpose,somenewinterpolantsofClement/Nedelectypeareintroduced and some interpolation error estimates are proved. Some regularity results for trans-mission problems are further revisited. Some numerical tests are presented which con rm our theoretical results.
Key Words edge elements.Maxwell equations, error estimator, piecewise coecien ts, AMS (MOS) subject classi cation65N30; 65N15, 65N50,
1 Setting of the problem LetO= IR2Rbe a bounded domain ofR3with a polygonal boundary∂O. The classical Maxwell equations are given by tB+curlE in= 0O, tDdlirucHv=D=Jnini,O,O(1) divB= 0 inO,
whereE,D,B,HandJare vector functions of positionxinR3and timetinR. EandH intensities,are the electric and magnetic eldDandB, are respectively the electric displacement and the magnetic induction.J(, t) is the source current density which is supposed to satisfy
divJ(, t) = 0 in ,t0.
By settingD=EandB=Hwhereandare positive, bounded, scalar functions, respectively called theelectric permittivityand themagnetic permeabilityweca, ndn relationships betweenEandHand obtain second-order Maxwell’s equations depending either on the magnetic eldHor on the electric eldE this paper, we arbitrary choose. In to eliminateHrather thanE.
1.1 Quasistatic electromagnetic elds in conductors
The computation of quasistatic electromagnetic elds in conductors usually employs the eddy current model [2]. In this case,Jis given byE+Jawhereis the conductivity of the body occupyingOandJa(, tcurrent density which is supposed to) is the source satisfy divJa(, t) = 0 in ,t0. This identity allows to transform (1) into tt2DB+ uclrcurlEt=H0=  ∂tEtJa t2tH E+lrulrcucEtH0==  ∂tEtJa tt2HE=+ 1tEcu+crlEurl( 1curlE) = tJa. For good conductors, we can assume that∂t2E= 0 and obtain the parabolic initial bound-ary value problem [4, 2]
t(E) +curl(curlE) = tJainO, E n on= 0∂O, E(, t= 0) =E0inO, whereEsionknuehtwnelectric eld,denotes the inverse of the magnetic permeability, andndenotes the unit outward normal vector along∂O.
Using a time discretization of the above problem we have to solve at each time step Maxwell’s equations cuurl(n=cu0rlu) + u=fnino,O,O(3)
whereusidle cofthtionctrieelemiaehtteixampporE, the coecien t is equal to /t (where tis the time step discretization) andfdepends onJaand the approximation of Ein the previous step. we may assume that Thereforefsatis es
divf= 0 inO.
1.2 Electromagnetic  elds in dielectrics
We now return to the time-dependent problem (1) and reduce it to the time-harmonic Maxwell system setting
b E(x, t) =<(exp( iωt)E(x)) b D(x, t) =<(exp( iωt)D(x)) b H(x, t) =<(exp( iωt)H(x)) b B(x, t) =<(exp( iωt)B(x)) b J(x, t) =<(exp( iωt)J(x)) (x, t) =<(exp( iωt)b(x)) 
b whereE(and similarly other hat variables) are now complex-valued functions depend-ing on the space variables but not on the time variable ([16]). We introduce the linear, inhomogeneous constitutive equations b b b b D=EandB=H b b b b and the constitutive relation for the currentsJ=E+Ja, where the vector functionJa b describes the applied current density. Asiωb= divJ, we arrive at the following time-harmonic system :  H+curlE= 0 b b b b div (EbH=)= i1ωJbdiv (E+Ja) b b b  E+E curla div (Hb) = 0 b b De ningEas02EandHas02Hwhere0and0respectively represents the electric permittivity and the magnetic permeability in vacuum, we obtain the second-order Maxwell system for the electric eldER3[16]: curl r1curlE 2rE=finO,(5) En= 0 on∂O,(6) b wherefdepends onJa,=ω 00=ωc 1is called thewavenumber,ω0 is the frequency of the electromagnetic wave andcis the speed of light in vacuum. Moreover,r
andrare the relative permeability and permittivity of the medium occupyingOde ned by :  r= d anr= . 00 We assume thatrandr Toare uniformly bounded from below and above. get the same system than before, we now set =ω2c 2rand=r 1. With these notations, our equations become cuurl(n=cu0rlu)  u=fnionO,,O(7)
whereucorresponds toE, the datumfis once more a multiple ofJand so is divergence free.
1.3 A common variational formulation
From now on, we reduce the problem (3) (or (7)) to a problem in the two-dimensional domain , namely assuming thatudepends only on thex1,x2variables, then the equations are reduced to : curl(curlu) +s u=fin , (8) ut on= 0,
wheret ,is the unit tangential vector alongs= 1 in the quasistatic case ands= 1 in the dielectric case. For the sake of simplicity, we assume that  is simply connected and that its boundary is connected. In the whole paper, we suppose thatand are piecewise constant, namely we assume that there exists a partitionPof  into of Lipschitz polygonal domains  a nite set1,  ,J such that, on each j,=jand = j, wherejand jare positive constants (see Fig. 1).
2❙ ✪ ✡ ❙ ✪5 1 ✡❙ ✪ 4. ✱❡ ✡ ✪ 6 ✱ ❡ ✡ 3Figure 1: Partition of the domain .
The variational formulation of (8) is well known and involves the space H0(curl, ) ={u[L2( )]2: curluL2( );ut= 0 on}
and the bilinear fo
rm a(u,v) =Z(curlucurlv+s uv)dx.
Forf[L2( )]2,)htweaefkroumalsatisfying(4 nisnidngonti(8ofon)cstsiuH0(curl,tahthcus)a(u,v) = (f,v),vH0(curl, ),(9) where (,) is the [L2( )]2-inner product. In the sequel, we assume thatais coercive onH0(curl, ), namely we assume that there exists >0 such that a(u,u) kuk 2,,uH0(curl, ( div ) : u) = 0,(10) wherekuk ,=Z|curlu|2+ |u|21/2. This coerciveness assumption guarantees that problem (8) has a unique solution by the Lax-Milgram lemma. In the quasistatic case, thanks to the positivity of and,a coercive-clearly satis es ness with = 1. In the dielectric case, the variational formulation is given by F(in rd1cuurlu,H0cuurlcr(vl), ) ωs2ucc h2(thraut,v) = (f,v),vH0(curl, ),(11) withusatisfying the divergence constraint div (ru) = 0. Ifω= 0, (11) has a unique solution. Otherwise, problem (11) enters within the framework of the Fredholm alternative and has a unique solution providedω2does not belong to the spectrum of the involved operator. In this paper, we assume thatωis small enough in order to guarantee the coerciveness ofa, given here by : a(u,u) =Z( r1|curlu|2 ω2c 2r|u|2)dx. It means that, if we denote by2Mthe smallest positive eigenvalue of the Maxwell system [16], we assume thatωc 1< M. Under this condition, we can estimate the optimal constantofcoercivenessand ndthat: 2 ω2c 2 = M+ω2c 2. 2
Problem(9)isapproximatedinaconforming niteelementspaceVhofH0(curl, ) based on a triangulationTof the domain made of isotropic triangles, the spaceVhis assumedtocontainlowerorderNedelecedgeelementspace(cf.[17]).Ifuhis the solution ofthediscretizationof(9)weconsiderecientandreliableresidualaposteriorierror estimators for the errore=u uhin theH0(curl, )-norm.
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