Uniform a posteriori error estimation for the heteregeneous Maxwell equations

Description

Uniform a posteriori error estimation for the heteregeneous Maxwell equations Sarah Cochez and Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis LAMAV Institut des Sciences et Techniques de Valenciennes F-59313 - Valenciennes Cedex 9 France Sarah.Cochez, February 14, 2006 Abstract We consider residual based a posteriori error estimators for the heteregeneous Maxwell equations with discontinuous coefficients in a bounded two dimensional 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 coefficients. For that purpose, some new interpolants of Clement/Nedelec type are introduced and some interpolation error estimates are proved. Some regularity results for trans- mission problems are further revisited. Some numerical tests are presented which confirm our theoretical results. Key Words Maxwell equations, error estimator, piecewise coefficients, edge elements. AMS (MOS) subject classification 65N30; 65N15, 65N50, 1 Setting of the problem Let O = ?? I ? R2 ?R be a bounded domain of R3 with a polygonal boundary ∂O. The classical Maxwell equations are given by ? ??? ?? ∂tB + curl E = 0 in O, divD = ? in O, ∂tD ? curlH = ?J in O, divB = 0 in O, (1) 1

• contain lower order


• source current


• along ∂o


• curl


• maxwell equations


• order maxwell system


• equations depending


• system setting

Sujets

• Université
• Saint Nicaise
• Maxwell's equations

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English

Extrait

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,[e-mail]

February 14, 2006

Abstract

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= IR 2 Rbe a bounded domain ofR 3 with a polygonal boundary∂O. The classical Maxwell equations are given by  ∂ t B+curlE in= 0O,   ∂ t D d l i rucH v = D= J  ni ni ,O ,O (1)   divB= 0 inO,

1

whereE,D,B,HandJare vector functions of positionxinR 3 and 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.

(2)

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+J a whereis the conductivity of the body occupyingOandJ a (
, tcurrent density which is supposed to) is the source satisfy divJ a (
, t) = 0 in ,∀t0. This identity allows to transform (1) into  ∂ ∂ t t 2 D B+  uclr curlE ∂ t = H 0 =  ∂ t E∂ t J a ⇔  ∂ ∂ t 2 t H  E +lru lr c uc∂ E t H 0= =  ∂ t E∂ t J a ⇔  ∂ ∂ t t 2 H E = +     ∂ 1 t E cu +c rlE url(  1 curlE) = ∂ t J a . For good conductors, we can assume that∂ t 2 E= 0 and obtain the parabolic initial bound- ary value problem [4, 2]

∂ t (E) +curl(curlE) = ∂ t J a inO, E n on= 0∂O, E(
, t= 0) =E 0 inO, 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  c u u  rl( n  = cu 0 rlu) +u=fni no ,O ,O∂ (3)

2

whereusidlecofthtionctrieelemiaehtteixampporE, the coecien tis equal to /
t (where
tis the time step discretization) andfdepends onJ a and the approximation of Ein the previous step. we may assume that Thereforefsatises

divf= 0 inO.

1.2 Electromagnetic elds in dielectrics

(4)

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+J a , where the vector functionJ a b describes the applied current density. Asiωb= divJ, we arrive at the following time- harmonic system :   iωH+curlE= 0 b b b b   div (E b H= )=   i 1 ω J b div (E+J a ) b b b  iωE+E curl a   div (H b ) = 0 b b DeningEas 0 2 EandHas 0 2 Hwhere 0 and 0 respectively represents the electric permittivity and the magnetic permeability in vacuum, we obtain the second-order Maxwell system for the electric eldE∈R 3 [16]: curl   r 1 curlE  2  r E=finO,(5) En= 0 on∂O,(6) b wherefdepends onJ a ,=ω  0  0 =ωc  1 is called thewavenumber,ω0 is the frequency of the electromagnetic wave andcis the speed of light in vacuum. Moreover, r

3

and r are the relative permeability and permittivity of the medium occupyingOdened by :   r = d  an r = .  0  0 We assume that r and r  Toare uniformly bounded from below and above. get the same system than before, we now set=ω 2 c  2  r and= r  1 . With these notations, our equations become  c u u  rl( n  = cu 0 rlu) u=fni onO, ,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 thex 1 ,x 2 variables, then the equations are reduced to :  curl(curlu) +su=fin ,  (8) u
t 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 set 1 ,


, J such that, on each j ,= j and= j , where j and j are positive constants (see Fig. 1).

❙ 2 ✡ ✡ ✪ ❙ ✪  ✡ ❙ ✪ 5 1  ✡ ❙ ✪ ❙ 4 ✡ ✡ ✪ . ✱❡ ✡  ✪ ✱ ❡ ✱ ✱ ❡ ✡ ✡ ✪ 6 ✱ ❡  ✡ ✱ 3 ❡ ✡ ✱ ✱ ❡ ❡ ✡ Figure 1: Partition of the domain .

The variational formulation of (8) is well known and involves the space H 0 (curl, ) ={u∈[L 2 ( )] 2 : curlu∈L 2 ( );u
t= 0 on}

4

and the bilinear fo

rm a(u,v) = Z (curlucurlv+su
v)dx.

Forf∈[L 2 ( )] 2 ,)htweaefkroumalsatisfying(4nisnidngonti(8ofon)cstsiu∈ H 0 (curl,tahthcus) a(u,v) = (f,v),∀v∈H 0 (curl, ),(9) where (
,
) is the [L 2 ( )] 2 -inner product. In the sequel, we assume thatais coercive onH 0 (curl, ), namely we assume that there exists >0 such that a(u,u)kuk  2 , ,∀u∈H 0 (curl, ( div ) :u) = 0,(10) wherekuk  , = Z |curlu| 2 +|u| 2  1/2 . 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   r d 1 c u ur ∈ lu, H 0 cu u rl cr( v l ) , )  ω s 2 u c c   h 2 ( t  h r a u t ,v) = (f,v),∀v∈H 0 (curl, ), (11) withusatisfying the divergence constraint div ( r u) = 0. Ifω= 0, (11) has a unique solution. Otherwise, problem (11) enters within the framework of the Fredholm alternative and has a unique solution providedω 2 does 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 (   r 1 |curlu| 2  ω 2 c  2  r |u| 2 )dx. It means that, if we denote by 2 M the smallest positive eigenvalue of the Maxwell system [16], we assume thatωc  1 <  M . Under this condition, we can estimate the optimal constantofcoercivenessandndthat:  2  ω 2 c  2 =   M +ω 2 c  2 . 2

Problem(9)isapproximatedinaconformingniteelementspaceV h ofH 0 (curl, ) based on a triangulationTof the domain made of isotropic triangles, the spaceV h is assumedtocontainlowerorderNedelecedgeelementspace(cf.[17]).Ifu h is the solution ofthediscretizationof(9)weconsiderecientandreliableresidualaposteriorierror estimators for the errore=u u h in theH 0 (curl, )-norm.

5