C .G.MAKRIDAKIS P - ESAIM: M2AN · PDF fileJOm MATHEUATICAL UODELUNG AND NUMERICAL ANALYSIS ... ( x , t) the electric and magnetic fields ... investigated for the two dimensional analogue

RAIROMODÉLISATION MATHÉMATIQUE

ET ANALYSE NUMÉRIQUE

CH. G. MAKRIDAKIS

P. MONKTime-discrete finite element schemes forMaxwell’s equationsRAIRO – Modélisation mathématique et analyse numérique,tome 29, no 2 (1995), p. 171-197.<http://www.numdam.org/item?id=M2AN_1995__29_2_171_0>

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Œ J O m MATHEUATICAL UODELUNG AND NUMERICAL ANALYSISfflMMH MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE

(Vol 29, n° 2, 1995, p 171 a 197)

TIME-DISCRETE FINITE ELEMENT SCHEMES FOR MAXWELL'SEQUATIONS (*)

by Ch. G MAKRIDAKIS C1) and P MONK (2)

Commumcated by M CROUZEIX

Abstiact — We analyze a family offully discrete finite element methods apphed to Maxwell'séquations To discretize in space we use the edge éléments of Nedelec which are particularlysmtable for discretizmg electro magnetic problems To discretize m Urne we use a farnily ofmethods based on rational approximations of the exponential We prove error estimâtes for thisscheme

Résumé — Nous étudions une famille de discrétisation complète des équations de MaxwellL'approximation en espace utilise les éléments d'arêtes de Nédelec, bien adaptés aux problèmesd'elettromagnetisme, la discrétisation en temps est basée sur une famille d'approximationsrationnelles de l'exponentielle Nous montrons des estimations d'erreur pour le schéma obtenu

1. INTRODUCTION

We shall analyze the use of fully discrete finite element schemes toapproximate the time dependent Maxwell équations on a bounded domainThe finite element scheme used to discretize m space is the standard Nédélecfamily of edge éléments for the electnc field and the correspondmg Raviart-Thomas-Nédélec éléments for the magnetic flux The advantages of theseéléments for electromagnetic computations are summarized for example m [3],To discretize m time, we use a family of implicit schemes based on rationalapproximations of the exponential of order 2, 3 or 4. Among others, a reasonof analyzmg the implicit schemes is that these schemes may be preferred insituations in which the mesh has some irregular tetrahedra (this is often thecase with meshes that are generated automatically, even though most tetra-hedra may be quite regular) We note however that our analysis can be

( ) Manuscript reeeived December 1, 1993C1) Depaitment of Mathematics , University of Crète, GR 714 09 Iraklion, Greece(2) Depaitment of Mathematical Sciences, University of Delaware, Newark DE 19716, USA

e mail monk@math udel edu Research supported m part by a grant from AFOSR

M2 AN Modélisation mathématique et Analyse numérique 0764-5 83X/95/02/S 4 00Mathematical Modellmg and Numencal Analysis {S) AFCET Gauthier Villars

172 Ch. G. MAKRIDAKIS, P. MONK

extended to include explicit schemes also, under of course, a restriction on thestability région of our method. For other implicit and explicit schemes for thetime discretization of the Maxwell équation we refer to [1, 17].

To state Maxwell's équations, let Q cz (R3 be a bounded, polyhedral, andconvex domain with boundary denoted by F and unit outward normal isdenoted by n. We remark that the assumption of convexity is necessary foranalysis, but not for the successful use of the method in practice. Theextension of the method to curved domains is possible but significantlycomplicates the présentation of the method (see [5] for details of how to defineedge éléments on a smooth domain). In the case of a smooth domain theconvexity assumption is not needed. We shall also assume that the materialcontained in Q is non-conducting. Some parts of the paper (in particular thesemi-discrete error analysis) can be extended trivially to include conductingmedia, but the fully discrete time stepping seheme must be modified in thatcase.

We suppose that Q is filled with a dielectric material have permitivity (ordielectric « constant ») E and permeability /J both of which may be three bythree matrix functions of position. We dénote by E = E(x, t) andH == H( x, t) the electric and magnetic fields respectively. These fields satisfythe Maxwell équations in Q :

eEt - V x H = J in Q , (la)

jMt + V x E = O in Q, (lb)

where

E, = | E and H, = | H .

In addition, for simplicity we shall assume that the boundary of Q is perfectlyconducting so that

n x E = 0 on F, (2)

Finally, we need to specify initial data so we suppose that functions Eo andHo are known and require that

E ( O ) = E O and H ( 0 ) = H0 in Q (3)

(where E ( 0 ) = E( - , 0 ) and H(0 ) = H( • , 0 ) ) . On physical grounds, wecan assume that

V • (juH0) = 0 in Q and (JJUQ) • n = 0 on F. (4)

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numerical Analysis

FINïTE ELEMENT SCHEMES FOR MAXWELL'S EQUATIONS 173

In order to handle the case where /J is not a constant function of x it is moreconvenient to use as dependent variables the electric field strength E and themagnetic flux B. The magnetic flux is given in terms of the magnetic field Hby the constitutive relation

B = juH. (5)

Using (5) in (1) to eliminate H, we arrive at the following equivalent équationsfor E and B :

eEt -Vx(ju~ 1 B ) = J in Ü, (6a)

B f + V x E = O in Q. (6b)

Our goal for this paper is to develop and analyze fully discrete finite elementapproximations to the fields ( E, B ) satisfying (2)-(6) for 0 ^ t ^ T whereT>0 .

In order to write down an appropriate variational formulation for theMaxwell system we will need s ui table spaces for the field and flux variables,Thus we recall the standard spaces :

tf(curl;fi) = {ue (L2(O))3 |V x u e ( L 2 ( O ) ) 3 } , (la)

#0(curl ; Q) = {u e H(curl ; fl)|u x n = Oon F} » (lb)

H(div;O) = {ue (L2(Q)f\V * u e L2(Q)} . (7c)

Let us now be more précise about the requirernents on the functions e, jua n d J . W e a s s u m e tha t s^(Cl(Ü)ff ju~ l e (Cl(Ü))9 a n dJ e C(0, T : (L (Q)) ) (the continuity assumptions are made to allow analy-sis and can be weakened considerably in practice), The functions e andfjT 1 are assumed to be positive definite and uniformly bounded (above andbelow) on Ü.

A weak formulation of the system (2)-(6) is

H0(curl ;f l) (8a)

(û"1Br,4i) + (VxE, / i~ 1 <l))=0 V<|>€ H(div;Q) . (8*)

Here B and E are still subject to the initial conditions (3). We will assume thatthe system (8) has a unique solution smooth enough for the purposes of ouranalysis. For existence uniqueness of the Maxwell équations see for ex-ample [10].

vol. 29, n° 2, 1995

174 Ch. G. MAKRIDÂKIS, P. MONK

To approximate (8) we use Nédélec's edge éléments and the Raviart-Thomas-Nédélec divergence conforming éléments [15] to construct finiteelement subspaces of jF/0(curl ; Q) and H(div ; Q). We detail this construc-tion, and the assumptions required on the mesh, in § 2. Hère it suffices to saythat if we are given the spaces U£ ° c Ho( curl ; Q ) and Vr

h <z H( div ; Q ) wecan discretize (8) in space in the obvious way to obtain the semi-discrèteproblem of finding ( Eh ( t), Bh ( t) ) e U f x V ^ such that, for0 < t T,

i f c A A A f (9a)

(^^B.^^ + iVxE^" 1^)^ VêV;. (96)

In addition (E ; ï, Bft) satisfy a discrete version of (3) so that

and

The first part of our paper § 2-§ 3 is devoted to analyzing the error in thissemi-discrète scheme. We remark that in a previous paper on this type ofscheme [14] error estimâtes for the corresponding semi-discrète scheme usingH in place of B and constant coefficients 8 and JJ were proved. The novelty ofthe analysis hère is that it allows variable matrix coefficients (but not y etdiscontinuous coefficients),

Having analyzed the semi-discrète scheme, we formulate a family of fullydiscrete time stepping schemes in § 4. We analyze the convergence of the fullydiscrete schemes in § 5, where we prove optimal order error estimâtes. Theseschernes belong to thc cîass of schemes inùoduced in [2] for the discretizationof second order hyperbolic équations. Their construction is based on a choiceof a rational approximation of the exponential with certain accuracy andstability properties. Related work includes [13] where similar time steppingschemes were used for the construction of mixed finite element fully discreteschemes for the équations of elasticity, and [1] where an implicit scheme isinvestigated for the two dimensional analogue of the Maxwell System con-sidered hère. For some computational results with edge/face éléments of thetype discussed hère see [12, 11].

2. PRELIMINAIRES

In this section we shall summarize the construction and properties ofNédélec's first family of edge finite éléments on a tetrahedral mesh [15]. Wenote that other families of edge éléments can also be constructed, and couldbe used for Maxwell's équations including Nédélec's second family ontetrahedra and the first family on cuboid meshes [15, 16].

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numerical Analysîs

FÎNITE ELEMENT SCHEMES FOR MAXWELL'S EQUATIONS 175

First let us define some notation. Whs(Q) will dénote the standard Sobolevspace of functions in LS(Q) having derivatives in LS(Q). SimilarlyHP(Q) is the standard Sobolev space of functions with p derivatives inL2(Q). In gênerai we shall dénote the norm on a metric space X by || • \\x

where X can be a space of vector functions. In the case of H(curl ; Q) (see(7)) the norm is defined by

+ II X ll||

In our analysis, we are going to use some weighted L2 spaces. We defineL2{Q) and L^- \{Q) to be the Standard L2 spaces with weights e and JJ~ l

respectively. We will dénote their inner products by

( u, v )fi = eu • v dV and ( u, v ) -1 = I fj~ l u • v dV

and the corresponding norms by

[|u||,?(fi) = ( u , u ) f and | | U | | L ; . I ( O ) = ( U , U ) ^ . .

Note that a conséquence of the propertLes of e and JJ~ x is that these norms areequivalent with the Standard L2 norm. We have already defined some spacesof vector functions in the introduction and so will not repeat them here.

Let { T ^ > 0 be a family of tetrahedral meshes of Q where h is the maximumdiameter of the tetrahedra in zh. We assume the meshes are regular andquasi-uniform [4].

In order to define the curl conforming space of Nédélec, we let Pr dénotethe standard space of polynomials of total degree less than or equal to r, andlet PF dénote the space of homogeneous polynomials of order r. Now wedefine Sr e (P,.)3 and Rr <= (P r)

3 by

For ex ample, if r = 1 then a polynomial p G R} has the formp(x) = a + f x x where a and p are constant vectors [16]. Following [15],for positive integer r, we define

K ={nh e H(cuû;Q)\uh\Ke Rr V/ê xh] .

vol. 29, n° 2, 1995


To define the degrees of freedom in U^ we defme the following moments. IfK G rh with genera! edge e and face ƒ and if t is a unit vector parallel to e :

u-tqds Vqe Pr_x{e) for the six edges e of K >, (10)

Mf(u) = \ | u x n q c M Vq e (P r _ 2 ( / ) ) 2 for all four faces ƒ of K \ ,

(11)

V q e (Pr_3(K))3\. (12)

u) = \ uxn-

These moments are defmed if u G ( WM( /(:) )3 for some s > 2. Nédélec [15]shows that the above three sets of degrees of freedom are Rr- uni sol vent andcurl conforming. Using these degrees of freedom we can defïne an interpolantdenoted r^ u e U^ for any function u for which (1Ö)-(12) are deflned. On eachK e xh we piek rh n\K G Rr such that

Me(u - rh u ) = Mf(u - rh u) = MK(n - rh u) = {0} .

To approximate functions in ^0(curl ; Q) we define

U£° = { u , E U ; | n x % = 0 on r] . (13)

The constraint n x u^ = 0 on f is easily implemented by taking the degreesof freedom associated with edges or faces on ƒ" to be zero [8].

The following estimate is known for r^ [15] (for other estimâtes includingestimâtes using weaker norms see [8, 5, 14]) : if u G (Hr* l(Q))3 then

\\v-TkV\\mcuAiO)* Chr\\u\\iHr+i{Q))>. (14)

Of crucial importance in our analysis is the fact U£° admits a discreteHelmholtz décomposition. Let

xh, <f>h\r = 0 } . (15)

Then V S ^ c U j 0 , and

; ) x (16)


FINITE ELEMENT SCHEMES FOR MAXWELL'S EQUATIONS 177

Next we deflne a space of divergence conforming fini te éléments. Thesefacial éléments are the Raviart-Thomas-Nédélec spaces [15]. Let

then we define

Ke Dr VZ e rh} .

An appropriate set of degrees of freedom for this space are the followingwhich are defined for any function v e ( / / ! ( ^ ) ) 3 where K e xh :

^(u)=\ u-nqds VqePr_x(f) for all four faces f of K [ ,J

(17)

MDK(u)= Uûqdx V q e (Pr_2(K)fy. (18)

These degrees of freedom are H( div ; Q ) conforming and unisolvent. Thus wecan define an interpolation operator wh : ( H

l ( Q ) )3 —» V^ by requiring that oneach K e zh

This interpolant satisfies the error estimate that if u e (Hr(Q))3 then (see[15])

(19)

Furthermore, if u is smooth enough that both interpolants are defined, thanw / ) Vxu = Vxr / i u.

An important property of the space U£ ° and Yrh is that

We define V£ = V x U^ ° and then can define the space V£± by theorthogonal décomposition

v; = v ; e M - . v ^ . (20)

vol. 29, n° 2, 1995


where the orthogonality is with respect to the L2- i(Q) inner product. This isanother discrete Helmholtz décomposition. Note that by virtue of the connec-tion between w^ and rA, if u is divergence free then wh u is divergence free.

We remark that V^ is a good space for approximating the magnetic flux Bbut it is usually more convenient to compute with the larger space \r

h.Nevertheless, we shall carry out some of our analysis in Vr

h . This is possiblesince by (20) we can write

where Bh e Yrh and B^ e V£ ± . Substituting this expansion in (9a) and (9b),

and using the orthogonality properties of Bh, we can see that( EA, Bh ) e U£ ° x V^ satisfles

, yh) - (ft' l B„ V x V A ) = ( J, vA) V V A e U^° (21a)

( ^ ' B ^ ^ + t V x E , ^ - 1 ^ ^ V * A e V ; . (216)

In the same way, we can see that

É£, = 0 sothat B ^ ( r ) = B ^ ( 0 ) for all t. (22)

In view of (22) we will see that it sufïïces to carry out the error analysis forthe approximation (21) to (8).

The error analysis of the semi-discrete and fully discrete problems rests onthe use of a suitable pair of projections in to the spaces U£ ° and \h. Following[14] we define first TIh : H( curl ; Q ) —> U£ ° by requiring that ifu G //(curl ; Q), then TIh u e U£ is the unique solution of

1 V V A e V , (23a)

( nh u, V0„ ) = ( u, V(j>h X V0A e S;- ° . (236)

(Here 77 corresponds to the solution operator for an appropriate discrete staticproblem.) Existence and error estimâtes for projections of this type (withconstant coefficients) have been studied in [15, 6, 7, 14]. Existence anduniqueness are proved by using the Babuska-Brezzi theory of mixed problemswhere we must use the Freidrichs type inequality proved in [9] for thecontinuous problem. This inequality can also be proved for the discreteproblem following Nédélec's analysis [15] provided the domain and coeffi-cients are such that the following problem of computing w G H( curl ; Q )such that



V x w = V x f in Q (24a)

V - (ew) = 0 in Q (24b)

w x n = 0 on r (24c)

has a unique solution obeying the a priori estimate

(25)

for 2 ^ s =£ sQ where s0 > 2, In this case the constant coefficient resultsalso hold for the variable coefficient problem. Our assumptions on E and ju(continuous differentiability) are suffîcient for (25) to hold, but are probablynot minimal. The result of this theory is that the following estimâtes hold :

(26)

In proving this estimate we use the estimate for the interpolant in (14). Weremark that an almost optimal estimate of this type, but with an improvednorm on the right hand side can be proved (see [14]) but we will not examinethat case here.

We also need a projection into the space of magnetic fïuxes. So we derkiePh : L*- i ( Q ) -» V^ to be the L*- i ( Q ) projection onto V^ so that for anyv e L2- \(Q), the function Ph v e V^ satisfies

Pk v, yA) = (AT ' V, V A ) V V A e % . (27)

We are only interested in the approximation properties of Ph on divergencefree vector fields so we shall only analyze this case. Let V =V x HQ(curl ; Q) then if v e V there is a function u e HQ(curl ; Q) such

that v = V x u . In view of the fact that Ph v e V£, we know that there isa function uh e U£° such that Ph v = V x u^ , we conclude that (27) maybe rewritten as

( V x uh , V h V i = ( V x u, yh V i , ^

which is exactly (23a). Thus if we make the arbitrary choice that uh will alsosatisfy the divergence condition (23&), we may conclude that if v e V, then

vol. 29, n° 2, 1995


By virtue of the fact that w^ maps V n (Hr(Q))3 into Vj, we may estimatethe error in the projection Ph using (19) as follows

(28)

3. ERROR ESTIMATES FOR THE SEMI-DISCRETE PROBLEM

It will prove convenient (when we come to analyze the fully discretescheme) to convert the variational équations (9) to operator form. For this weintroducé two discrete operators C and C as follows :Let C ; H(curl ;O) -> Vh be deflned by

(Cu, <t>„V i = ( V x u, 4>A V ' V4>* G K - (29)

Since V x U j ° c V^, we have that

Cuh = V x uh if uh e lij;0 , (30)

and so

CV?hz%, VVA eVf . (31)

Let C : L2~ i (Q) —» U^° be the operator defined as follows : for^ ( O ) , the function Cve U '° is the solution of the variational

problem

( Cv, 4>A) = ( v, V x <j>, V . V0A E ü f . (32)

Note that we have used the Standard (L2(Ü) )3 inner product on the left handside here. Using the weighted inner-product notation as above we may write(91?) as

( * M , v * V i + (v x E A ,v*V • = o , ;t

Hence using (29) and (30), we obtain the operator équation

Kt +CEA = 0 . (33)

To obtain an operator form for (9a) we need two more projection operators.Let Ph be the standard (L2(Q) )3 orthogonal projection onto U£° . We defineA„ :Vr

h° -»U?° by

K un =P^mh)' w h e r e uh e vh° •

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelîing and Numerical Analysis


(Note hère that we can extend Ah : (L2e(Q) )3 -» U£° as A^ u = PA(eu) for

ue (L](Lwritten asu G (L2(Q))3 .) Using A^ and the curl operator C, équation (9a) can be

A f t E , , -CBh =PhJ (34)

and for simplicity we define Jh = Ph J.From the above analysis, we conclude that the semidiscrete problem (9) can

be written as :

\Kr -CBh=Jh, (35a)

Bht +CEh = 0 , (35b)

where Eh(t) e U£° and Bh(t) e V^ for each t.Analogously, using the fact that CEh e V^ and the fact that CÈ^= y we

obtain the operator form of (21) : find (EA , Bh) : [0, T] -» U^'° x V^ suchthat

Bh t + CEh = O . (36Z?)

To write the above équations more compactly, let the operator matrix %h

) A"1 C\

c o )• (37>

The operator %h maps U£ x V^ into itself, and because of the orthogonalityin the définition of V^ (see (31)), we can conclude that

We may now rewrite (35) as

H Bh ) = ( h i ' (38)

and rewrite (36) as

A | ^ J _ C g J „ A j _ | h h \ ^ty

The operator %h has useful properties that we describe next. In order todo this, we defme the following inner product on vector functions

), /= 1,2 by

vol. 29, n° 2, 1995


with the associated norm

GMO-G)))"In the above inner-product, %h is anti-symmetric since

£) • (tl))) = V

But from the définitions of C and C, if §h e U£° and \\rh e \rh we have

( Cà>, , \i/i ) - i = ( V x ó . , \if, ) - i = (<t>, , C\|/. ) .

Putting the above two équations together proves the antisymmetry of %h :

(«:: ) • ( : : ) ) ) - ( ( ( : : ) •<:) ) )

(41)

It folllows that if ( tyh , x\th ) e U '° xVj then

(42)

Now we shall state and prove our theorem concerning the approximation errorin the semidiscrete problem (9).

THEOREM 3.1 : Let (E, B) a solution of (S) such that

C(O,T:(Hr+ l and Be C(0, T: h\ , (O))



and let (~Eh(t),Bh(t)) e Uri° xVj be a solution of(_9) then the followingerror estimate holds with constant C independent of h, T and J for0 s= t^T:

K -

+ hr\ | |E,(s) | | ,H ,+ .,o„3«fc). (43)

Remark : This theorem also holds for the more gênerai problem of approxi-mating the solutions E and B of

using an obvious extension of (9) provided the necessary smoothness of E ispresent (hère the conductivity a is a non-negative function of x).

To prove this theorem, we first prove that (43) holds for (21) and then showthat this implies (43) in gênerai. To dérive this intermediate result we shallcompare the solution of (21) with the couple (£, Ç) defined by

l 4 4 )

where I7h and Ph are defined in (23) and (27) respectively. Our first lemmadérives the équation satisfied by (£,, Ç).

LEMMA 3.2 : Under the hypotheses of Theorem 3.1 the pair (Ç, Ç) satisfies

where Ph e is the L£(Q) orthogonal projection onto U£ .

vol. 29, n° 2, 1995


Proof : We prove (45) component by component, starting with the loweréquation. Using (86), (23), (27) and (29) we have that for any Vfh e \r

h

0 = ( B , , ** )„ - .+ ( V x E, v j , -

* ' + ( V x 77A E,

Now since Pft commutes with time differentiation Ph B, = Çf and from (30)we see that Ct, <= Yr

h. Thus we have proved that

Ç, + C^ = 0 . (46)

Next we analyze the first équation in (45). For all ^h e U£ we have (usingthe fact that Vx«j»s e V^ ) that

( CÇ, <!>„) = (Ç, V x ^ V , = (P h B,Vx ^ V , = (B, V x <!>„)„- , =

so that

C^ = CB . (47)

Furthermore, using <t> = ^ e U£° in (8a) shows thatPh(sTLt)-CB = PhJ.

Using the above results,

Ah %t - C Ç - A , Ç, -CB = Ah %, +PhJ-Ph(eEt)

where in the last equality we have used the définition of Ah . We will showbelow that

where Ph e is the L2F(Q) projection onto U£° . Using (48) we have

(49)



Taken together équations (46) and (49) give (45).It remains to prove (48). Note first that Aj^.o is symmetrie and positive

definite and therefore its inverse in U£° , A " \ exists and is symmetrie andpositive definite too. ïf UG (L (Q)) and <|>A e U£ , the properties of thevarious projections show that

(K l K »><U = (K ».K ! W = W " 0 . K * 0.)

Thus (48) is proved. D

LEMMA 3.3: Under the hypotheses of Theorem 3.1, let (E, B) be asufficiently smooth solution of (8) and let (~Eh,Éh) G U^ X V [ be asolution of (21) then the following error estimate holds with constant Cindependent of h, T and J :

||(B„ -

hr\ ||E, | , + 1 3 ^ ) . (50)

Proof : Subtracting (45) from (39) we have that if

then

(51)

Now let %t dénote the solution operator for the problem of ComputingW, (t) e U''° x V', such that

| W , - ^ W / i = 0 , and w A (0)=Wj with

vol. 29, n° 2, 1995


Thus Wfe ( f ) = %t Wh . By virtue of (42) and a standard energy argument wehave that

iiiw, ( om = «fis, w j ; m = m w); m.

Duhamel's principle applied to (51) gives us that

and hence

C* / P E — E \

inê(oiii « mê( o)in + J ( *•* ' ' JBut by (26)

t -n„Et\\Liia)

ds. (52)

Thus

i /O

(53)

and after standard manipulations we have eompleted the proof of thislemma. D

Proof [of Theorem 3.1] : We recall that B. ( r ) = B.(f) + B ^ ( 0 whereB h ( 0 e V ; and Bi ( / ) e(d/dt) B^" (t) = 0. But since :

0

Therefore we have that

Further we have shown thatV £ X we have

(54)

Now let

e = sothat 0 = 0+Bt


FINITE ELEMENT SCHEMES FOR MAXWELL'S EQUATIONS

Equations (51) and (54) give

187

The same argument as in the proof of the last lemma, via DuhammeFsprinciple, shows that 9 satisfies (52) so that (53) holds for 9. Standardmanipulations now prove the desired estimate. D

4. FULLY DISCRETE SCHEMES

In this section we shall describe how to discretize the semi-discrete prob-lem (9) in time. It will prove convenient to work from the semi-discreteéquations in operator form given by (38).

To discretize (38) we use a rational approximation of the exponential as in[13]. Let

w h e r e a n d

where/?j,/?2, ql and q2 are constants chosen so that firstly there exists a v with1 ^ v ^ 4 with

\r(z)-ez\ ^c\z\v+1

for all z in a neighborhood of zero and z e C and secondly such that

| r ( z ) | ^ 1 for all ze iR. (55)

By the correct choice of p and q we can obtain a number of interestingschemes [13]

Method

Backward EulerCrank-NicolsonPadéPadé

V

1234

- 1- 1/2-2 /3- 1/2

00

1/61/12

Pi

01/21/31/2

Pi

000

1/12

vol. 29, n° 2, 1995


As a resuit of these assumptions, we know that if y(t) is v+ 1 timesdifferentiable and k > 0 then

+px kyXO +p2k2y"(O + O(kv + l ylv+l)) . (56)

Now (38) implies that

o

- A , " 1 CC 0 \{V

B J +- CA' l C/\B* ) \ - CA~ l J„

(57)

Motivated by (56) and using (57) to replace second derivative terms, we arriveat the a fully discrete approximation to (8). Let (Ej.BjJ) E U£° X V [ bea given approximation to (E(fn), B(fn) ) where tn — nk. We define(Ej + l , B^ + ! ) as the solution of

<58)

where

'l-k2q2A~x CC kq.A'1 C \

kaC , k2 f A - ' C ( 5 9 a )

- kqx C I-k q2 CA. C I

I-k2p.A~} CC fcp.A."1 C \, * , , - (59b)

-kpxC I-k2p2CA,;x C

and

:2 A ~ l ( q Jn + l — p J" )•wïl + 1 -rn \

h -p2jft)



Remark : Our error analysis covers explicit schemes too. In particular, thescheme (58) is explicit if qx= q2 = 0. In this case (55) should be replacedby | r (z) | ^ 1, z e ( - ia, ia), a > 0. Then the stability of the schemein this case will be ensured provided that kh~ l remains smalL

5. ERROR ANALYSIS OF FULLY DISCRETE SCHEMES

This section is devoted to proving the following theorem which givesconvergence results for the fully discrete method. We also will state and provea corollary of this theorem that covers the practically useful case.

THEOREM 5.1 : Suppose that E and B are solutions of (8) with the regu-

larity :

E e CA(0, T: (Hr+ l(Q)f) n Cv + '(O, T: L2(Q))

Be C(0, r 1 lThen if

there is a constant C independent of k and h such that for 0 ^ tn ^ T thefollowing estimât e holds where III • III is the norm defined in (40) :

)rtlll ^ c | lHI0rt III ^ c | III0°III + k\Uh 0°lll + k2 \\\%2h

+ t

sup i | ^ | »))}(61)

Remark : If e is constant, the term khr l does not appear in the aboveestimate. For gênerai £, if we take k = O(h), the bound in the theorem willbe of order O(hr+kv).

The next corollary summarizes the results in a particularly practical case :

COROLLARY 5.2 : Suppose the conditions of Theorem 5.1 hold. In additionsuppose that E^ = rh E(0) and B°h =wh B(0). Then B£ e V^ for eachn. In addition the following error estimate holds for 0 =S tn ^ T providedE and B are smooth enough :

vol. 29, n° 2, 1995

190 Ch. G. MAKRIDAKIS, R MONK

SUD[ff

Remark : This case, in which we simply interpolate the initial data, is easyto use in practice. Furthermore, the magnetic flux is exactly divergence freeat each time step.

To avoid the term (hr + khr ~ 1 + k2 hr' ~ 2) in the error estimate we couldchoose B£ =P^B(0) and E°h =77Ê(0) but this choice is very costly toimplement.

Proof [of Corollary 5.2] : Since B(0) e V, we know that wh B(0)and so is divergence free. Now we show that sinceeach n. As we have seen if B^ = B^ + B^€6A(0,B^) r = 0 or

ti-O-

with^ thenBh e

^ forthen

(62)

Therefore, if we write

for each n where Bnh e V^ and B"' ± e V£ ± then we have the following

équations :


FTNITE ELEMENT SCHEMES FOR MAXWELL1 S EQUATIONS 191

Also (31) implies that the second component of &*n is in V^ so

k2CA-l(q2rk+l-p2rh)eVr

k. (64)

Using (58) and (62)-(64) we get :

/E" + ' \ / o \ /EW +1 \ /E" +1 \

x) + \^x •tpiÂfrU ) + * f t ^ i ™ - * ' ) - ^ - ( 6 5 )

This équation implies that

BrU=B^ (66)

and proves the resuit. Note also that

/T?n + 1 \ / X?n \

I '"Ij 1 / •"h V

9l( I = £f I I + S rt (67)

where 91, SP and 3Fn are given by (59)-(60).The remainder of the corollary is proved by using (61) and estimating the

errors at t = 0 and t = tn using (14), (19), (28) and (26) together with thefact that since the mesh is quasi-uniform we may estimate

\\Kh yll! *£ Ch~ l\\\\y\\\, Vy e U ' ° x V^ . a

We start the proof of Theorem 5.1 by proving a consistency estimate for thefully discrete scheme. To do this we let JJh and Ph be the projections definedin (23) and (27) and let

(68)

where E and B are the exact solutions of (8). Let us define yn and an by

( Jn J = 9ll\ + t J - Sff \) . (69)

LEMMA 5.3 : Suppose all the conditions of Theorem 5.1 are satisfied. Let( £n, Çn ) ^^ öfj defined in (68) araJ ( y", an ) be as defined in (69) then we canwrite

( \ / \ / n \ / M \

n \ / K« +1 e« \ / Y \ / y \

vol. 29, n° 2, 1995


where

(71)

and

w/iere W" + ! anii Wrt satisfy the estimate (80).

Proof : By expanding (69) using the définitions of 8ft and £ƒ we may write

We study each term on the right hand side of this expansion. First we studythe term multiplying k on the right hand side. So we define

U J l -«„«--ft» j - (74)

Using (46) we have :

Cr = - C • (75)Now using (47) we have that CÇ" = CB( fn ), and hence from the secondéquation of (8) we have that for \j/A e U£°

M2 AN Modélisation mathématique et Analyse numériqueMathematica! Modelling and Numerical Analysis

FINÏTR ELEMENT SCHEMES FOR MAXWELL'S EQUATIONS 193

where Ph is the (L 2 (Q ) )3 projection on U£ and Ph g is the LF(Q) projection

°h is the ( L ( Q ) ) projection on U£ and Ph g

on U£ °. Now since Ph £ E, e U£ ° we have

(76)

The identities (75) and (76) in (74) give (71).Now we estimate the term multiplying k2 in (73). To this end we define

By using (75) and differentiating (76) with respect to t we have

A; ' CCK = - Al ' CC = -P A i ,E K ( ï B ) + A; ' j ; , . (78)

Also by (76), (29) and the définition of Ç we have

CA; ' cç" = C P M E . c r j - C A ; ' J :

- /) E,(tn) + C E , ( O - CA; > rh

- /) E , ( O - pfcB„(/B) - CA; X rh

= C(Phe-I)Et(tn)-t;"tt-CA-irh

= W" - C, - CA~h ' J2 (79)

where W" = C(PA c - / ) E,(rn) and W" can be estimated as follows. Usingthe définition of C we have

^ ^ ) l | W " l l t ; . 1 ( O ) .

So that for some positive constant C,

HW'V, (e ) ^ C I K P ^ - / ) ^ ^ ) ! ! ^ ^ , ^ ) . (80)

vol. 29, n° 2, 1995


The equalities (78) and (79) can be combined in (77) to give (72). DOur next lemma shows that the time stepping scheme is stable.

LEMMA 5.4: For any ¥ e Uj°xVj,

MSfcVIII. (81)

Proof : F rom (42) we have that %h has pure imaginary eigenvalues and sincethe rational function r satisfies (55) we have the desired resuit. D

Proof [of Theorem 5.1] : Let us recall the définition

*•©©Using (69)-(73), (74), (71), (72) and (67) we have the estimate :

( 8 2>

where W is defined after (79). We define D" \ D"'2 and D"'3 to be respec-tively the first second and third terms on the right hand side of (82). Each ofthese is estimated separately.

We estimate D"' ' first using the mean value theorem and the error estimatefor nh in (26) :

sup \\Et{s)\\Hr^,Q). (83)

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelhng and Numencal An al y si s

FÎNITE ELEMENT SCHEMES FOR MAXWELL'S EQUATIONS 195

The relation (56) gives

iiiDn'2in ^ ŒV + 1 ( sup r V ! + F ( J ) +

^ V T T C O I I 2 V (84)

The estimation of Dn' is more troublesome. Since the grid is quasi-uniform,we can easily show that

) « C * r " 1 I I V l l i r - ( O ) - (85)

In view of the fact that the finite element interpolant approximates a functionto O(hr) in the //(curl ; Q) norm, one might hope to improve (85) withhr ~ 1 replace by hr. Using (85) we have

HlDn'3!ll ^ Ck2hr~ l sup ||E,(5)||ff,+ .(f l ) . (86)

Now (82) and (81) and the estimâtes (83), (84) and (86) give :

r + khr'l)s^ SUD

supup Y v + f ( j ) + sup FfA+.] Il dtv + ' lliî(O) * e tv t +.] Il

^ IHSft©nlll + Ck((hr + khr'1) sup

kv+l( sup | | ^ ^ f ( * ) | | + sup ||\ " t ' A . . ] Il dtv + l WLHQ) * • [»..t. • il II

Iterating this estimate we prove that

« G i l < IIIS710OIII + Cnkf (hr+khr~l)s s\yg^ \\E,(s)||H, + . ( O ) +

( llytij , M I h v + 1 R II

sup r v + r 0 ) + sup F-TTT(*)

( l l ^ v + ] i? Il l !^ v + l R II \ \

sup a y + f(j) + sup i - _ f ( , ) )).

vol. 29, n° 2, 1995


But using the antisymmetry of %h in the ( ( • , • ) ) inner product [13] weobtain

"IH2 = ( ( &\ 0n ) ) + k\ q\-2q2 )((<8A &\ %h ©")) +

4 \ ® \ % 2h 0 n ) ) . (87)

It is easy to see that (55) implies that q[ - 2 q2 5= 0, [13]. Therefore we haveproved that

h)s$\xf>t \\Et(s)\\Hr+i(Q) +

+ sup | | ^ n * ( * ) | | ^ V (88)Q) s * rf',] II d t V + l H L ; - I ( / /

Use of (87) in (88) complètes the estimate. D

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[1] J. ADAM, A. SERVENIÈRE, J. NÊDÉLEC and P. RAVIART, 1980, Study of an implicitschcrne for integrating Maxwell's équations, Comp. Meth. AppL Mech. Eng., 22,327-346.

[2] G. BAKER and J. BRAMBLE, 1979, Semidiscrete and single step fully discreteapproximations for second order hyperbolic équations, RAÏRO Anal. Numen, 13,75-100.

[3] A. BOSSAVIT, 1990, Solving Maxwell équations in a closed cavity, and thequestion of « spurious » modes, IEEE Trans. Mag., 26, 702-705.

[4] P. ClARLET, 1978, The Finite Element Method for Elliptic Problems, vol. 4 ofStudies in Mathematics and It's Applications, Elsevier North-Holland, New York.

[5] F. DUBOIS, 1990, Discrete vector potential représentation of a divergence freevector field in three dimensional domains : numerical analysis of a modelproblem, SIAM J. Numer. Anal., 27, 1103-1142.

[6] V. GlRAULT, 1988, Incompressible finite element methods for Navier-Stokeséquations with nonstandard boundary conditions in R , Math. Comp., 51, 53-58.

[7] V. GIRAULT, 1990, Curl-conforming finite element methods for Navier-Stokeséquations with non-standard boundary conditions in R , in The Navier-Stokeséquations. Theory and Numerical Methods, Lecture Notes, 1431, Springer, 201-218.

[8] V. GIRAULT and P. RAVIART, 1986, Finite Element Methods for Navier-StokesEquations, Springer-Verlag, New York.



[9] M. KRÎZEK and P. NEITTAANMÂKI, 1989, On time-harmonic Maxwell équationswith nonhomogeneous conductivities : solvability and Fis-approximation, Apli-kace Matematiky, 34, 480-499.

[10] R. LEIS, 1988, Initial Boundary Value Problems in Mathematical Physics, John

Wiley, New York.

[11] K. MAHADEVAN and R. MiTTRA, 1993, Radar cross section computations ofinhomogeneous scatterers using edge-based finite element method in frequencyand time domains, Radio Science, 28, 1181-1193.

[12] K. MAHADEVAN, R. MITTRA and P. M. VAIDYA, 1993, Use of Whitney's edge andface éléments for efficient finite element time domain solution of Maxwell* séquations, Preprint.

[13] C. G. MAKRIDAKIS, 1992, On mixed finite element methods in linear elastody-

namics, Numer. Math., 61, 235-260.

[14] P. MONK, 1993, An analysis of Nédélec's method for the spatial discretization ofMaxwell's équations, J. Comp. Appl. Math., 47, 101-121.

[15] J. NÉDÉLEC, 1980, Mixed finite éléments in [R3, Numer. Math., 35, 315-341.

[16] J. NÉDÉLEC, Éléments finis mixtes incompressibles pour l'équation de Stokes

dans R3, Numer. Math., 39, 97-112.[17] K. YEE 1966, Numerical solution of initial boundary value problems involving

Maxwell's équations in isotropic media, IEEE Trans, on Antennas and Propa-gation, AP-16, 302-307.

vol. 29, n° 2, 1995

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