Time-discrete finite element schemes for Maxwell's equations

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M2AN - M

ODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

C H . G. M AKRIDAKIS

P. M ^ONK

Time-discrete finite element schemes for Maxwell’s equations

M2AN - Modélisation mathématique et analyse numérique, tome 29, n^o2 (1995), p. 171-197

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fflMMH MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE

(Vol 29, n° 2, 1995, p 171 a 197)

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

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's équations To discretize in space we use the edge éléments of Nedelec which are particularly smtable for discretizmg electro magnetic problems To discretize m Urne we use a farnily of methods based on rational approximations of the exponential We prove error estimâtes for this scheme

Résumé — Nous étudions une famille de discrétisation complète des équations de Maxwell L'approximation en espace utilise les éléments d'arêtes de Nédelec, bien adaptés aux problèmes d'elettromagnetisme, la discrétisation en temps est basée sur une famille d'approximations rationnelles 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 to approximate the time dependent Maxwell équations on a bounded domain The finite element scheme used to discretize m space is the standard Nédélec family of edge éléments for the electnc field and the correspondmg Raviart- Thomas-Nédélec éléments for the magnetic flux The advantages of these éléments for electromagnetic computations are summarized for example m [3], To discretize m time, we use a family of implicit schemes based on rational approximations of the exponential of order 2, 3 or 4. Among others, a reason of analyzmg the implicit schemes is that these schemes may be preferred in situations in which the mesh has some irregular tetrahedra (this is often the case with meshes that are generated automatically, even though most tetrahedra may be quite regular) We note however that our analysis can be

( ) Manuscript reeeived December 1, 1993

C1) 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 00

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

To state Maxwell's équations, let Q cz (R3 be a bounded, polyhedral, and convex domain with boundary denoted by F and unit outward normal is denoted by n. We remark that the assumption of convexity is necessary for analysis, but not for the successful use of the method in practice. The extension of the method to curved domains is possible but significantly complicates the présentation of the method (see [5] for details of how to define edge éléments on a smooth domain). In the case of a smooth domain the convexity assumption is not needed. We shall also assume that the material contained in Q is non-conducting. Some parts of the paper (in particular the semi-discrete error analysis) can be extended trivially to include conducting media, but the fully discrete time stepping seheme must be modified in that case.

We suppose that Q is filled with a dielectric material have permitivity (or dielectric « constant ») E and permeability /J both of which may be three by three matrix functions of position. We dénote by E = E(x, t) and H == H( x, t) the electric and magnetic fields respectively. These fields satisfy the 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 perfectly conducting so that

n x E = 0 on F, (2) Finally, we need to specify initial data so we suppose that functions Eo and Ho 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, we can assume that

V • (juH0) = 0 in Q and (JJUQ) • n = 0 on F. (4) M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

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

B = juH. (5)

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

eE^t -Vx(ju~¹B ) = J in Ü, (6a)

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

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

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

tf(curl;fi) = { u e ( L2( O ) )3| V x u e ( L2( 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, ju a n d J . W e a s s u m e t h a t s^(Cl(Ü)ff ju~ l e (Cl(Ü))9 a n d J 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 and fjT 1 are assumed to be positive definite and uniformly bounded (above and below) on Ü.

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

H0( c u r l ; f l ) (8a) (^u"1Br,4i) + ( V x E , / i ~1< l ) ) = 0 V<|>€ H(div;Q) . (8*) Here B and E are still subject to the initial conditions (3). We will assume that the system (8) has a unique solution smooth enough for the purposes of our analysis. For existence uniqueness of the Maxwell équations see for example [10].

vol. 29, n° 2, 1995

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To approximate (8) we use Nédélec's edge éléments and the Raviart- Thomas-Nédélec divergence conforming éléments [15] to construct finite element subspaces of jF/0(curl ; Q) and H(div ; Q). We detail this construction, and the assumptions required on the mesh, in § 2. Hère it suffices to say that if we are given the spaces U£ ° c H^o( curl ; Q ) and V^rh <z H( div ; Q ) we can discretize (8) in space in the obvious way to obtain the semi-discrète problem of finding ( Eh ( t), Bh ( t) ) e U f x V ^ such that, for 0 < t ^ T,

i f c A A A

f (9a)

( ^ ^ B . ^ ^ + i V x E ^ "1^ ) ^ V^eV;. (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 this semi-discrète scheme. We remark that in a previous paper on this type of scheme [14] error estimâtes for the corresponding semi-discrète scheme using H in place of B and constant coefficients 8 and JJ were proved. The novelty of the analysis hère is that it allows variable matrix coefficients (but not y et discontinuous coefficients),

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

2. PRELIMINAIRES

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

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysîs

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First let us define some notation. Whs(Q) will dénote the standard Sobolev space of functions in LS(Q) having derivatives in LS(Q). Similarly HP(Q) is the standard Sobolev space of functions with p derivatives in L2(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 define L2{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 are equivalent with the Standard L2 norm. We have already defined some spaces of 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 maximum diameter of the tetrahedra in zh. We assume the meshes are regular and quasi-uniform [4].

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

For ex ample, if r = 1 then a polynomial p G R} has the form p ( 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/^e xh] .

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To define the degrees of freedom in U^ we defme the following moments. If K G r^h 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 (Pr _ 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 ( W^M( /(:) )³ for some s > 2. Nédélec [15]

shows that the above three sets of degrees of freedom are R^r- uni sol vent and curl conforming. Using these degrees of freedom we can defïne an interpolant denoted r^ u e U^ for any function u for which (1Ö)-(12) are deflned. On each K e x^h we piek r^h n\^K G R^r 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 degrees of freedom associated with edges or faces on ƒ" to be zero [8].

The following estimate is known for r^ [15] (for other estimâtes including estimâtes 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 discrete Helmholtz décomposition. Let

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

Then V S ^ c U j⁰, and

; ) ^x (16)

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

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Next we deflne a space of divergence conforming fini te éléments. These facial 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 following which are defined for any function v e ( / /!( ^ ) )3 where K e xh :

^(u)=\ u-nqds VqeP

r

_

x

(f) for all four faces f of K [ ,

J

(17) MDK(u)= U^uqdx V q e (Pr_2(K)fy. (18) These degrees of freedom are H( div ; Q ) conforming and unisolvent. Thus we can define an interpolation operator wh : ( Hl ( Q ) )3 —» V^ by requiring that on each 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, than w/ )V x u = V x r/ 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 the orthogonal décomposition

v; = v ; e

M

- . v ^ . (20)

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where the orthogonality is with respect to the L²- i(Q) inner product. This is another 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 B but it is usually more convenient to compute with the larger space \rh. Nevertheless, we shall carry out some of our analysis in Vrh . This is possible since 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) VV A e U^° (21a) ( ^ ' B ^ ^ + t V x E , ^ - 1 ^ ^ V *Ae 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 for the approximation (21) to (8).

The error analysis of the semi-discrete and fully discrete problems rests on the 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 if u G //(curl ; Q), then TIh u e U£ is the unique solution of

1 VV Ae 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 static problem.) Existence and error estimâtes for projections of this type (with constant coefficients) have been studied in [15, 6, 7, 14]. Existence and uniqueness are proved by using the Babuska-Brezzi theory of mixed problems where we must use the Freidrichs type inequality proved in [9] for the continuous problem. This inequality can also be proved for the discrete problem 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

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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 =£ s^Q where s⁰ > 2, In this case the constant coefficient results also hold for the variable coefficient problem. Our assumptions on E and ju (continuous differentiability) are suffîcient for (25) to hold, but are probably not 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). We remark that an almost optimal estimate of this type, but with an improved norm on the right hand side can be proved (see [14]) but we will not examine that case here.

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

P^k v, y^A) = (AT ' V, V A) VV A e % . (27) We are only interested in the approximation properties of P^h on divergence free vector fields so we shall only analyze this case. Let V = V x H^Q(curl ; Q) then if v e V there is a function u e H^Q(curl ; Q) such that v = V x u . In view of the fact that P^h v e V£, we know that there is a function uh e U£° such that Ph v = V x u^ , we conclude that (27) may be 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 also satisfy the divergence condition (23&), we may conclude that if v e V, then

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

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3. ERROR ESTIMATES FOR THE SEMI-DISCRETE PROBLEM

It will prove convenient (when we come to analyze the fully discrete scheme) to convert the variational équations (9) to operator form. For this we introducé 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%, V^VA eVf . (31)

Let C : L2~ i (Q) —» U^° be the operator defined as follows : for

^ ( O ) , the function C v e 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 hand side here. Using the weighted inner-product notation as above we may write (91?) as

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

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

Kt +CE^A = 0 . (33)

To obtain an operator form for (9a) we need two more projection operators.

Let P^h be the standard (L²(Q) )³ orthogonal projection onto U£° . We define A„ :Vrh° -»U?° by

K un =P^mh)' w h e r e uh e vh° •

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelîing and Numerical Analysis

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(Note hère that we can extend A^h : (L^2e(Q) )³ -» U£° as A^ u = P^A(eu) for ue (L](L

written as

u G (L²(Q))³ .) Using A^ and the curl operator C, équation (9a) can be

A^{f t}E , , -CB^h =P^hJ (34)

and for simplicity we define J^h = P^h J.

From the above analysis, we conclude that the semidiscrete problem (9) can be written as :

\Kr -CB^h=J^h, (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^ such that

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

To write the above équations more compactly, let the operator matrix %^h ) A"1 C\

c o )• ( 3 7 >

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

We may now rewrite (35) as

H B

^h

)

⁼

(

^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 to do this, we defme the following inner product on vector functions

), / = 1,2 by

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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^'° x V j then

(42)

Now we shall state and prove our theorem concerning the approximation error in 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))

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and let (~Eh(t),Bh(t)) e Uri° x V j be a solution of(_9) then the following error estimate holds with constant C independent of h, T and J for 0 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 is present (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 show that this implies (43) in gênerai. To dérive this intermediate result we shall compare the solution of (21) with the couple (£, Ç) defined by

l 4 4 )

where I7^h and P^h are defined in (23) and (27) respectively. Our first lemma dérives 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

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Proof : We prove (45) component by component, starting with the lower équation. Using (86), (23), (27) and (29) we have that for any Vf^h e \^rh

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

* ' + ( V x 77A E,

Now since P^ft commutes with time differentiation P^h B, = Ç^f and from (30) we see that Ct, <= Y^rh. Thus we have proved that

Ç, + C^ = 0 . (46) Next we analyze the first équation in (45). For all ^^h e U£ we have (using the fact that Vx«j»s e V^ ) that

( CÇ, <!>„) = (Ç, V x ^ V , = ( PhB , V x ^ V , = (B, V x <!>„)„- , =

so that

C^ = CB . (47) Furthermore, using <t> = ^ e U£° in (8a) shows that Ph(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 show below that

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

(49)

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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 and positive definite too. ïf U G (L (Q)) and <|>A e U£ , the properties of the various 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 a sufficiently smooth solution of (8) and let (~Eh,Éh) G U^ X V [ be a solution of (21) then the following error estimate holds with constant C independent 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 Computing W, (t) e U''° x V', such that

| W , - ^ W

/ i =

0 , and w

A

( 0 ) = W j with

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186 Ch. G. MAKRIDAKIS, P. MONK

Thus W^fe ( f ) = %^t W^h . By virtue of (42) and a standard energy argument we have that

iiiw, ( o m = «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 ( • ' ' J**

But by (26)

t -n„Et\\Liia)

ds. (52)

Thus

i / O

(53) and after standard manipulations we have eompleted the proof of this lemma. D

Proof [of Theorem 3.1] : We recall that B. ( r ) = B . ( f ) + B ^ ( 0 where B^h( 0 e V ; and Bi ( / ) e

(d/dt) B^" (t) = 0. But since :

0

Therefore we have that

Further we have shown that V £^X we have

(54) Now let

e =

sothat 0 = 0+

Bt

(18)

Equations (51) and (54) give

187

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

4. FULLY DISCRETE SCHEMES

In this section we shall describe how to discretize the semi-discrete problem (9) in time. It will prove convenient to work from the semi-discrete équations 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 with 1 ^ 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 interesting schemes [13]

Method Backward Euler Crank-Nicolson Padé

Padé

V

1 2 3 4

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

0 0 1/6 1/12

Pi

0 1/2 1/3 1/2

Pi

0 0 0 1/12

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As a resuit of these assumptions, we know that if y(t) is v + 1 times differentiable 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 arrive at the a fully discrete approximation to (8). Let (Ej.BjJ) E U£° X V [ be a given approximation to (E(fn), B(fn) ) where tn — nk. We define ( E j + l , B^ + ! ) as the solution of

< 5 8 )

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)

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Remark : Our error analysis covers explicit schemes too. In particular, the scheme (58) is explicit if qx= q2 = 0. In this case (55) should be replaced by | r ( z ) | ^ 1, z e ( - ia, ia), a > 0. Then the stability of the scheme in 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 gives convergence results for the fully discrete method. We also will state and prove a 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 l

Then if

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

)rtlll ^ c | l

HI0rt 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 above estimate. For gênerai £, if we take k = O(h), the bound in the theorem will be 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 addition suppose that E^ = rh E ( 0 ) and B°h =wh B(0). Then B£ e V^ for each n. In addition the following error estimate holds for 0 =S tn ^ T provided E and B are smooth enough :

vol. 29, n° 2, 1995

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190

SUD

[ff

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

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

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 since

each n. As we have seen if B^ = B^ + B^

€6A(0,B^)r = 0 or

ti-O-

with

^ then Bh e

^ for then

(62) Therefore, if we write

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

(22)

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

k²CA-^l(q²r^k+^l-p²r^h)eV^rk. (64) Using (58) and (62)-(64) we get :

/E" +

' \ / o \

^/E^W⁺¹

\

^/E"⁺¹

\

x) ⁺ \^ ^x •tpi^AfrU ) + f t ^ i ™ - * ' ) - ^ -*

( 6 5 )

This équation implies that

B r

U

= 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 the fact that since the mesh is quasi-uniform we may estimate

\\Kh yll! *£ Ch~ l\\\\y\\\, V y e U^' ° x V^ . a We start the proof of Theorem 5.1 by proving a consistency estimate for the fully discrete scheme. To do this we let JJh and Ph be the projections defined in (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 can write

(

\ / \ / n \ / M \

n \ / K« +1 e« \ / Y \ / y \

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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 study the 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 second équation of (8) we have that for \j/A e U£°

M2 AN Modélisation mathématique et Analyse numérique Mathematica! Modelling and Numerical Analysis

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where Ph is the ( L2( 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,(t

ⁿ

) + C E , ( O - CA; > r

^h

- /) E , ( O - p

^fc

B„(/

^B

) - CA;

^X

r

^h

= 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. Using the 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)

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

LEMMA 5.4: For any ¥ e U j ° x V j ,

MSfcVIII. (81) Proof : F r o m (42) w e h a v e that %h has pure imaginary eigenvalues and since the rational function r satisfies (55) w e have t h e desired resuit. D

Proof [of T h e o r e m 5.1] : Let us recall the définition

*•©©

U s i n g (69)-(73), ( 7 4 ) , (71), (72) and (67) w e 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 of these is estimated separately.

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

sup \\Et{s)\\Hr^,Q). (83) M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal An al y si s

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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"1I I V l l i r - (O) - (85) In view of the fact that the finite element interpolant approximates a function to O(hr) in the //(curl ; Q) norm, one might hope to improve (85) with hr ~ 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 F

fA+.] Il dtv + ' lliî(O) *e t v t +.] Il

\ " t ' A . . ] Il dtv + l WLHQ) * • [»..t. • il II

Iterating this estimate w e prove that

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

(

l l y t i j , M I hv + 1R II sup r v + r 0 ) + sup F - T T T ( * )

(

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

sup

^{a y +}

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

(27)

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

4 \ ® \ % 2 h 0 n ) ) . (87)

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

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

+ sup | | ^ n * ( * ) | | ^ V (88)

Q) s * rf',] II d tV + l H L ; - I ( / / Use of (87) in (88) complètes the estimate. D

REFERENCES

[1] J. ADAM, A. SERVENIÈRE, J. NÊDÉLEC and P. RAVIART, 1980, Study of an implicit schcrne 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 discrete approximations for second order hyperbolic équations, RAÏRO Anal. Numen, 13, 75-100.

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

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

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

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

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

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

M² AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

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[9] M. KRÎZEK and P. NEITTAANMÂKI, 1989, On time-harmonic Maxwell équations with 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 of inhomogeneous scatterers using edge-based finite element method in frequency and time domains, Radio Science, 28, 1181-1193.

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