Numerische Mathematik

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1 Numer. Mat. ( : Numerisce Matematik c Springer-Verlag 1999 Electronic Edition Fully discrete finite element approaces for time-dependent Maxwell s equations P. Ciarlet, Jr 1, Jun Zou 2, 1 Ecole Nationale Supérieure des Tecniques Avancées, 32, boulevard Victor, F Paris Cedex 15, France; ciarlet@ensta.fr 2 Department of Matematics, Te Cinese University of Hong Kong, Satin, N.T., Hong Kong; zou@mat.cuk.edu.k Received February 3, 1997 / Revised version received February 27, 1998 Summary. A fully discrete finite element metod is used to approximate te electric field equation derived from time-dependent Maxwell s equations in tree dimensional polyedral domains. Optimal energy-norm error estimates are acieved for general Lipscitz polyedral domains. Optimal L 2 -norm error estimates are obtained for convex polyedral domains. Résumé. On résout, dans un domaine polyédrique, les équations de Maxwell temporelles. Une métode par éléments finis discrète en temps et en espace est proposée pour calculer le camp électrique. Une estimation d ordre optimal est obtenue pour l erreur en norme-énergie dans le cas général. Pour la norme L 2, on obtient une estimation optimale dans le cas d un polyèdre convexe. Matematics Subject Classification (1991: 65N30, 35L15 1. Introduction Many problems in sciences and industry involve te solutions of Maxwell s equations, for example, problems arising in plasma pysics, microwave devices, diffraction of electromagnetic waves. In tis paper, we are interested in te numerical solution of time-dependent Maxwell s equations in a bounded polyedral domain in tree dimensions. In te literature, one can find a great deal of work on numerical approximations to time-dependent Maxwell s Te work of tis autor was partially supported by Hong Kong RGC Grant No. CUHK 338/96E Correspondence to: J.Zou page 193 of Numer. Mat. ( :

2 194 P. Ciarlet, Jr, J. Zou equations and also analyses on te convergence of numerical scemes for stationary Maxwell s equations and related models. We refer readers to Raviart [21], Assous et al [5], Hewett-Nielson [16], Degond-Raviart [11], Ambrosiano-Brandon-Sonnendrücker [2] and Ciarlet-Zou [8], etc. But to our knowledge, it seems tat tere are few existing works on te convergence analysis for semi-discrete or fully discrete numerical metods for te timedependent Maxwell systems. In [18], Monk obtained error estimates for a semi-discrete finite element approximation to te time-dependent Maxwell s equations using Nédélec s elements, from wic our current paper was initiated. Furtermore, in [17] Makridakis-Monk proposed a fully discrete finite element sceme and obtained te error estimates under strong regularities on te solutions. Tis sceme involves solving coupled non-symmetric and indefinite linear algebraic systems of bot electric and magnetic fields. Te purpose of te current paper is to analyse te convergence of a simple fully discrete finite element sceme for te electric field equation derived from Maxwell s equations by eliminating te magnetic field. Te sceme is a fully discrete version of te semi-discrete sceme studied in [18], and it is constructed in a way tat involves only solving a symmetric and positive definite linear algebraic system. One of our major interests ere is to investigate te convergence order of te fully discrete sceme witout making use of strong regularities on te solutions, wic is certainly of practical importance. Under appropriate assumptions on te regularity of te continuous solutions, we derive for te concerned fully discrete sceme te optimal energy-norm error estimates for general polyedral domains and optimal L 2 -norm error estimates for convex polyedral domains. We now introduce te Maxwell s equations to be considered in te paper. Let Ω be a bounded Lipscitz continuous polyedral domain in R 3, E(x,t and H(x,t te electric and magnetic fields respectively. Ten Maxwell s equations can be formulated as follows: (1 (2 εe t + σe curl H = J in Ω (0,T, µh t + curl E =0 in Ω (0,T, were ε(x and σ(x are te dielectric constant and te conductivity of te medium respectively, wile µ(x and J(x,t are te magnetic permeability of te material in Ω and te applied current density respectively. Here, te subscript t denotes te time derivative. It is assumed tat tese coefficients are piecewise smoot, real, bounded and positive, tat is, tere exist ε 0 > 0 and µ 0 > 0 suc tat, for all x Ω, (3 ε 0 ε(x, µ 0 µ(x, and 0 σ(x. Moreover, tese coefficients ε(x, µ(x and σ(x may be discontinuous. We assume tat te boundary of te domain Ω is a perfect conductor, tat page 194 of Numer. Mat. ( :

3 Fully discrete scemes for Maxwell s equations 195 is, (4 E n =0 on Ω (0,T. We supplement Maxwell s equations wit te initial conditions E(x, 0 = E 0 (x and H(x, 0 = H 0 (x, x Ω. Instead of solving te coupled system (1-(2 wit bot te electric and magnetic fields as unknowns, we eliminate te magnetic field H, by taking te time derivative of (1 and using (2, to obtain te second order electric field equation: (5 εe tt + σe t + curl ( 1 µ curl E =J t, in Ω (0,T, wit te boundary condition still being (4 but te previous initial conditions being replaced by (6 E(x, 0 = E 0 (x and E t (x, 0 = E 1 (x, were E 1 (x =ε 1 (J(x, 0 + curl H 0 (x σ(xe 0 (x. Remark 1.1 We ave implicitly assumed tat te electromagnetic field is generated by a current wit density J, witout any carge density: i.e. te medium is locally electrically neutral, and div J =0. In te more general case, te carge conservation equation reads: ρ t + div J =0, were ρ is te carge density. Terefore, if σ =0, we derive from (1 and (2 tat div (εe = ρ, and div (µh = 0, wen tese relations old for te initial data. In tis case, one may consider a saddle-point approac, like in Raviart [21] or Ciarlet-Zou [8], were Darwin s model of approximation to Maxwell s equations was studied. We end tis section wit te introduction of some notations used in te paper. We define H(div ; Ω ={v (L 2 (Ω 3 ; div v L 2 (Ω}, H(div 0; Ω ={v H(div ; Ω; div v =0}, H(curl ; Ω ={v (L 2 (Ω 3 ; curl v (L 2 (Ω 3 }, H α (curl ; Ω ={v (H α (Ω 3 ; curl v (H α (Ω 3 }, H 0 (curl ; Ω ={v H(curl ; Ω; v n =0on Γ }, page 195 of Numer. Mat. ( :

4 196 P. Ciarlet, Jr, J. Zou were α is a nonnegative real number. H(div ; Ω, H(curl ; Ω and H α (curl ; Ω are equipped wit te norms ( 1/2, v 0,div = v div v 0 2 ( 1/2, v 0,curl = v curl v 0 2 ( 1/2. v α,curl = v 2 α + curl v α 2 Here and in te sequel of te paper, 0 will always mean te (L 2 (Ω 3 - norm (or L 2 (Ω-norm if only scalar functions are involved. And in general, we will use α and α to denote te norm and semi-norm in te Sobolev space (H α (Ω 3 (or H α (Ω if only scalar functions are involved. We refer to Adams [1] and Grisvard [15] for more details on Sobolev spaces. C will always denote a generic constant wic is independent of bot te time step τ and te finite element mes size. 2. Fully discrete finite element scemes We consider discretizing te electric field Caucy problem (5-(6 by te implicit backward difference sceme in time togeter wit Nédélec s finite elements in space. Let us first triangulate te space domain Ω and assume tat T is a sape regular triangulation of Ω wit a mes size made of tetraedra. An element of T is denoted by K, and te diameters of K and its inscribed ball are denoted by K and ρ K respectively. As usual, we let = max K T K. As te triangulation is sape regular, we ave K /ρ K C (cf. Ciarlet [6]. We ten introduce te following Nédélec s H(curl ; Ω-conforming finite element space V = {v H(curl ; Ω; v K (P 1 3, K T } were P 1 is te space of linear polynomials. It was proved in Nédélec [20] tat any function v in V can be uniquely determined by te degrees of freedom in te moment set M E (v on eac element K T. Here M E (v is defined as follows: M E (v ={ (v τ qds; q P 1 (e on any edge e of K}, e were τ is te unit vector along te edge e. From [3], Lemma 4.7, we know tat te integrals required in te definition of M E (v make sense for any v X p (K, wit p>2, were X p (K ={v (L p (K 3 ; curl v (L p (K 3 ; v n (L p ( K 3 }. page 196 of Numer. Mat. ( :

5 Fully discrete scemes for Maxwell s equations 197 Tus we can define, for any v H 1/2+δ (curl ; Ω 3 wit δ>0 (wic implies tat curl v (L p δ(k 3 and v (L p δ( K 3 for some p δ > 2 wic depends on δ, an interpolation Π v of v suc tat Π v V and Π v as te same degrees of freedom (defined by M E (v asv on eac K T. In order to take te boundary condition E n =0on Ω into account, we define a subspace of V : V 0 = {v V ; v n =0 on Ω}. Tis can be done simply by zeroing te degrees of freedom wic correspond to te boundary edges. Next we divide te time interval (0,T into M equally-spaced subintervals by using nodal points 0=t 0 <t 1 < <t M = T wit t n = nτ, and denote te n-t subinterval by I n =(t n 1,t n ]. For a given sequence {u n } M n=0 L2 (Ω or (L 2 (Ω 3, we introduce te first and second order backward finite differences: τ u n = un u n 1 τ, τ 2 u n = τ u n τ u n 1. τ For a continuous mapping u :[0,T] L 2 (Ω or (L 2 (Ω 3, written as u C(0,T;(L 2 (Ω 3 subsequently, we define u n = u(,nτ for 0 n M. Using te above notation, our fully discrete finite element approximation to te electric field equations (5-(6 is formulated as follows: (7 E 0 = Π E 0, E 0 E 1 = τπ E 1, and for n =1, 2,,M, find E n V 0 suc tat (8 (ε τ 2 E n, v+(σ τ E n, v+(1 µ curl En, curl v =( τ J n, v, v V 0. Obviously, for eac n =1, 2,,M, it is clear tat, by Lax-Milgram teorem, te system (8 as a unique solution E n as its left-and side defines a symmetric positive definite bilinear form in H(curl ; Ω wit respect to E n. In addition, as (8 is symmetric and positive definite, it can be solved by te well-known conjugate gradient metod. Remark 2.1 Instead of te first order backward difference in time used in te fully discrete sceme (7-(8, one can also use some second order difference page 197 of Numer. Mat. ( :

6 198 P. Ciarlet, Jr, J. Zou approximation in time, e.g. te Crank-Nicolson sceme. In tis case, te wole discrete system can be taken as te following: (9 E 0 = Π E 0, E 1 E 1 =2τΠ E 1, and for n =0, 1,,M 1, find E n+1 V 0 suc tat (εδτ 2 E n, v+(σδ 2τ E n, v+(1 µ curl Ēn, curl v =(δ 2τ J n, v, (10 v V 0. were δτ 2 u n =(u n+1 2u n +u n 1 /τ 2, ū n =(u n+1 +u n 1 /2, δ 2τ u n = (u n+1 u n 1 /(2τ. Note tat te sceme preserves te symmetry and te positive definiteness. Te first unknown E 1 can be solved by using te initial approximation in (9 and (10 for n =0, and te resultant linear system is also symmetric and positive definite. Wit tis sceme we can acieve similar convergence results as obtained in te paper, see Remark Interpolation properties Tis section is devoted to some basic approximation properties of te finite element interpolant Π defined in Sect. 2, wic will be needed in te later error estimates for te finite element sceme (7-(8. First of all, we know te following properties of Π : for any u (H 2 (Ω 3, (11 u Π u 0 C 2 u 2, wile for any u H 1 (curl ; Ω,weave (12 curl (u Π u 0 C curl u 1. Te estimate (11 can be found in Girault [13] (Teorem 3.1 and Nédélec [20] (Proposition 3. Te estimate (12 was proved by Monk [18] (Lemma 2.3. Te estimates (11 and (12 stand for functions wic are appropriately smoot, i.e. for functions in (H 2 (Ω 3 or H 1 (curl ; Ω. But usually te solutions of te Maxwell system considered in te paper may not ave suc kind of regularity, especially wen te domain Ω is not convex and only Lipscitz continuous. Next we are going to present some approximation properties of te interpolant Π under weak assumptions on regularity. We first sow a similar result to (12 but for te L 2 -norm. Comparing wit te estimate (12 for te curl operator, we lose one error order. Similar results were obtained in [12] for a different finite element (see Remark 3.3. page 198 of Numer. Mat. ( :

7 Fully discrete scemes for Maxwell s equations 199 Lemma 3.1 We ave u Π u 0 C u 1,curl, u H 1 (curl ; Ω. Te proof of te lemma is omitted, since it can be inferred from tat of Lemmas 3.2 and 3.3 (see [9] for a detailed proof. Lemma 3.2 We ave, for 1/2 <α 1, u Π u 0 C α u α,curl, u H α (curl ; Ω. Remark 3.1 α > 1/2 is needed for te definition of te moments in M E (v,φ. Proof. For any element K T, let x = B K ˆx + b K be te affine mapping between K and te reference element ˆK, and we define (cf. Nédélec [20], (13 u(x =(B K 1 û(ˆx or û(ˆx =B Ku(x, were B K is te transpose of te matrix B K. Let ˆΠ be te interpolant on te reference element ˆK, ten (14 u Π u 2 L 2 (K B K (B K 1 2 û ˆΠû 2 L 2 ( ˆK. Trougout te paper, A means det(a for any square matrix A. Let us now bound û ˆΠû L 2 ( ˆK. For tat, let ê (respectively ˆF beany edge (respectively face of ˆK. Forp>2and p suc tat 1/p +1/p =1, on any edge ê of ˆK we define (15 ˆv Mê = sup ˆφ P 1 (ê 3 Mê(ˆv, ˆφ ˆφ W 1 1/p,p (ê were Mê(ˆv, ˆφ = ê (ˆv ˆτ ˆφds. Using te norm equivalence in finite dimensional spaces, we ave ˆΠû L 2 ( ˆK C ˆΠû Mê = C û Mê ê ˆK ê { ˆK C ĉurl û L p ( ˆK + ˆF } û ˆn L p ( ˆF, ˆK were te last inequality is obtained by integration by parts and te standard extension and lifting tecniques (cf. Lemma 4.7 of [3]. Tis implies û ˆΠû { L 2 ( ˆK C ĉurl û L p ( ˆK + û L 2 ( ˆK + ˆF } û ˆn L p ( ˆF ˆK (16 C { } ĉurl û H α ( ˆK + û H α ( ˆK. page 199 of Numer. Mat. ( :

8 200 P. Ciarlet, Jr, J. Zou As te left and side does not cange wen replacing û by û plus any constant, we ave û ˆΠû { ĉurl } L 2 ( ˆK C û H α ( ˆK + inf û + ˆp ˆp P 0 ( ˆK H 3 α ( ˆK. Note { ŵ H α ( ˆK = ŵ(ˆx ŵ(ŷ 2 } 1/2 ˆK ˆK ˆx ŷ 3+2α dˆxdŷ, it is clear tat ŵ + ˆp H α ( ˆK = ŵ H α ( ˆK for all ˆp P 0( ˆK 3. From tis point, one can easily adapt te proof of Teorem 14.1 in [6] to obtain te norm equivalence in te quotient space H α /P 0. Ten one as û ˆΠû { ĉurl } (17 L 2 ( ˆK C û H α ( ˆK + û H α ( ˆK. Tere remains to bound te rigt-and side in (17. Noting tat x y B K B 1 K (x y, we deduce û 2 H α ( ˆK B K 5+2α B 1 K 2 u 2 H α (K. Similarly we ave (see [9] for details ĉurl û 2 L 2 ( ˆK C B K 4 B 1 K curl u 2 L 2 (K, and ĉurl û 2 H α ( ˆK C B K 7+2α B 1 K 2 curl u 2 H α (K. Tis wit (17 sows (for B K small û ˆΠû 2 L 2 ( ˆK C max( B K 5+2α B 1 K 2, B K 4 B 1 K u 2 H α (curl ;K. Using te bounds on B K and te sape regularity of T, we get from (14 tat u Π u 2 L 2 (K C2α K u 2 H α (curl ;K. Remark 3.2 Te following lemma is an improvement on te results obtained in Nédélec [20] (Propositions 1 and 2 and Monk [18] (Lemma 2.3, were only integers α 1 were considered. Lemma 3.3 For 1/2 <α 1, we ave curl (u Π u 0 C α curl u H α (Ω, u H α (curl ; Ω. page 200 of Numer. Mat. ( :

9 Fully discrete scemes for Maxwell s equations 201 Proof. We follow Nédélec [20] for te notation used below. Let Θ be te H(div 0; Ω-conforming space of degree 0: Θ = {v H(div 0; Ω; v K (P 0 3, K T }, and let r be te corresponding interpolant to Θ wit te moment set: M F (v ={ (v n qdσ; q P 0 (F on any face F of K}. F Tere are four degrees of freedom attaced to tis finite element, but as div v =0by definition, teir sum (wit q =1 is equal to 0. We can sow (18 r (curl u = curl (Π u. Indeed, curl (Π u (P 0 3, div (curl (Π u=0and F curl (Π u n dσ = = = F F F curl F (Π u dσ = Π u τds F u τds= curl F (u dσ curl u n dσ. Hence (19 curl (u Π u 0 = (curl u r (curl u 0, tis wit te following result (20 gives te lemma. Next, we sow tat for any element K and 1/2 <α 1, F (20 w r w L 2 (K C α w H α (K, w H(div 0; Ω H α (Ω 3. To prove tis, we replace te degrees of freedom in M F (v (cf. [19] or [20] by { 2 (v nqdσ; q K (P 0 3, K T }. B K F Ten te following transformation x = B K ˆx + b K, w(x =B K ŵ(ˆx, preserves te interpolation and divergence, i.e. ˆrŵ(ˆx =r w(x, div ŵ(ˆx = div w(x, were ˆr is te reference interpolant on ˆK. page 201 of Numer. Mat. ( :

10 202 P. Ciarlet, Jr, J. Zou Now consider te moment M ˆF (ŵ, ˆφ = ˆF (ŵ ˆn ˆφdˆσ. Noting tat div w =0implies div ŵ =0, we ave by integration by parts M ˆF (ŵ, ˆφ = div ŵ ˆφ dˆx + ŵ grad ˆφ dˆx ˆK ˆK = ŵ grad ˆφ dˆx (21 ˆK C ŵ L p ( ˆK ˆφ W 1 1/p,p ( ˆF were p is again te conjugate number of p and ˆφ te extension by zero from W 1 1/p,p ( ˆF into W 1 1/p,p ( ˆK combined wit a lifting operator from W 1 1/p,p ( ˆK onto W 1,p ( ˆK. Using (21, we can bound ŵ M ˆF = sup ˆφ (P 0 ( ˆF 3 M ˆF (ŵ, ˆφ ˆφ W 1 1/p,p ( ˆF by ŵ L p ( ˆK. Tis wit te norm equivalence in finite dimensional spaces gives ˆrŵ L 2 ( ˆK C ˆF ˆK ˆrŵ M ˆF = C ˆF ˆK ŵ M ˆF tat implies (22 C ŵ L p ( ˆK 3 C ŵ H α ( ˆK, ŵ ˆrŵ L 2 ( ˆK C ŵ H α ( ˆK. As replacing ŵ by ŵ plus any constant does not cange te left and side, we come to ŵ ˆrŵ L 2 ( ˆK C ŵ H α ( ˆK. Tus we finally derive w r w 2 L 2 (K B K 2 B K ŵ ˆrŵ 2 L 2 C2α ( ˆK K w 2 H α (K, tis proves (20. Remark 3.3 All te results of tis paper are also valid for certain oter first order, H(curl ; Ω-conforming, Nédélec s elements, e.g. te element defined in [19], i.e. eac element function as te form v K R 1 (K = {a K + b K x, (a K, b K R 6 }, wit te related degrees of freedom. Te crucial step for te validity is to establis Lemmas for tis element. In fact, Lemma 3.1 was establised in ([12], Teorem 3.2. Lemmas can be extended to include tis first order element by means of similar page 202 of Numer. Mat. ( :

11 Fully discrete scemes for Maxwell s equations 203 tecniques as used in te present paper. Note, in addition, tat iger order finite elements are not considered ere as we do not assume, for te solutions, a stronger regularity tan H α (curl ; Ω wit 1/2 <α 1 in Subsect. 4.1 (energy-norm error estimates, or tan H 2 (Ω in Subsect. 4.2 (L 2 -norm error estimates. 4. Finite element error estimates We are now going to derive te error estimates for te fully discrete finite element sceme (7-(8 bot in te energy-norm and te L 2 -norm. Trougout tis section, E n and E n will denote te solutions of te electric field equations (5-(6 and te finite element approximation (8 at time t = t n. For te error analysis, we need te solution function E to be defined also in te interval [ 2τ,T] in terms of te time variable t. Tis can be done by extending E wit some regularity from te time interval [0,T] to te interval [ 2τ,T]. So we sall always implicitly assume tat E is well defined in terms of time variable t on te interval [ 2τ,T]. Furtermore, to acieve te optimal energy-norm error estimates for te concerned fully discrete finite element sceme, we introduce an important projection operator P : H 0 (curl ; Ω V 0 defined by (23 a(p u, v =a(u, v, v V 0 were a(u, v is te scalar product associated wit 0,curl. Obviously, P is well-defined in H 0 (curl ; Ω. By te definition of te projection P in (23, we easily see tat, for α>1/2, u P u 0,curl u Π u 0,curl, (24 u H 0 (curl ; Ω H α (curl ; Ω. Later on, we will need te following identity k k (25 (a m a m 1 b m = a k b k a 0 b 0 a m 1 (b m b m 1 m=1 m=1 and te following estimates for B = H 1 (curl ; Ω or B =(H α (Ω 3 wit α 0, (26 (27 τ u n 2 B 1 τ 2 τ u n 2 B 1 τ (28 τ u n t τ 2 u n 2 B Cτ t n t n 1 u t (t 2 B dt, t n t n 2 u tt (t 2 B dt, t n t n 2 u ttt (t 2 B dt, u H 1 (0,T; B, u H 2 (0,T; B, u H 3 (0,T; B. page 203 of Numer. Mat. ( :

12 204 P. Ciarlet, Jr, J. Zou 4.1. Energy-norm error estimates Tis subsection is devoted to te estimate on te energy-norm error for E n E n. For te purpose, we first analyse te error ηk = Ek P E k, for 1 k n. Once we ave estimates for η n, we can easily get te error estimates for E n E n by te triangle inequality, te projection properties (24 and te interpolation properties discussed in Sect. 3. We conduct our analysis only for te constant coefficients case, i.e., we assume ε(x, µ(x and σ(x are all constants. It is straigtforward to extend te analysis to te non-constant or elementwise constant case by simply keeping tese coefficients inside te integrals or norms and bounding tem by taking teir maximum or minimum values if necessary. To analyse η k, we multiply te equation (5 by v/τ V 0 and integrate ten te resultant over Ω in space and over I k in time to obtain ε ( τ E k t, v+σ( τ E k, v+ 1 ( curl E dt, curl v =( τ J k, v, τµ I k (29 v V 0. Now subtracting (29 from (8 and making some rearrangements, we ave ε ( τ 2 η k, v+ 1 µ (curl ηk, curl v+σ( τ η k, v ( ( = ε τ (E k t τ P E k, v + σ τ (E k P E k, v + 1 ( curl (E P E k dt, curl v, v V 0 τµ. I k Ten taking v = τ τ η k = ηk ηk 1 above and using a(a b a 2 /2 b 2 /2, for any real numbers a and b, yield ( ε στ τ η k τ η k 2 0 ε 2 τ η k ( 1 + 2µ curl ηk curl ηk 1 2µ 2 0 ( τσ ( τ E k P τ E k, τ η k ( +τε τ (E k t τ E k, τ η k ( +τε τ 2 E k P τ 2 E k, τ η k + 1 ( curl (E E k dt, curl τ η k µ I k + τ ( curl (E k P E k, curl τ η k µ page 204 of Numer. Mat. ( :

13 Fully discrete scemes for Maxwell s equations 205 (30 : 5 (I i. i=1 Next, we will estimate (I i for i =1, 2, 3, 4, 5 one by one. First for (I 1, using Caucy-Scwarz inequality, we ave (I στ τ η k στ τ E k P τ E k ( 2 στ τ η k Cστ τ E k Π τ E k 2 0,curl (by ( στ τ η k Cστ 2 τ E k 2 1,curl (by (12 and Lemma στ τ η k Cσ I 2 E t 2 1,curl dt (by (26. k For te estimation of (I 2, by writing τ ( k into te integral of form I ( k t dt and using Caucy-Scwarz inequality, we easily come to (I τε τ η k Cετ 2 t k t k 2 E ttt 2 0 dt. Te estimate for (I 3 is acieved using te same tecnique as used for (I 1, (I ετ τ η k Cε 2 I k E tt 2 1,curl dt. To analyse (I 4, we use Green s formula and te boundary condition to derive (I 4 = 1 ( curl curl (E E k, τ η k dt µ I k = 1 t k ( curl curl E t, τ η k dt dt, µ ten by Caucy-Scwarz inequality we obtain I k (I τε τ η k τ 2 t 2εµ 2 I k curl curl E t 2 0 dt. Finally, we estimate (I 5. By te definition of P in (23, we ave (I 5 = τ µ ( curl (E k P E k, curl τ η k = τ ( E k P E k, τ η k, µ page 205 of Numer. Mat. ( :

14 206 P. Ciarlet, Jr, J. Zou ten applying te Caucy-Scwarz inequality and (12, (24 and Lemma 3.1, we come to (I 5 τ µ Ek P E k 0 τ η k 0 τ µ 2 τ η k Cτ 2 E k 2 1,curl. Tis completes all te estimates for (I i in (30. Now summing bot sides in (30 over k =1, 2,,n and making use of te previous estimates for (I i (1 i 5, lead to ε 2 τ η n µ curl ηn 2 0 Cm 0 (E(τ d(η 0 n ( (31 +Cτ τ η k curl η k 2 0, were C is a constant depending on te coefficients ε, σ and µ, and m 0 (E is an a priori bound of E of te following form m 0 (E = max 0 t T E(t 2 1,curl + + T τ E ttt 2 0 dt, T 0 ( E tt 2 1,curl + curl curl E t 2 0 dt wile d(η 0 is te initial error d(η 0 = ε 2 τ η µ curl η0 2 0, wic can be analysed as follows: First by te definitions of E 0 and te projection P,weave curl η 0 0 = curl (Π E 0 P E 0 0 = curl P (Π E 0 E 0 0 Π E 0 E 0 0,curl C E(0 1,curl. Ten for te first term in d(η 0, by definition of η0, E0 know and E 1,we τ η 0 = τ 1 (τπ E t (0 P E(0 + P E( τ = τ 1 P (E( τ E(0 + τe t (0 + P (Π E t (0 E t (0, using te property of te projection P, we derive τ η 0 0 τ 1 E( τ E(0 + τe t (0 0,curl + Π E t (0 E t (0 0,curl Cτ sup E tt 1 + C E t (0 1,curl. ( τ,0 page 206 of Numer. Mat. ( :

15 Fully discrete scemes for Maxwell s equations 207 Terefore, we get te estimates for te initial error d(η 0 : d(η 0 Cτ2 max 1 n M sup ( τ,0 E tt C 2 ( E(0 2 1,curl + E t(0 2 1,curl. Ten substituting tis into (31 and applying te well-known discrete Gronwall s inequality, we conclude tat ( τ (E n P E n curl (E n P E n 2 0 Cm 0 (E(τ Finally, applying te triangle inequality to τ E n En t =( τ E n P τ E n +(P τ E n τ E n +( τ E n E n t and E n En =(E n P E n +(P E n E n, we ave proved te following energy-norm error estimates Teorem 4.1 Let E and E n be te solutions of te electric field equations (5-(6 and te finite element approximation (7-(8 at time t = t n, respectively. Assume tat E H 2 (0,T; H 0 (curl ; Ω H 1 (curl ; Ω H 3 (0,T;(L 2 (Ω 3. Ten we ave ( τ E n En t curl (E n En 2 0 C (τ max 1 n M were C is a constant independent of bot te time step τ and te messize. Remark 4.1 Te error estimate in Teorem 4.1 is optimal bot in terms of time step size τ and mes size as we ave used only te H 1 (curl ; Ω- regularity in space and H 3 (0,T-regularity in time. If te solution E as no so muc regularity in space as in Teorem 4.1 (cf. Costabel [10], Assous et al [4], we ten ave te following weaker error estimates Teorem 4.2 Let E and E n be te solutions of te electric field equations (5-(6 and te finite element approximation (7-(8 at time t = t n, respectively. Assume tat for some 1/2 <α<1, E H 2 (0,T; H 0 (curl ; Ω H α (curl ; Ω H 3 (0,T;(L 2 (Ω 3. Ten we ave ( τ E n En t 2 0+ curl (E n En 2 0 C (τ 2 +τ 2 2(α 1 + 2α. max 1 n M page 207 of Numer. Mat. ( :

16 208 P. Ciarlet, Jr, J. Zou Proof. Te proof is almost identical to te one for Teorem 4.1 by replacing H 1 (curl ; Ω by H α (curl ; Ω, Lemma 3.1 by Lemmas , (12 by Lemma 3.3. Te only remaining term we ave to re-estimate is te term (I 4 in (30 as we now ave no regularity curl (curl E t in (L 2 (Ω 3 as used in tat proof. Instead we can bound te term as follows: by assumption we ave E H 1 (0,T; H α (curl ; Ω,socurl E H 1 (0,T;(H α (Ω 3 and ten curl (curl E H 1 (0,T;(H α 1 (Ω 3, were H α 1 (Ω is te dual space of H 1 α (Ω (note tat tis is true as, wen 0 < 1 α<1/2, H 1 α (Ω =H0 1 α (Ω. Terefore, by Green s formula and te inverse inequality we ave (I 4 = 1 curl curl (E E k, τ η k dt µ I k H α 1,H1 α = 1 t k curl (curl E t, τ η k µ I k t H α 1,H dt dt 1 α τ curl (curl E t µ H α 1 (Ω τ η k H 1 α (Ωdt I k Cτ α 1 τ η k 0 curl (curl E t H α 1 (Ωdt I k 1 4 τε τ η k Cτ 2 I 2(α 1 curl (curl E t 2 H α 1 (Ω dt. k Tat completes te proof of Teorem 4.2. Remark 4.2 It is easy to see tat if we take τ to be te same magnitude as, ten te error estimate in Teorem 4.2 is of optimal order, i.e. O( 2α, in te sense of te space regularity we ave used. Remark 4.3 Teorems can be extended to te Crank-Nicolson sceme (9-(10. Te major difference in proving te related teorems is to derive a similar equation as (29 and to coose an appropriate test function v. For te former, we may multiply equation (5 at time t = t n by v/τ V 0 and ten integrate te resulting equation over Ω. For te latter, we may coose te test function v = η n+1 η n 1 =(η n+1 η n +(ηn ηn L 2 -norm error estimates Tis subsection is dedicated to te derivation of te L 2 -norm error estimates. Te basic tecnique used ere is borrowed from Girault [13] and Monk [18], were Nédélec s finite element metods were applied for stationary Navier- Stokes equations and semi-discrete scemes wit Nédélec s finite elements for time-dependent Maxwell s equations, respectively. page 208 of Numer. Mat. ( :

17 Fully discrete scemes for Maxwell s equations 209 As usual, we assume te domain Ω is a convex polyedron in order to acieve te optimal L 2 -norm error estimates. If te domain Ω is a general Lipscitz polyedron, we ave te following L 2 -norm error estimates: for E H 3 (0,T;(L 2 (Ω 3 and E H 2 (0,T; H 0 (curl ; Ω H 1 (curl ; Ω, max 1 n M En En 0 C (τ +, wic can be obtained immediately by applying te triangle inequality to te relation n E n En =(Π E 0 E 0 +τ ( τ E k τ E k =(Π E 0 E 0 +τ n ( τ E k Ek t +τ n (E k t τ E k and te results in Teorem 4.1. A similar result, wen E is only in H 2 (0,T; H 0 (curl ; Ω H α (curl ; Ωand H 3 (0,T;(L 2 (Ω 3, for 1/2 <α<1, is max 1 n M En En 0 C α, by using Teorem 4.2 and taking τ to be of order O(. But if te domain Ω is convex and te solution E is more regular, we can expect iger convergence order. We now sow tis is possible. We will consider only te case tat te coefficients ε and µ are constants (we take 1 for simplicity and σ =0. We need te following decomposition (cf. Ciarlet [7] (32 L 2 (Ω 3 = M M were M and M are two spaces defined by and M = {v = z; z H 1 0 (Ω} M = {v L 2 (Ω 3 ;(v, z =0 z H 1 0 (Ω} = H(div 0; Ω. Te last equality can be proved directly from definition. As Ω is convex, we ave H 0 (curl ; Ω M H 1 (Ω 3 (see [14]. To introduce a discrete version of (32, we define a finite element space S = {z H 1 0 (Ω; z K P 2 K T }. An L 2 (Ω projection Q onto S will be needed in our later analysis, i.e. we define Q : L 2 (Ω S as follows (Q z, v =(z, v, z L 2 (Ω,v S. page 209 of Numer. Mat. ( :

18 210 P. Ciarlet, Jr, J. Zou Now we can introduce te following discrete decomposition (33 V 0 = M M were M and M are two spaces defined by M = {v = z ; z S } and M = {v V 0 ;(v, z =0 z S }. Note tat, by definition, M (resp. M is te kernel of te curl operator in H 0 (curl ; Ω (resp. V 0. Before discussing our main results in tis section, we introduce some operators wic are very useful in te later L 2 -norm error estimates. We first define an operator T. For any u M, T u H 0 (curl ; Ω M and satisfies (curl (T u, curl v =(u, v, v H 0 (curl ; Ω M. Foragivenu, T u can be regarded also as te solution, in variable w,tote following system (cf. Girault [13]: (34 curl (curl w = u in Ω, (35 div w =0 in Ω, (36 w n =0 on Γ. Te problem (34-(36 is a particular case of te system in (w,p: (37 curl (curl w + p = g in Ω, (38 div w =0 in Ω, (39 w n =0 on Γ, (40 p =0 on Γ, for a given g in (L 2 (Ω 3. According to Lemma 4.1 in [13], te system (37- (40 as a unique solution w H 0 (curl ; Ω H(div 0; Ω and p H0 1(Ω. In addition, te following properties old: w H 1 (Ω 3 and w 1 C curl w 0 C g 0. curl w H 1 (Ω 3 and curl w 1 C g 0. p 1 C g 0. Clearly, if g M, ten p =0: tis is te case wit g = u. Te discrete version T : M M of te operator T can be defined by (curl (T u, curl v =(u, v, u M, v M. We now state some properties of te operators T and T. Te results (i and (ii were establised by Monk in [18], so we focuse on te proof of (iii and (iv. Te property (iii is an improvement over te result in [18]. page 210 of Numer. Mat. ( :

19 Fully discrete scemes for Maxwell s equations 211 Lemma 4.1 Let Ω be a convex polyedron and suppose T u H 2 (Ω 3, for u H 0 (curl ; Ω H(div 0; Ω. Ten tere exists a constant C independent of and u suc tat (i curl ((T T u 0 C curl (T u 1, (ii curl (T u 0 C u 0, (iii (T T u 0 C 2 T u 2, (iv T 1/2 u 0 C u 0. Proof. Notice tat by using Weber s result [22], one easily gets tat curl w 0 defines a norm in H 0 (curl ; Ω H(div 0; Ω wic is equivalent to its canonical norm. Now, as p =0, w = T u is te solution to: (curl w, curl v =(u, v, v H 0 (curl ; Ω, and w = T u M also satisfies te discrete variational formulation: (curl w, curl v =(u, v, v V 0 H 0(curl ; Ω. Terefore, curl (w w 0 curl w 0 C u 0. We first prove te fourt inequality. We see T 1/2 u 2 0 =(T u, u =(w, u =(w w, u+(w, u. Define q H0 1 (Ω by ( q, µ =(w, µ, µ H0 1 (Ω. Note tat q and u belong to ortogonal subspaces of L 2 (Ω 3. Set v = w q: curl v = curl w, div v =0, v n =0on Γ. Terefore w q w H 0 (curl ; Ω H(div 0; Ω and w q w 0 C curl (w w 0 C curl w 0 C u 0. Also, w 0 C curl w 0 C u 0, wic proves (iv. Now we turn to te tird inequality: take any g in L 2 (Ω 3 and solve (37-(40. Ten (g, w w = (curl w, curl (w w+( p, w w = (curl (w w, curl (w w+( (p p, w w = (curl (w w, curl (w w+( (p p, w y (41 +( (p p, y w, w, y V 0, p S. page 211 of Numer. Mat. ( :

20 212 P. Ciarlet, Jr, J. Zou Set z = w y and split up z into z = q + v, wit q H0 1(Ω defined by ( q, µ =(z, µ, µ H0 1 (Ω. As usual v = z q is an element of H 0 (curl ; Ω H(div 0; Ω.Asz belongs to V 0, it as sufficient regularity for defining Π z = z. Terefore, it can also be split up into z = Π v + q, wit q S. Remark 4.4 To prove te splitting one can write Π ( q as q, ten use (18 wic leads to curl (Π ( q = 0: tus tere exists q H 1 (Ω suc tat Π ( q = q. Moreover, by definition, Π ( q is in V 0,so its restriction to any tetraedron K belongs to (P 1 3. Putting tese togeter sows tat q is an element of S. Now, te estimate of ( (p p, z proceeds as in Girault [13]: coose for p te H 1 0 -projection of p into S, ence (42 ( (p p, z p 1 Π v v 0. We are going to estimate Π v v 0. First, following [3], ˆv ˆΠ { ˆv L 2 ( ˆK C ĉurl ˆv L p ( ˆK + ˆv L 2 ( ˆK + ˆF } ˆv ˆn L p ( ˆF. ˆK But curl v = curl z and (13 preserves te curl. Terefore, ĉurl ˆv = ĉurl ẑ wic belongs to a finite dimensional space. Tus, ĉurl ˆv L p ( ˆK Ĉ ĉurl ˆv L 2 ( ˆK. Using te norm equivalence in te quotient space H 1 /P 0, we come to ˆv ˆΠ { ĉurl } ˆv L 2 ( ˆK C ˆv L 2 ( ˆK + ˆv H 1 ( ˆK. Tanks to te estimates wic bound te rigt-and side (cf. [9] and [6], it follows from (14 tat Tus v Π v L 2 (K C K v H 1 (K. v Π v 0 C v 1 = C curl v 0 C curl z 0 = C curl (w y 0. We ten ave from (42 tat ( (p p, z C p 1 { curl (w w 0 + curl (w y 0 }. page 212 of Numer. Mat. ( :

21 Fully discrete scemes for Maxwell s equations 213 Terefore, te last two terms in (41 are bounded by p 1 {C( curl (w w 0 + curl (w y 0 + w y 0 }, y V 0. By assumption, w belongs to H 2 (Ω: coosing y = Π w gives (43 w y 0 C 2 w 2, curl (w y 0 C curl w 1. And, wit te elp of (i, we obtain te final bound for te last two terms in (41, tat is C 2 p 1 w 2 C 2 g 0 w 2. To conclude, as w belongs to H 1 (curl ; Ω H 0 (curl ; Ω, we can coose w = Π w, and using (43 and (i, we ave te final bound for te first term in (41, (curl (w w, curl (w w C 2 g 0 w 2. Tis proves (iii. Now we are in a position to derive te main results in tis section, i.e. te L 2 -norm error estimate. By means of te decomposition (32, for any J(t L 2 (Ω, t (0,T, we can write (44 J = J 1 + z, J 1 M,z H 1 0 (Ω. Tis decomposition enables us to separate our error estimates into two parts, i.e. Teorems 4.3 and 4.4. Te general principles of te proofs for tese two teorems are similar to tose of te proofs for Teorems 4.1 and 4.2 in [18]. But as our sceme is fully discrete, a lot of tecnical details need to be treated newly. We will give only an outline for eac proof but refer to [9] for details. First we sow Teorem 4.3 Let E and E n be te solutions to (5-(6 and (7-(8. Assume tat J(t M for t (0,T, and tat E 0 and E 1 belong to M (H 2 (Ω 3. Moreover we assume tat E C 1 (0,T;(H 2 (Ω 3 C 2 (0,T;(H 1 (Ω 3 H 3 (0,T;(L 2 (Ω 3. Ten we ave E n E n 0 C (τ + 2 for n =1, 2,,M were te constant C is independent of τ and but may depend on te a priori bounds on E. page 213 of Numer. Mat. ( :

22 214 P. Ciarlet, Jr, J. Zou Proof. By te assumption, it is easy to derive E(t M. Using (33, we can write E k = Ek + z k wit Ek M and zk S, for 1 k n. Ten (45Ek E k =(Ek E k z k ηk zk, for 1 k n. It suffices to estimate te L 2 -norms for bot η n and zn. We first analyse z n. Taking v = τ z k M for any z k S in (8 and te Caucy-Scwarz inequality, we get τ z n 2 0 τ z n τ z But by ortogonality, we can write τ z =( τ E 0, τ z 0 =( τ E 0 τ E 0, τ z 0, from wic and te initial condition (7 we can sow (46 τ z 0 0 C( 2 E τ sup E tt (t 1. t ( τ,0 Similarly, we ave z 0 0 C 2 E 0 2. Using tis, (46 and te identity we derive tat n z n = z0 + τ τ z k, (47 z n 0 C 2 ( E E 0 2 +Cτ sup E tt (t 1. t ( τ,0 Next we estimate η k in (45. Taking v = T φ in (8 wit φ M, we ave, for 1 k n, ( 2 τ T E k,φ +(E k,φ =( τ T J k,φ, φ M. (48 A similar relation to (48 for te continuous solution E can be derived by multiplying (5 by v = τ 1 Tφwit φ M and ten integrating over I k in time and over Ω in space: (49 ( τ T E k t,φ+(ẽk,φ=( τ T J k,φ, φ M, were Ẽk = τ 1 I E(t dt. k Note for any φ M, we can write φ = φ + z 1 wit φ M and z 1 H0 1 (Ω. Using tis relation we obtain from (49 tat (50 ( τ T E k t,φ +(Ẽk,φ =( τ T J k,φ, φ M. page 214 of Numer. Mat. ( :

23 Fully discrete scemes for Maxwell s equations 215 Subtracting (48 from (50 and making some arrangements, we derive for any φ M tat (51 ( τ 2 T η k,φ +(η k,φ =((T T τ J k,φ +( τ 2 T E k T τ E k t,φ +(E k Ẽk,φ, ten taking φ = τ E k gives ( τ 2 T η k, τ η k +(ηk, τ η k =(T τ 2 η k, τ E k +(η k, τ E k +((T T τ J k, τ E k +(Ẽk E k, τ E k +( τ T E k t τ 2 T E k, τ E k. Note tat, (52 olds actually for any φ V 0 by ortogonality. Ten, adding te previous equation to te equation (52 wit φ = Π τ E k V 0, we get ( τ 2 T η k, τ η k +(ηk, τ η k =(T τ 2 η k, τ E k Π τ E k +(η k, τ E k Π τ E k +( (T T τ J k, τ E k Π τ E k +(Ẽk E k, τ E k Π τ E k +( τ T E k t τ 2 T E k, τ E k Π τ E k 5 : (I i, i=1 wic implies, using a(a b 1 2 a2 1 2 b2, for any a, b R, { 1 1/2 T 2 τ η k } 1/2 T 2 τ η k { ηk } 5 (52 2 ηk τ (I i. Te terms (I i, i =1,, 5 can be estimated one by one (cf. [9] for details. Wit tese estimates, we ten sum up (52 over k =1,,n and finally obtain T 1/2 τ η n η n 2 0 T 1/2 τ η η C(E(τ C(E(τ Tis wit (45 and (47 implies te desired estimate of Teorem 4.3. Now we turn to te second part of te L 2 -norm error estimate, i.e. we are going to prove i=1 page 215 of Numer. Mat. ( :

24 216 P. Ciarlet, Jr, J. Zou Teorem 4.4 Let E and E n be te solutions to (5-(6 and (7-(8. Assume tat J is in H 1 (0,T; M (H 2 (Ω 2, and tat E 0 and E 1 belong to M (H 2 (Ω 3. Moreover we assume tat Ten we ave E C 2 (0,T;(H 2 (Ω 3 H 3 (0,T;(L 2 (Ω 3. E n E n 0 C (τ + 2 for n =1, 2,,M were te constant C is independent of τ and but may depend on te a priori bounds on E. Proof. By te assumption, it is easy to get E(t M. SoweaveE(t = z(t wit z H 1 0 (Ω. Using (33, we can write Ek = Ek + z k wit E k M and zk S, for 1 k n. Note E 0 M, soe 0 = z 0 for some z 0 H0 1(Ω, tus we ave E0 = Π E 0 = Π z 0 = z 0 for some z 0 S, i.e. E 0 =0. Similarly we ave E 1 =0. By means of tese decompositions, we may write n t k E n E n =(E0 E 0 + (E t τ E k dt t k 1 =(E 0 Π E 0 + t k n t k t k 1 (z t τ z k dt n τ E k dt. t k 1 wic, wit te fact E(t = z(t, tus ( (z t τ z k T 1/2 0 τ E tt 0dt 2 + (z k t τ z k 0, 0 implies n n E n E n 0 τ (zt k τ z k 0 + τ τ E k 0 (53 ( +C 2 T 1/2. E τ E tt 0dt 2 It remains to estimate te first and second terms in te rigt side of (53. Let us first estimate te term τ E k 0. Taking v = τ E k M in (8, and using ortogonality and J(t = g(t for some g H0 1 (Ω, weget ( 2 τ E k, τ E k + (curl E k, curl τ E k =( τ g k, τ E k, 0 page 216 of Numer. Mat. ( :

25 Fully discrete scemes for Maxwell s equations 217 wic implies { τ E k τ E k { 1 2 curl Ek Ek } } τ( τ g k, τ E k. Summing up te above equations over k =1,,n, we obtain τ E n curl E n 2 0 2τ 2τ τ 2 n ( τ g k, τ E k n ( ( τ g k Q τ g k, τ E k n T τ E k C 4 g t 2 3dt. Tat means by Gronwall s inequality and J(t = g tat 0 (54 T τ E n curl E n 2 0 C 4 J t 2 2dt. 0 Next, we are going to estimate te term (z k t τ z k 0 in (53. Taking v = y k for any yk S in (29 and ten subtracting it from (8 yields using tis, we obtain ( τ (z k t τ z k, yk =0, yk S, 1 2 (zk t τ z k (zk 1 t τ z k τ( τ (zt k τ z k, (zk t y k. summing te equations over k =1,,n and using (25 gives for any y k S, (zt n τ z n 2 0 n (zt 0 τ z τ ( (zt k τ z k, τ (zt k y k +( (zt n τ z n, (zn t y n ( (z0 t τ z 0, (z0 t y 0. (55 page 217 of Numer. Mat. ( :

26 218 P. Ciarlet, Jr, J. Zou Now taking y k = Q τ z k, te terms in (55 can be bounded as follows: (z k t y k 2 0 Cτ t k t k 1 E tt 2 0dt + C 4 t k τ (zt k y k 2 0 Cτ E ttt 2 0dt + C 4 t k 1 and (zt 0 τ z 0 0 C 2 E 1 2. Substituting tese estimates into (55, we ave (56 (z n t τ z n 2 0 C(E(τ Teorem 4.4 ten follows from (53, (54 and (56. max E t 2 2dt, 0 t T max E tt 2 2, 0 t T Acknowledgements. Te autors wis to tank te two anonymous referees for many constructive comments. Te autors also tank Prof. Vivette Girault for many elpful discussions and for providing te improved error estimate (iii of Lemma 4.1. References 1. Adams, R.A. (1975: Sobolev spaces. Academic Press, New York 2. Ambrosiano, J.J., Brandon, S.T., Sonnendrücker, E. (1995: A finite element formulation of te Darwin PIC model for use on unstructured grids. J. Comput. Pysics 121, Amrouce, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in treedimensional nonsmoot domains. To appear in Mat. Met. Appl. Sci. 4. Assous, F., Ciarlet, Jr, P., Sonnendrücker, E.: Resolution of te Maxwell equations in a domain wit reentrant corners. To appear in RAIRO Mat. Modelling Numer. Anal. 5. Assous, F., Degond, P., Heintzé, E., Raviart, P.-A., Segré, J. (1993: On a finite-element metod for solving te tree-dimensional Maxwell equations. J. Comput. Pysics 109, Ciarlet, P. (1991: Basic error estimates for elliptic problems. In: P. Ciarlet and J.-L. Lions, eds., Handbook of Numerical Analysis, Volume II, pp Nort Holland 7. Ciarlet, Jr, P. (1993: A decomposition of L 2 (Ω 3 and an application to magnetostatic equations. Mat. Models Met. App. Sci. 3, Ciarlet, Jr, P., Zou, J. (1997: Finite element convergence for te Darwin model to Maxwell s equations. RAIRO Mat. Modelling Numer. Anal. 31, Ciarlet, Jr, P., Zou, J. (1996: Fully discrete finite element approaces for time-dependent Maxwell equations. Researc Report MATH (105, Department of Matematics, Te Cinese University of Hong Kong 10. Costabel, M. (1990: A remark on te regularity of solutions of Maxwell s equations on Lipscitz domains. Mat. Met. Appl. Sci. 12, Degond, P., Raviart, P.-A. (1992: An analysis of te Darwin model of approximation to Maxwell s equations. Forum Mat. 4, Dubois, F. (1990: Discrete vector potential representation of a divergence-free vector field in tree-dimensional domains: Numerical analysis of a model problem. SIAM J. Numer. Anal. 27, page 218 of Numer. Mat. ( :

27 Fully discrete scemes for Maxwell s equations Girault, V. (1988: Incompressible finite element metods for Navier-Stokes equations wit nonstandard boundary conditions in IR 3. Mat. Comp. 51, Girault, V., Raviart, P.-A. (1986: Finite element metods for Navier Stokes equations. Springer Verlag, Berlin 15. Grisvard, P. (1985: Elliptic problems in nonsmoot domains. Pitman Advanced Publising Program, Boston 16. Hewett, D.W., Nielson, C. (1978: A multidimensional quasineutral plasma simulation model. J. Comput. Pys. 29, Makridakis, C.G., Monk, P. (1995: Time-discrete finite element scemes for Maxwell s equations. RAIRO Mat. Modelling Numer. Anal. 29, Monk, P. (1992: Analysis of a finite element metod for Maxwell s equations. SIAM J. Numer. Anal. 29, Nédélec, J.-C. (1980: Mixed finite elements in IR 3. Numer. Mat. 35, Nédélec, J.-C. (1986: A new family of mixed finite elements in IR 3. Numer. Mat. 50, Raviart, P.-A. (1993: Finite element approximation of te time-dependent Maxwell equations. Tecnical Report GdR SPARCH #6, Ecole Polytecnique, France 22. Weber C. (1980: A local compactness teorem for Maxwell s equations. Mat. Met. Appl. Sci. 2, page 219 of Numer. Mat. ( :