DISCONTINUOUS GALERKIN DISCRETIZATIONS OF OPTIMIZED SCHWARZ METHODS FOR SOLVING THE TIME-HARMONIC MAXWELL S EQUATIONS

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1 Electronic Transactions on Numerical Analysis. Volume 44, pp , 205. Copyrigt c 205,. ISSN ETNA DISCONTINUOUS GALERKIN DISCRETIZATIONS OF OPTIMIZED SCHWARZ METHODS FOR SOLVING THE TIME-HARMONIC MAXWELL S EQUATIONS MOHAMED EL BOUAJAJI :, VICTORITA DOLEAN ;, MARTIN J. GANDER, STÉPHANE LANTERI :, AND RONAN PERRUSSEL Abstract. We sow in tis paper ow to properly discretize optimized Scwarz metods for te time-armonic Maxwell s equations in two and tree spatial dimensions using a discontinuous Galerkin (DG) metod. Due to te multiple traces between elements in te DG formulation, it is not clear a priori ow te more sopisticated transmission conditions in optimized Scwarz metods sould be discretized, and te most natural approac, at convergence of te Scwarz metod, does not lead to te monodomain DG solution, wic implies tat for suc discretizations, te DG error estimates do not old wen te Scwarz metod as converged. We present ere a consistent discretization of te transmission conditions in te framework of a DG weak formulation, for wic we prove tat te multidomain and monodomain solutions for te Maxwell s equations are te same. We illustrate our results wit several numerical experiments of propagation problems in omogeneous and eterogeneous media. Key words. computational electromagnetism, time-armonic Maxwell s equations, Discontinuous Galerkin metod, optimized Scwarz metods, transmission conditions. AMS subject classifications. 65M55, 65F0, 65N22. Introduction. Discontinuous Galerkin (DG) metods ave received a lot of attention over te last decade since tey combine te best of bot finite-element and finite-volume metods. Te approximation of eac field is done locally at te level of eac mes element by using a local basis of functions, and te discontinuity between neigboring elements is treated using a finite-volume flux. A ricer representation of te solution is given at te price of increasing te total number of degrees of freedom as a result of te decoupling of elements. Te literature on tese metods applied to different types of equations is ric, and we will focus on contributions concerning Maxwell s equations. A complete istorical introduction wit a large panel of references can be found in te milestone book on DG metods by Hestaven and Warburton [28]. Teoretical results on DG metods applied to te time-armonic Maxwell s equations ave been obtained by several autors. Most of tese use te second-order formulation of te Maxwell s equations. An alternative is to use te first-order formulation as in [25, 26, 27] based on te teory of Friedrics systems. In a large part of te literature on time-armonic problems, a mixed formulation is used (see [30, 35]), but DG metods for te non-mixed formulation, like interior penalty tecniques [5, 29] and local discontinuous Galerkin metods [5], ave also been studied. A numerical convergence study of discontinuous Galerkin metods based on centered and upwind fluxes and nodal polynomial interpolation applied to te first-order time-armonic Maxwell system in te two-dimensional case can be found in [0]. Like for all oter discretizations of te time-armonic Maxwell s equations, it is also difficult to solve linear systems obtained by DG discretizations wit iterative metods. Due to te indefinite nature of te problems, classical iterative solvers fail as in te Helmoltz Received September 0, 204. Accepted August 2, 205. Publised online on November 3, 205. Recommended by U. Langer. INRIA Sopia Antipolis-Méditerranée, Sopia Antipolis Cedex, France ({Moamed.El_bouajaji,Stepane.Lanteri}@inria.fr). Université de Nice Sopia-Antipolis, Laboratoire J.-A. Dieudonné, Nice, France (dolean@unice.fr). Section de Matématiques, Université de Genève, CP 64, 2 Genève, Switzerland (Martin.Gander@mat.unige.c). CNRS, Université de Toulouse, Laboratoire plasma et conversion d énergie, 307 Toulouse Cedex 7, France (perrussel@laplace.univ-tlse.fr). 572

2 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS 573 case [8]. Després defined in [8] a first provably convergent domain decomposition algoritm for te Helmoltz equation. Tis algoritm was extended to Maxwell s equations in [9]. Even better transmission conditions were proposed in [6, 7, 23] based on optimized Scwarz teory [9, 20] wit an application to te second-order Maxwell system in []. An entire ierarcy of optimized Scwarz metods for te first-order Maxwell s equations can be found in [] wit complete asymptotic results for te optimization. DG discretizations of optimized Scwarz metods for time-armonic Maxwell s equations were proposed first in [5]. In te sort proceedings paper [7], te autors proposed a different DG discretization of te transmission conditions for te TM formulation of Maxwell s equation in two spatial dimensions and stated an equivalence teorem of te decomposed DG solution wit te monodomain DG solution witout a proof. Te purpose of our manuscript is to prove tis teorem and also to present a consistent DG discretization for Maxwell s equations in tree spatial dimensions togeter wit an equivalence teorem wic is more involved to prove tan in te two-dimensional case. Classical finite-element based non-overlapping and non-conforming domain decomposition metods for te computation of multiscale electromagnetic radiation and scattering problems can be found in [3, 32, 33, 34, 36, 37]. Tey do not need any special treatment for te discretization of te optimized transmission conditions. For DG discretizations, owever, even for te Poisson equation, te discretization of transmission conditions needs to be done wit care [22, 24], and classical block Jacobi metods are not equivalent to classical Scwarz metods for DG discretizations [2]. Tis paper is organized as follows: in Section 2 we present te tree-dimensional timearmonic Maxwell s equations as a first-order system and introduce te notation for wat follows. In Section 3 we state te classical and optimized Scwarz algoritm at te continuous level for te first-order Maxwell system in 3D. In Section 4, we introduce a weak formulation for te first-order system and use a DG approximation to obtain discrete subdomain problems. We ten sow tat wile te DG discretization of te classical Scwarz metod is very natural, te optimized transmission conditions are more tricky to discretize, and we present for te tree-dimensional Maxwell s equations a consistent discretization of te transmission conditions, for wic we prove tat te monodomain and multidomain formulations are equivalent. Next we also prove te equivalence result for te two-dimensional TM formulation announced in te proceedings paper [7]. We finally provide in Section 5 results of several numerical experiments for bot omogeneous and eterogeneous propagation problems to illustrate te performance of te optimized Scwarz metods as solvers for DG-discretized Maxwell s equations. Section 6 contains a brief conclusion. 2. Te time-armonic Maxwell system. Te time-armonic Maxwell s equations in a omogeneous medium are given by (2.) iωεe curlh `σe 0, iωµh `curle 0, were te positive real parameter ω is te pulsation of te armonic wave, σ is te electric conductivity, ε is te electric permittivity, µ is te magnetic permeability, and te unknown complex-valued vector fields E and H are te electric and magnetic fields. In te omogeneous case, to simplify notation, we can rewrite equation (2.) as (2.2) i ωe curlh ` σe 0, i ωh `curle 0, were ω : ω? εµ and σ : σ a µ ε. Collecting te variables into one big vectorw : pe,hq, we can rewrite (2.2) as a first-order system, G 0 W `G x B x W `G y B y W `G z B z W 0,

3 574 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL were and wit G x : 03ˆ3 N x N T x 0 3ˆ3 j p σ G 0 : `i ωqi3ˆ3 0 3ˆ3, 0 3ˆ3 i ωi 3ˆ3 j j j, G y : 03ˆ3 N y Ny T, G 0 z : 03ˆ3 N z 3ˆ3 Nz T, 0 3ˆ3» N x : fi» 0 0 fl, N y : 0 0 fi» fl, N z : 0 0 fi 0 0fl For a general vectorn pn x,n y,n z q, we can define te matrices G n : 03ˆ3 j N n Nn T 0 3ˆ3, and N n :» 0 n z n y n z 0 n x n y n x 0 Te skew-symmetric matrixn n allows us to define te cross-product between a vector V and te vector n, fi fl. V ˆn N n V and n ˆV N T nv. Moreover, if te vector n is normalized, we also ave Nn 3 N n. Using tis notation, te matrices G l, wit l standing for tx,y,zu, are in fact G l G el, weree l, l,2,3, are te canonical basis vectors. We consider ere a total field formulation, tat is, we are interested in te unknown vector W W inc `W sc, werew inc represents te incident field andw sc represents te scattered field by an obstacle wit boundaryγ m or in an inomogeneous medium. Our goal is to solve te boundary-value problem wose strong form is given by G 0 W ` ÿ G l B l W 0, in Ω, lptx,y,zu pm Γm G n qw 0, on Γ m, pm Γa G n qpw W inc q 0, on Γ a. Here te matrices M Γm and M Γa are used for taking into account te boundary conditions of te problem imposed on te metallic boundary Γ m and te absorbing boundaryγ a, j j M Γm 03ˆ3 N n Nn N Nn T n and M 0 Γa G n T 0 3ˆ3 3ˆ3 0 3ˆ3 NnN T. n In wat follows we will use te matrices Gǹ and Gń, wic denote te positive and negative parts ofg n according to its diagonalization. We note tat G n Gǹ Gń, and te definition ofgǹ and Gń can be deduced from tose ofg n and G n by (2.3) Gń 2 pg n G n q and Gǹ 2 pg n ` G n q.

4 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS Continuous classical and optimized Scwarz algoritms. We decompose te computational domainωinto two non-overlapping subdomainsω andω 2. We denote byσte interface betweenω and Ω 2, byw j te restriction of W to te subdomain Ω j, and bynte unit outward normal vector toσpointing fromω toω 2. Scwarz algoritms compute at eac iteration step n 0,,2,... a new approximation W n` j from a given approximation Wj n, j,2, by solving (3.) G 0 W n` lptx,y,zu G 0 W n` 2 G l B l W n` 0, in Ω, pgń `S Gǹ qw n` pgń `S Gǹ qw n 2, onσ, lptx,y,zu G l B l W n` 2 0, in Ω 2, pgǹ `S 2 Gń qw n` 2 pgǹ `S 2 Gń qw n, on Σ, weres ands 2 are differential operators. WenS ands 2 are equal to zero, te algoritm is called classical Scwarz algoritm, and it uses classical transmission conditions. It as been sown in [] tat tese classical conditions ave te meaning of imposing Diriclet conditions on caracteristic (incoming) variables in eac subdomain. Since Gń j Nn Nn T N n 2 Nn T NnN T j I3ˆ3 (3.2) Nn n 2 Nn T Nn T N n, Gǹ j Nn Nn T N n 2 NnN T j Nn I3ˆ3 (3.3) Nn T N n, n 2 N T n te classical transmission conditions are also equivalent to imposing impedance conditions, N T n (3.4) n` GńW GńW 2 n ðñ B n pe n`,h n` q B n pe n 2,H n 2 q, n` GǹW2 GǹW n ðñ B n pe2 n`,h2 n` q B n pe n,h n q, were te impedance operator is given by (3.5) B n pe,hq : N n N T ne N n H and for te subdomain Ω 2 we ave used te fact tat Gǹ G n. Te classical Scwarz algoritm as been torougly tested in [4] for te solution of te tree-dimensional timearmonic Maxwell s equations discretized by low-order DG metods. In te second-order formulation of Maxwell s equation, te classical Scwarz metod uses te impedance condition (3.6) Bn peq p ˆE ˆnq ˆn`i ωe ˆn; see [9]. Tis impedance condition is equivalent to using te condition (3.7) Bn peq p ˆE ˆnq i ωn ˆ pe ˆnq, wic is just a rotation by 90 degrees of (3.6) but is more adapted to variational formulations; see, for example, [4]. Condition (3.7) is equivalent to (3.5) if we express H by Maxwell s equation as a function of ˆ E. Te equivalence between te first- and second-order formulation as been illustrated in [2, 3].

5 576 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL (3.8) As in (3.4), we also ave te equivalences ðñ ðñ pgń `S Gǹ qw n` pgń `S Gǹ qw2 n pb n ` S B n qpe n`,h n` q pb n ` S B n qpe n 2,H n 2 q, pgǹ `S 2 Gń qw n` 2 pgǹ `S 2 Gń qw n pb n ` S 2 B n qpe n` 2,H n` 2 q pb n ` S 2 B n qpe n,h n q. Here S and S 2 denote differential operators wic are approximations of te transparent operators, and S and S 2 are defined to guarantee te above equivalence. In [6], an entire ierarcy of optimized algoritms, defined by te coice of S j, j,2, was obtained from te transparent operators. Using (3.2) and (3.3), te optimized transmission conditions (3.8) become N n N T ne n` N n H n` ` S pn n N T ne n` `N n H n` q N n N T ne n 2 N n H n 2 ` S pn n N T ne n 2 `N n H n 2 q, N n N T ne n` 2 `N n H n` 2 ` S 2 pn n N T ne n` 2 N n H n` 2 q N n N T ne n `N n H n ` S 2 pn n N T ne n N n H n q. 4. Discontinuous Galerkin approximation. We now present a weak formulation and a DG discretization of te Scwarz algoritms (3.) and sow ow te optimized transmission conditions are properly discretized in a DG framework. 4.. Weak formulation. We denote by T a triangulation of te domainω, byγ 0, Γ m, and Γ a, te sets of purely internal, metallic, and absorbing faces, by K an element of T, and byf K X K te face sared by two neigboring elementsk and K. On eac facef, we define te average twu and te tangential trace jump W of W by twu : 2 pw K `W Kq and W : G nk W K `G n KW K. For two vector-valued functionsuand V in pl 2 pdqq 6, we introduce te inner products ż ż pu,vq D : U Vdx, xu,vy F : U Vds, D for D being a domain of R 3 and F a two-dimensional face. For simplicity, we skip te index fort, i.e., we write in wat follows p, q : p, q T ÿ p, q K. KPT On te boundaries we define $ «ff & η F N nk Nn T K N nk M F,K : Nn % T witη F 0, if F belongs toγ m, K 0 3ˆ3 G nk if F belongs toγ a. We tus obtain a weak formulation of te problem, pg 0 W,Vq ` ÿ G l B l W,V ÿ x W, tvuy F ` 2 W, V D F lptx,y,zu F PΓ 0 F PΓ 0 ` 2 pm F,K G nk qw,v D F 2 pm F,K G nk qw inc,v D, F F PΓ m YΓ a F PΓ a were we used an upwind flux discretisation [4, equation (4.4)]. F

6 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS Discretization of te subdomain problems and te classical Scwarz algoritm. LetP p pdq denote te space of polynomial functions of degree at mostpon a domaind. For any element K P T, let D p pkq pp p pkqq 6. Te discontinuous finite-element spaces we use are ten defined by! D p V P pl 2 pωqq 6 V K P D p P T ). Approximate solutionswand test functionsvfor te discretized problem will be taken in te space D p. Let Γ Σ be te set of faces on te interface Σ, Γ j 0 be te set of faces in te interior of eac subdomainω j, andγ j b be te set of faces of eac subdomain wic lie on te real boundary BΩ. For any face F K X K, note also tatg 2 n K G 2 n K G nk G n K. Ten, for eac subdomainω andω 2, te weak form can be written as ÿ pg 0 W,V q ` G l B l W,V (4.) pg 0 W 2,V 2 q ` l Γ 0 B F 2 p G n K G nk q pw W 2 q,v F PΓ Σ ÿ G l B l W 2,V 2 l F PΓ Σ Γ 2 0 Γ b Γ 2 b F 0, B F ` Gn 2 G K n pw2 K W q,v 2 0, F were, for simplicity, we ave replaced some terms on te faces tat do not play any particular role in wat follows by a. For any face F K X K on Σ, let n denote te normal on Σ directed from Ω towards Ω 2, and if K and K are elements of Ω and Ω 2, ten we ave n K n n K. Te classical algoritm, wic uses caracteristic transmission conditions, corresponds in tis DG formulation to a simple relaxation of te coupling flux terms in te coupled formulation (4.): starting from initial guessesw 0 andw2, 0 te iteratesw n` j are computed from Wj n,j,2, by solving on Ω and Ω 2 te subproblems (4.2) `G0 W n`,v ` ÿ l l G l B l W Gń pw n` W2 n q,v DF 0, F PΓ Σ ÿ `G0 W2 n`,v 2 ` G l B l W2 n`,v Gǹ pw2 n` W n q,v 2 DF 0, F PΓ Σ were we used again (2.3) to simplify te notation. Te relaxation in (4.2) is completely natural n` in te context of a DG discretization: we simply replaced te occurrence of te fluxgńw from outside te subdomain by te flux from te neigboring subdomain GńW 2 n at te Γ 0 Γ 2 0 Γ b Γ 2 b

7 578 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL n` previous iteration and vice versa te occurrence ofgǹw2 bygǹw n. Tis corresponds precisely to using te transmission conditions in (3.) wits j 0,j,2, namely (4.3) GńW n` GńW n 2 GǹW n` 2 GǹW n, and tus it naturally guarantees tat, at convergence of te associated classical Scwarz algoritm, te monodomain DG solution is obtained. Suc a simple replacement is, owever, not possible for te optimized transmission conditions,s j 0. Te DG discretization wic seems natural for te transmission conditions using te variables available in eac subdomain, namely (4.4) n` n` GńW `S GǹW GńW 2 n `S GǹW 2, n n` n` GǹW2 `S 2 GńW2 GǹW n `S 2 GńW, n leads to an obtained solution of te Scwarz algoritm wic is different from te monodomain DG solution. Te solver sould, owever, never cange te solution sougt, and suc a discretization is terefore to be avoided. We sow in te next section ow to properly discretize optimized transmission conditions in te framework of DG discretizations Discretization of optimized transmission conditions. In order to correctly introduce optimized transmission conditions (3.) wit a non-zeros j into te DG discretization, we first write explicitly wat transmission conditions te classical relaxation in (4.2) corresponds to. To do so, te subdomain problems solved in (4.2) are not allowed to depend on variables of te oter subdomain anymore since te coupling will be performed wit te transmission conditions, and we tus need to introduce additional unknowns, namely W n` 2,Ω on Ω and W n`,ω 2 onω 2, in order to write te classical Scwarz iteration wit local variables only, i.e., (4.5) `G0 W n`,v ` `G0 W n` 2,V 2 ` ÿ l ÿ F PΓ Σ ÿ l F PΓ Σ G l B l W n`,v Γ 0 Γ b A E Gń pw n` W n` 2,Ω q,v G l B l W2 n`,v 2 F Γ 2 0 Γ 2 b A E Gǹ pw2 n` W n`,ω 2 q,v 2 Comparing wit te classical Scwarz algoritm (4.2), we see tat in order to obtain te same algoritm, te transmission conditions for (4.5) need to be cosen as (4.6) GńW n` 2,Ω GńW n 2, F GǹW n`,ω 2 GǹW n, wic we ave already encountered wen explicitly stating te relaxation as a replacement in (4.3). But one as to be careful wen keeping tese variables since tey represent te outside traces at te interface, not te inside traces of te elements! Te transmission condition (4.6) implies tat in te limit, wen te algoritm converges, te so-called coupling conditions (4.7) GńW 2,Ω GńW 2, GǹW,Ω2 GǹW, will be satisfied, were we dropped te iteration index to denote te limit quantities. Tese are te conditions wic imply te equivalence of te converged solution to te monodomain 0, 0.

8 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS 579 DG solution. Wen using te Scwarz algoritm (4.5) wit te optimized transmission conditions (3.), we terefore propose to use DG discretizations of te strong relations (4.8) n` n` GńW2,Ω `S GǹW GńW 2 n `S GǹW,Ω n 2, n` n` GǹW,Ω 2 `S 2 GńW2 GǹW n `S 2 GńW 2,Ω n, wic are substantially different from te transmission conditions (4.4) since tey use additional variablesw 2,Ω andw,ω2, wic in principle belong to te traces at te interfaceσof te neigboring subdomain and are not available in te formulation (4.4). We now prove tat wit te transmission conditions (4.8), at convergence of te associated Scwarz algoritm, te same coupling conditions as (4.7) old, and tus te optimized Scwarz metod converges to te monodomain solution of te cosen DG discretization. First, from (3.2) and (3.3), note tat relation (4.7) is equivalent to We now introduce te auxiliary variables N n N T ne 2,Ω N n H 2,Ω N n N T ne 2 N n H 2, N n N T ne,ω2 `N n H,Ω2 N n N T ne `N n H. Λ 2,Ω : N n N T ne 2,Ω N n H 2,Ω, Λ 2 : N n N T ne 2 N n H 2, Λ,Ω2 : N n N T ne,ω2 `N n H,Ω2, Λ : N n N T ne `N n H. Tese variables represent traces belonging to a trace finite-element space ) M p!η P pl 2 pσqq 3 η F P pp p pf qq 3, pη nq F P Σ. Note tatm p consists of vector-valued functions wose normal component is zero on any face F P Σ. At convergence of te classical Scwarz algoritm and ence for te monodomain DG solution, we see from (4.7) tat tese trace variables ave to satisfy (4.9) Λ 2,Ω Λ 2, Λ,Ω2 Λ. From (4.8) and (4.9), we ave to find for te optimized transmission conditions a suitable DG discretization of te relations (4.0) Λ 2,Ω ` S Λ Λ 2 ` S Λ,Ω2, Λ,Ω2 ` S 2 Λ 2 Λ ` S 2 Λ 2,Ω. We terefore need to give now te precise expressions used in optimized Scwarz metods for te operators S j, j,2. Several coices for tese operators ave been proposed in [6] based on Fourier analysis under te assumption tat te interface is a plane: tey are secondorder differential operators in te tangential direction of te interface, wose Fourier symbols are given in Table 4., weref denotes te Fourier transform andkis te Fourier parameter in te tangential direction of te interface. Te matrix-valued operators Q sj are given by j Bττ Q sj B τ2τ 2 σs j 2B ττ 2, B τ2τ 2 B ττ σs j 2B ττ 2 and te division by k 2 indicates an integral operation. We explain below ow tis integration can be avoided in te implementation. Every coice in Table 4. leads to a different transmission condition and tus a different optimized Scwarz algoritm. Note tat te operator Q sj

9 580 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL TABLE 4. Symbols of te different operators for 3D Maxwell s equations. Algoritm Fp S j q 0 2 s i ω ps`i ωqp k 2`s σq Fp Q s q, 3 k 2 2 ω 2`2i ω σ`p2i ω` σqs Fp Q s q, 4 s j i ω ps j`i ωqp k 2`s j σq Fp Q sj q, s P C s P C s j P C 5 k 2 2 ω 2`2i ω σ`p2i ω` σqs j Fp Q sj q, s j P C can be rewritten in a more natural form for Maxwell s equations, j j Bττ Q sj B ττ 2 Bτ2τ ` 2 B ττ 2 σs B ττ 2 B τ2τ 2 B ττ 2 B j I ττ loomoon τ τ ` looooomooooon τ ˆ τ ˆ σs j I, S TE S TM wereidenotes te identity operator,τ j, j,2, are two independent vectors in te tangent plane to te interface, τ denotes te gradient in te tangent plane to te interface, τ is te divergence in te tangent plane, and τ ˆ is te two-dimensional curl operator in te tangent plane. Te operatorss TM and S TE satisfy te remarkable relation τ I S TE S TM, were τ is te Laplace-Beltrami operator, and tey act mainly on te transverse electric and transverse magnetic part of te solution; see [2, 3] for a more detailed explanation. To avoid an integral relation in te transmission condition, one as to multiply te entire transmission conditions by te operator symbol in te denominator and ten obtains second-order differential transmission conditions. Tese second-order differential transmission conditions are equivalent to te transmission conditions (4.0) and are of te form (4.) P pλ 2,Ω Λ 2 q Q s pλ,ω2 Λ q, P 2 pλ,ω2 Λ q Q s2 pλ 2,Ω Λ 2 q, were, for example for Algoritms 2 and 4 indicated in Table 4., we ave (4.2) Pj : s j `i ω s j i ω p τ ` σs j qi, s j P C, and for Algoritms 3 and 5 in Table 4., we ave (4.3) P j : p τ 2 ω 2 `2i ω σ `2i ωs j ` σs j qi S TE S TM ` p 2 ω 2 `2i ω σ `2i ωs j ` σs j qi, s j P C. We see tat even toug tese transmission conditions ave been derived in [6] assuming tat te interface is planar, teir reformulation allows us to use tem also for non-planar interfaces obtained for example by an automatic mes partitioning tool in te context of DG discretizations.

10 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS 58 TABLE 4.2 Asymptotic convergence factor and optimized coice of te parameters in te transmission conditions for te 3D Maxwell s equations. wit overlap,l Algoritm ρ parameters 4 3 `9 ω4 σ none p ω σ q {6 3 p p ω 4 σ 2 q ? 2 p ω σq 0 5 p p ω σq p ω4 σ 2 q p2 ω σq{3 2 3 p 22 5 p ω 4 σ 2 q p p ω4 σ 2 q6,p witout overlap,l 0,p 2 p ω σq2 5 ω2 σ C 3 3 none p ω σq 4?? C? p p ω σq 4 C p ω4 σ 2 q C p2 ω σq 8 4 C p ω 4 σ 2 q C 3 3 p p2 ω σq 8C p p ω 4 σ 2 q26c ? p 24 7 p ω 4 σ 2 q 4C ? 2p ω4 σ 2 q ,p 2 p2 ω σq3{8 C {4 2 4,p p ω 4 σ 2 q26c It remains to coose te parameters s j, j,2, in (4.2) and (4.3) to complete te definition of te corresponding optimized Scwarz metod. Tese parameters are selected by a minimization of te associated contraction factors for a model problem suc tat te performance of te metod is optimized, and we sow for completeness in Table 4.2 te optimized values from [6] adapted to te notation in tis manuscript. Having defined all te components in te transmission conditions (4.), we now explain ow to discretize te five variants in a consistent fasion using a DG discretization: let pη j q j be a basis of M p. On te interface Σ we define te matrices pm Σ q i,j : ÿ F PΣxη i,η j y F, pk Σ q i,j : ÿ F PΣx τ ˆη i, τ ˆη j y F ` x τ η i, τ η j y F epbσ ÿ epbσ ÿ epbσ ż ż ÿ α e kpt,2u e η i τ k η j τ k tt τ η i uu η j n e,τ η i n e,τ τ η j (( ż tt τ ˆη i uu η j ˆn e,τ η i ˆn e,τ (( τ ˆη j, e

11 582 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL and pa Σ q i,j : ÿ F PΣx τ ˆη i, τ ˆη j y F x τ η i, τ η j y F epbσ epbσ ÿ epbσ ż ż ż ÿ α e kpt,2u e e η i τ k η j τ k tt τ η i uu η j n e,τ η i n e,τ τ η j ((, tt τ ˆη i uu η j ˆn e,τ η i ˆn e,τ τ ˆη j ((, were te positivity of te discretized operator is guaranteed for sufficiently large α, BΣ denotes te set of interior edges of Σ, and tt uu denote te jump and te average at an edge e of te values at neigboring triangles, and n e,τ is te outward normal on e in te tangent plane. Ten matrix K Σ stems from te discretization of τ using a symmetric interior penalty approac [2, 3]. Note tat te operator τ as to be taken in vector form since it is applied to pλ 2, Ω Λ 2 q, wic is a discretization of a vector quantity. M Σ is an interface mass matrix wit te same dimensions as te interface stiffness matrix K Σ, and A Σ represents te discretization of te operator j Bττ B τ2τ 2 2B ττ 2. 2B ττ 2 B τ2τ 2 B ττ Ten te DG discretization of (4.) for te Algoritms 2 and 4 is s `i ω s i ω pk Σ ` σs M Σ qpλ 2,Ω Λ 2 q pa Σ σs M Σ qpλ,ω2 Λ q, s 2 `i ω s 2 i ω pk Σ ` σs 2 M Σ qpλ,ω2 Λ q pa Σ σs 2 M Σ qpλ 2,Ω Λ 2 q, and for te Algoritms 3 and 5 we get (4.4) pk Σ `α M Σ qpλ 2,Ω Λ 2 q pa Σ σs M Σ qpλ,ω2 Λ q, pk Σ `α 2 M Σ qpλ,ω2 Λ q pa Σ σs 2 M Σ qpλ 2,Ω Λ 2 q, wereα j 2i ωpi ω ` σq `2i ωs j ` σs j. In te following teorem we will only treat te case of Algoritms 3 and 5; similar tecniques can be applied for Algoritms 2 and 4. THEOREM 4. (DG discretization of Algoritms 3 and 5). If s and s 2 are suc tat s j p j p `iq wit p j a strictly positive real number for j,2, and σpp p 2 q 0, ten te relations (4.9) and (4.4) are equivalent. Proof. We first observe tatiα j 2 ω σ `2 ωp j ` σp j ą 0. Let us denote U Λ,Ω2 Λ, U 2 Λ 2,Ω Λ 2. Multiplying te first relation in (4.4) on te left by ŪT 2 and te second by ŪT and summing tem up, we get Ū T 2 pk Σ `α M Σ qu 2 `ŪT pk Σ `α 2 M Σ qu ŪT 2 pa Σ σs M Σ qu `ŪT pa Σ σs 2 M Σ qu 2.

12 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS 583 Since K Σ is symmetric and non-negative,m Σ is symmetric and positive definite, and A Σ is symmetric, all te quantities ŪT j M ΣU j, ŪT j K ΣU j, andūt A Σ U 2 `ŪT 2A Σ U are real. In tis case, by taking te imaginary part of te previous relation, we get (4.5) Iα Ū T 2M Σ U 2 `Iα 2 Ū T M Σ U ` σips Ū T 2M Σ U `s 2 Ū T M Σ U 2 q 0. In order to simplify notation and by using tatm Σ is symmetric positive definite, we introduce te norm }U} 2 M Σ : ŪT M Σ U induced by te Hermitian product pu,u 2 q MΣ ŪT 2M Σ U. Since by definition pu 2,U q MΣ pu Ğ,U 2 q MΣ, we see tat IpŪT 2M Σ U q 2i ppu,u 2 q MΣ pu 2,U q MΣ q IpŪT M Σ U 2 q, RpŪT 2M Σ U q 2 ppu,u 2 q MΣ ` pu 2,U q MΣ q RpŪT M Σ U 2 q, Ips Ū T 2M Σ U q p prpu,u 2 q MΣ `IpU,U 2 q MΣ q, Ips 2 Ū T M Σ U 2 q p 2 prpu 2,U q MΣ `IpU 2,U q MΣ q p 2 prpu,u 2 q MΣ IpU,U 2 q MΣ q. Also, letp p `δ and p 2 p δ, and suppose tatδ ě 0. Ten (4.5) becomes (4.6) ô ô 2 ωp σ `p q}u 2 } 2 M Σ `2 ωp σ `p 2 q}u } 2 M Σ ` σpp `δq}u 2 } 2 M Σ ` σpp δq}u } 2 M Σ ` σpp `δq RpU,U 2 q MΣ `IpU,U 2 q MΣ ` σpp δq RpU,U 2 q MΣ IpU,U 2 q MΣ 0 2 ωp σ `p q}u 2 } 2 M Σ `2 ωp σ `p 2 q}u } 2 M Σ ` σp }U 2 } 2 MΣ ` }U } 2 MΣ `2RpU,U 2 q MΣ ` σδ }U 2 } 2 MΣ }U } 2 MΣ `2IpU,U 2 q MΣ 0 2 ωp σ `p q}u 2 } 2 M Σ `2 ωp σ `p 2 q}u } 2 M Σ ` σp}u `U 2 } 2 M Σ ` σδ }U 2 } 2 MΣ }U } 2 MΣ `2IpU,U 2 q MΣ 0. We tus see tat if σ 0 orδ 0, wic means tatp p 2 (Algoritm 3 from Table 4.2), ten te last form of (4.6) leads to te conclusion tatu j 0 since all te terms are positive, wic proves te equivalence between (4.4) and (4.9) Te two-dimensional case. As in te tree-dimensional case, we can rewrite (4.8) and (4.7) by introducing te auxiliary variables (see [7] for more details) (4.7) Λ 2,Ω : E 2,Ω N n H 2,Ω, Λ 2 : E 2 N n H 2, Λ,Ω2 : E,Ω2 `N n H,Ω2, Λ : E `N n H, belonging to te trace space M p η P L 2 pσq η F P P p pf P Σ (. Ten (4.7) becomes (4.8) Λ 2,Ω Λ 2 and Λ,Ω2 Λ. From (4.8) and (4.7), we see tat for te optimized transmission conditions, we ave to find a suitable DG discretization of te relations (4.9) Λ 2,Ω ` S Λ Λ 2 ` S Λ,Ω2 and Λ,Ω2 ` S 2 Λ 2 Λ ` S 2 Λ 2,Ω.

13 584 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL If we focus on te second-order transmission conditions, (4.9) becomes (4.20) p B 2 τ `i ω σ 2 ω 2 `2i ωs qpλ 2,Ω Λ 2 q ` p B 2 τ `i ω σqpλ,ω2 Λ q 0, p B 2 τ `i ω σ 2 ω 2 `2i ωs 2 qpλ,ω2 Λ q ` p B 2 τ `i ω σqpλ 2,Ω Λ 2 q 0. Let pη j q j be a basis ofm p. We define te matrices pm Σ q i,j : ÿ F PΣxη i,η j y F, pk Σ q i,j : ÿ xb τ η i, B τ η j y F α n rrrrη i ssss n rrrrη j ssss n F PΣ npσ 0 ÿ ttb τ η i uu n rrrrη j ssss n rrrrη i ssss n ttb τ η j uu n, npσ 0 were positiveness is guaranteed for sufficiently large α n, Σ 0 denotes te set of interior nodes ofσ, rrrr ssss n and tt uu n denotes te jump and te average at a nodenof te values on neigboring segments. Te matrix K Σ comes from te discretization of B 2 τ using a symmetric interior penalty approac [3]. Te DG discretization of (4.20) is ten (4.2) pk Σ `α M Σ qpλ 2,Ω Λ 2 q p K Σ i ω σm Σ qpλ,ω2 Λ q, pk Σ `α 2 M Σ qpλ,ω2 Λ q p K Σ i ω σm Σ qpλ 2,Ω Λ 2 q, witα j 2 ω 2 `ip ω σ `2 ωs j q. As in te tree-dimensional case,k Σ is symmetric and nonnegative definite, andm Σ is symmetric and positive definite. A similar result to Teorem 4. can be obtained also in 2D: THEOREM 4.2 (DG discretization for te second-order conditions in 2D). If s and s 2 are suc tat s j p j p ` iq wit p j a strictly positive real number for j,2, ten te relations (4.8) and (4.2) are equivalent. Proof. We first note tatiα j ω σ `2 ωp j ą 0. Setting U Λ,Ω2 Λ, U 2 Λ 2,Ω Λ 2, and multiplying te first relation in (4.2) on te left by ŪT 2, te second by ŪT, and adding tem up, we obtain by taking te imaginary part p ω σ `2 ωp j qpūt M Σ U `ŪT 2M Σ U 2 q ω σpūt 2M Σ U `ŪT M Σ U 2 q. By rearranging te terms using te norm, we get 2 ωp j p}u } 2 M Σ ` }U 2 } 2 M Σ q ` ω σ}u `U 2 } 2 M Σ 0. From tis last equation, we see tat U j 0 since all te terms are positive, wic proves te equivalence between (4.2) and (4.8). 5. Numerical results. We illustrate te performance of te optimized Scwarz algoritms discretized using a DG metod in two dimensions. We consider te TM formulation of Maxwell s equations, i.e., E p0,0,e z q T andh ph x,h y,0q T. We can ten rewrite te algoritm in (3.) by using tatw pe z,h x,h y q, T and te correspondingg-matrices are σ `i ω G 0 0ˆ2 0 2ˆ i ωi 2ˆ2 j, G x j 0 Nex Ne T, G x 0 y j 0 Ney Ne T, y 0

14 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS 585 TABLE 5. Symbols of te different operators for 2D Maxwell s equations. Algoritm Fp S j q 0 2 s i ω s`i ω, s P C k 3 2`i ω σ k2 2 ω 2`i ω σ`2i ωs, s P C 4 sj i ω s j`i ω, s j P C 5 k 2`i ω σ k 2 2 ω 2`i ω σ`2i ωs j, s j P C TABLE 5.2 Asymptotic convergence factor and optimal coice of te parameters in te transmission conditions for 2D Maxwell s equations. witout overlap Algoritm ρ parameters ω2 σ C 3 3 none p ω σq 4?? C? p p ω σq 4 C p ω4 σ 2 q C p2 ω σq 8 4 C p ω 4 σ 2 q C 3 3 p p2 ω σq 8C p p ω 4 σ 2 q 26C ? p 24 7 p ω 4 σ 2 q 4C , p 2 p2 ω σq3{8 C {4 2 4,p p ω 4 σ 2 q26c weren n pn y, n x q T. We present in Table 5. te corresponding Fourier symbols of S j in te two-dimensional case, wic were derived from te 3D results given in [6]. Te parameterss pp `iq,s p p `iq, ands 2 p 2 p `iq are solutions of specific min-max problems solved in [6], and teir asymptotic beavior in te omogeneous non-overlapping case is displayed in Table 5.2 togeter wit te corresponding convergence factors. Te constant C is defined suc tat k max C is te igest numerical frequency tat can be represented by te discretization metod on a mes wit mes size. Te Fourier symbols of te operators in Algoritms, 2, and 4 are constants, terefore teir expression is te same in te pysical space. In tis case, (3.8) can be written in te 2D situation considered ere as (5.) E n` N n H n` ` S pe n` `N n H n` q E n 2 N n H n 2 ` S pe n 2 `N n H n 2 q, E n` 2 `N n H n` 2 ` S 2 pe n` 2 N n H n` 2 q E n `N n H n ` S 2 pe n N n H n q. Tis is not te case for Algoritms 3 and 5, wic lead to second-order transmission conditions because a factor of k 2 appears in te corresponding Fourier symbols. As in te 3D case, we need to rewrite te transmission conditions: te S j are operators wit Fourier symbols Fp S j q q jpkq r j pkq wit q j pkq pk 2 `i ω σq, r j pkq k 2 2 ω 2 `i ω σ `2i ωs j. We observe tat te numerator and denominator,f pq j q andf pr j q, are partial differential

15 586 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL operators in te tangential direction, F q j B ττ i ω σ, F r j B ττ 2 ω 2 `i ω σ `2i ωs j. In tis case, we multiply te transmission conditions on bot sides by te denominator, and ten te interface iteration (5.) can be rewritten as F r pe n` N n H n` q `F q pe n` `N n H n` q F r 2 pe n` 2 `N n H n` 2 q `F q 2 pe n` 2 N n H n` 2 q similarly to te general 3D case as we explained in (4.). F r pe n 2 N n H n 2 q `F q pe n 2 `N n H n 2 q, F r 2 pe n `N n H n q `F q 2 pe n N n H n q, 5.. Plane wave in a omogeneous conductive medium. We first consider te propagation of a plane wave in a omogeneous conductive medium. Te computational domain is Ω p0,q 2, and σ 0.5. We use DG discretizations wit several polynomial orders denoted by DG-P k, witk,2,3,4, and impose on BΩ Γ a an incident wave» W inc k ỹ ω k x ω fi ffi fle ik x, and k kx k y j «b ω i σ ω 0 Te domain Ω is decomposed into te two subdomains Ω p0,0.5q ˆ p0,q and Ω 2 p0.5,q ˆ p0,q. Te goal of tis first test problem is to retrieve numerically te asymptotic beavior of te convergence factors of te optimized Scwarz metods wen discretized using DG and to compare wit te teoretical convergence factors of Table 5.2. Te iteration numbers to reduce te relative residual by six orders of magnitude are given in Table 5.3, were also in parenteses te iteration numbers are included for te use of te Scwarz metods as preconditioners for a Krylov metod, wic is BiCGStab in our case. We clearly see tat tere is a ierarcy of faster and faster algoritms, and teir asymptotic beavior corresponds well to te analysis as one can see from Figure Plane wave in a multi-layer eterogeneous medium. We study te performance of te optimized Scwarz algoritms in te case of a eterogeneous propagation medium. Te model problem we consider is te propagation of a plane wave in a multi-layer conductive medium, as displayed in Figure 5.2 on te left. We decompose te computational domain Ω p,q 2 into two subdomains Ω p0,0.5q ˆ p0,q and Ω 2 p0.5,q ˆ p0,q; see Figure 5.2 on te rigt. Te electromagnetic caracteristics of te medium are given in Table 5.4. We test ere te metod DG-P,2,3,4 were te interpolation degree is fixed for eac element of te mes according to te local wavelengt; see te last column in Table 5.4. In Table 5.5, we again present te iteration numbers obtained by te various optimized Scwarz algoritms for reducing te relative residual by six orders of magnitude and in parenteses te corresponding iteration numbers wen te Scwarz metods are used as preconditioners. In Figure 5.3, we plot tese iteration numbers as a function of te mes size as well as te corresponding teoretical asymptotic iteration number counts, wic sows tat even in suc a layered medium, were our analysis is not valid any more, te Scwarz algoritms still beave asymptotically as te constant medium teory indicates. We finally display in 5.4 te real part of te electric field for tis scattering problem. ff.

16 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS 587 TABLE 5.3 Wave propagation in a omogeneous medium. Iteration count as a function of wen te optimized Scwarz metods are used as iterative solvers and in parenteses wen used as preconditioners DG-P, ω 2π Algoritm 383 (6) 396 (2) 5434 (27) (35) Algoritm 2 30 (9) 43 () 62 (3) 92 (8) Algoritm 3 29 (9) 40 (0) 59 (3) 8 (8) Algoritm 4 28 (0) 34 (0) 43 (2) 52 (7) Algoritm 5 28 (9) 32 (9) 38 (0) 45 (5) DG-P 2, ω 0{3π Algoritm 573 (2) 2288 (24) 0520 (29) (35) Algoritm 2 37 () 53 (2) 77 (6) (8) Algoritm 3 35 (0) 48() 69 (6) 95 (7) Algoritm 4 30 (0) 36 (2) 45 (4) 55 (6) Algoritm 5 29 (9) 33 (0) 39 (3) 49 (4) DG-P 3, ω 3{3π Algoritm 650 (2) 3025 (25) 7900 (30) (5) Algoritm 2 40 () 58 (4) 84 (6) 22 (2) Algoritm 3 38 () 5 (3) 75 (5) 05 (9) Algoritm 4 3 (0) 38 (3) 47 (5) 57 (9) Algoritm 5 30 (9) 33 () 39 (3) 47 (6) DG-P 4, ω 6π Algoritm 072 (29) 638 (38) (5) (64) Algoritm 2 50 (2) 73 (5) 06 (8) 54 (2) Algoritm 3 47 () 69 (4) 98 (8) 39 (20) Algoritm 4 37 (2) 47 (4) 59 (7) 7 (9) Algoritm 5 34 (0) 42 (2) 5 (5) 60 (7) TABLE 5.4 Caracteristic parameters of te medium for te model problem of scattering of a plane wave in a multi-layer domain. Layer i ε i σ i µ i DG-P i Scattering of a plane wave by a conductive dielectric cylinder. Te final model problem we consider is te scattering of a plane wave by a dielectric conductive cylinder wit radiusr m. Te computational domain is obtained by artificially restricting te domain to a cylinder wit radius r.6 m and using te Silver-Müller condition on te artificial boundary. We use a non-uniform triangular mes wic consists of 2078 vertices and 3958 triangles; see Figure 5.5. Te relative permittivity of te inner cylinder is set to ε r 2.25 and its electric conductivity to σ 0.0, wile vacuum is assumed for te rest of te domain. Te frequency we consider is F=300 MHz. Numerical simulations are performed

17 588 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL Iterations 0 2 Alg. 2 O( /2 ) Alg. 3 O( 3/7 ) Alg. 4 O( /4 ) Alg. 5 O( 3/3 ) Iterations 0 2 Alg. 2 O( /2 ) Alg. 3 O( 3/7 ) Alg. 4 O( /4 ) Alg. 5 O( 3/3 ) 0 0 Iterations 0 2 Alg. 2 O( /2 ) Alg. 3 O( 3/7 ) Alg. 4 O( /4 ) Alg. 5 O( 3/3 ) Iterations 0 2 Alg. 2 O( /2 ) Alg. 3 O( 3/7 ) Alg. 4 O( /4 ) Alg. 5 O( 3/3 ) 0 0 FIG. 5.. Asymptotic beavior of te iteration numbers from Table 5.3 as a function of te mes size for te DG-P, DG-P 2, DG-P 3, and DG-P 4 discretizations. FIG Domain configuration for te model problem of scattering of a plane wave in a multi-layer domain. using decompositions into 4 and 6 subdomains; for an example see Figure 5.5. In Table 5.6 we display te iteration numbers for te various optimized Scwarz metods for reducing te relative residual by six orders of magnitude. Here, DG-P,2,3,4 stands for a non-uniform-order DG discretization, i.e., te interpolation order is defined on an elementwise basis: small elements use low-order sape functions and large elements use ig-order ones. We note tat te optimized algoritms improve substantially te convergence of te classical Scwarz

18 DISCONTINUOUS GALERKIN METHODS FOR THE MAXWELL EQUATIONS 589 TABLE 5.5 Scattering of a plane wave in a multi-layer domain. Iteration count as a function of wen te optimized Scwarz metods are used as iterative solvers and in parenteses wen used as preconditioners Algoritm 727 (3) 2974 (4) 973 (52) (70) Algoritm 2 08 (2) 53(25) 220 (30) 35 (33) Algoritm 3 0 (20) 38 (23) 97 (27) 267 (30) Algoritm 4 87 (8) 03 (22) 28 (25) 57 (28) Algoritm 5 84 (6) 96 (20) 3 (22) 40 (25) TABLE 5.6 Scattering of a plane wave by a dielectric conductive cylinder. Iteration count vs. mes size. DG-P DG-P 2 DG-P 3 DG-P 4 DG-P,2,3,4 Algo. # of domains # of domains # of domains # of domains # of domains Iterations 0 2 Alg. 2 O( /2 ) Alg. 3 O( 3/7 ) Alg. 4 O( /4 ) Alg. 5 O( 3/3 ) 0 2 FIG Asymptotic beavior of te iteration numbers from Table 5.5 as a function of te mes size. algoritm (Algoritm in te table) and also tat te gain between bot te optimized and te classical algoritms seems to sligtly increase wit te interpolation order. Finally, we also observe, as could be expected, a dependence of te iteration count on te number of subdomains since we are not using any coarse grid correction in tese experiments. 6. Conclusions. In tis paper we ave sown ow optimized Scwarz metods can be properly discretized in te framework of DG-metods suc tat at convergence, te result of te underlying DG monodomain solution is recovered. Te key idea is to introduce additional trace variables on eac subdomain interface representing te DG-traces of te neigboring subdomain interface traces and ten to use bot traces appropriately to discretize te optimized transmission conditions. We ave tested te performance of te DG-discretized Scwarz

19 590 M. EL BOUAJAJI, V. DOLEAN, M. J. GANDER, S. LANTERI, AND R. PERRUSSEL FIG Real part of te electric field for te scattering of a plane wave in a multi-layer domain y FIG Mes and subdomain decomposition for te scattering problem of a plane wave by a dielectric conductive cylinder. x metods on many numerical scattering experiments, bot for omogeneous and eterogeneous media and in various pysical configurations and for various decompositions. Our numerical results indicate tat te asymptotic performance of tese algoritms obtained at a teoretical level for omogeneous media and constant coefficients well predicts te performance of te algoritms wen discretized using DG-discretizations, bot in omogeneous and eterogeneous media and for very general decompositions.

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