DG discretization of optimized Schwarz methods for Maxwell's equations

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DG discretization of optimized Schwarz methods for

Maxwell’s equations

Mohamed El Bouajaji, Victorita Dolean, Martin Gander, Stephane Lanteri,

Ronan Perrussel

To cite this version:

Mohamed El Bouajaji, Victorita Dolean, Martin Gander, Stephane Lanteri, Ronan Perrussel. DG

discretization of optimized Schwarz methods for Maxwell’s equations. 2013. �hal-00830274�

for Maxwell’s equations

Mohamed El Bouajaji1, Victorita Dolean2,3, Martin J. Gander3, St´ephane Lanteri1, and Ronan Perrussel4

1 Introduction

In the last decades, Discontinuous Galerkin (DG) methods have seen rapid growth and are widely used in various application domains (see [13] for an historical intro-duction). This is due to their main advantage of combining the best of finite element and finite volume methods. For the time-harmonic Maxwell equations, once the problem is discretized with a DG method, finding robust solvers is a difficult task since one has to deal with indefinite problems. From the pioneering work of Despr´es [5] where the first provably convergent domain decomposition (DD) algorithm for the Helmholtz equation was proposed and then extended to Maxwell’s equations in [6], other studies followed. Preliminary attempts to obtain better algorithms for this kind of equations were given in [3, 4, 12], where the first ideas of optimized Schwarz methods can be found. Then, the advantage of the optimization process was used for the second order Maxwell system in [1]. Later on, an entire hierarchy of optimized transmission conditions for the first order Maxwell’s equations was proposed in [9, 11] . For the second order or curl-curl Maxwell’s equations second order optimized transmission conditions can be found in [14, 15, 16, 17]. We study here optimized Schwarz DD methods for the time-harmonic Maxwell equations dis-cretized by a DG method. Due to the particularity of the latter, DG discretization ap-plied to more sophisticated Schwarz methods is not straightforward. In this work we show a strategy of discretization and prove the equivalence between multi-domain and single-domain solutions. The proposed discrete framework is then illustrated by some numerical results in the two-dimensional case.

We consider time-harmonic Maxwell’s equations in a homogeneous medium written as a first order system (see [10] for more details)

G0W + Gx∂xW + Gy∂yW + Gz∂zW = 0, (1) where W = E H  , G0= (σ + iω)I3×3 03×3 03×3 iωI3×3 

with E, H the complex-valued electric and magnetic fields, ω the angular frequency of the time-harmonic wave, σ the electric conductivity. For a general vector n =

Inria Sophia Antipolis-M´editerran´ee, France [email protected] · Univer-sity of Nice-Sophia Antipolis, France [email protected] · UniverUniver-sity of Geneva, Switzer-land [email protected] · CNRS, Universit´e de Toulouse, Laplace, France [email protected]

2 El Bouajaji et al. nxnynz
, we also define the matrices

Gn=  03×3 Nn N_nT 03×3  and Nn=   0 nz −ny −nz 0 nx ny −nx 0  .

Then, for l ∈ {x, y, z}, we have that Nl= Nel and Gl= Gel, where el, l = 1, 2, 3 are

the canonical basis vectors. Our goal is to solve the boundary-value problem G0W + Gx∂xW + Gy∂yW + Gz∂zW = 0 in Ω ,

(MΓm− Gn)W = 0 on Γm and (MΓa− Gn)(W − Winc) = 0 on Γa,

(2)

where Wincis a given incident field, while MΓm and MΓa are trace operators defined

on the metallic and absorbing boundaries Γmand Γa(see [10] for more details)

MΓm =  03×3 Nn −NT n 03×3  and MΓa= |Gn| =  N_nN_nT 03×3 03×3 NnTNn  .

The matrices G+_n and G−_n are the positive and negative parts of Gn based on its

diagonalization and we have that |Gn| = G+n − G−n.

2 Continuous classical and optimized Schwarz algorithms

We now decompose the domain Ω into two non-overlapping subdomains Ω1and

Ω2, and denote by Σ the interface between Ω1and Ω2, by Wj the restriction of

W to Ωj and by n the unit outward normal vector to Σ directed from Ω1to Ω2.

Schwarz algorithms consist in computing iteratively Wn+1_j from Wn

j, for j = 1, 2 G0Wn+11 + Gx∂xWn+11 + Gy∂yWn+11 + Gz∂zWn+11 = 0, in Ω1, (G−_n+ S1G+n)Wn+11 = (G − n+ S1G+n)Wn2, on Σ , G0Wn+12 + Gx∂xWn+12 + Gy∂yWn+12 + Gz∂zWn+12 = 0, in Ω2, (G+_n+ S2G−n)Wn+12 = (G+n+ S2G−n)Wn1, on Σ , (3)

where S1and S2are differential operators. When S1= S2= 06×6, the interface

con-ditions become the positive and negative flux operators G+_n and G−_n, and the clas-sical Schwarz algorithmis obtained. Applying G+_n (respectively G−_n) to a vector W means to select the characteristic variables associated to out-going (respectively in-coming) waves, which is very natural considering the hyperbolic nature of the problem, see [9] (section 3.1). We note that

Algorithm 1 2 3 4 5 F ( ˜Sj) 0 −s−iω_s+iω − k 2_+iωσ k2_−2ω2_{+iωσ +2iωs} − sj−iω sj+iω − k2+iωσ k2_−2ω2_{+iωσ +2iωs} j

Table 1 Five different choices for the symbols of the operators in the transmission conditions (6) leading to five different optimized Schwarz algorithms

G−_n = −NnN T n Nn NT n −NnTNn  =  I3×3 −NT n  −NnNnT Nn
, G+_n = NnN T n Nn N_nT N_nTNn  =  I3×3 N_nT  NnNnT Nn
. (4)

Thus the classical transmission conditions are equivalent to impedance conditions, G−_nWn+1₁ = G−_nWn

2⇔Bn(En+1₁ , Hn+1₁ ) =Bn(En2, Hn2),

G+_nWn+1₂ = G+_nWn₁⇔B−n(En+12 , Hn+12 ) =B−n(En1, Hn1).

(5)

withBn(E, H) = NnTE − NnTNnH. For Ω2we have used the fact that G+n = −G − −n.

The classical Schwarz algorithm is adopted in [10] together with low order DG methods in the 3D case. Along the lines of (5), we have the equivalences

(G−_n+ S1G+n)Wn+11 = (G − n+ S1G+n)Wn2 ⇔ (Bn+ ˜S1B−n)(En+1₁ , Hn+1₁ ) = (Bn+ ˜S1B−n)(En2, Hn2), (G+_n+ S2G−n)Wn+12 = (G+n+ S2G−n)Wn1 ⇔ (B−n+ ˜S2Bn)(En+12 , H n+1 2 ) = (B−n+ ˜S2Bn)(En1, Hn1), (6)

where ˜S1and ˜S2denote differential operators which are approximations of the

trans-parent operators. From these transtrans-parent operators we can obtain a hierarchy of op-timized algorithms with appropriate choices for ˜S1and ˜S2[11]. The operators S1

and S2are eventually defined to guarantee the equivalences in (6).

If we consider the TM formulation of Maxwell’s equations, that is with E = ( 0 0 Ez)T and H = ( HxHy0 )T, then W = ( EzHxHy)T, Nn= ( ny−nx)T, and

G0=  σ + iω 01×2 02×1 iωI2×2  , Gx=  0 Nex N_eT_x 0  and Gy=  _{0 N} ey N_eT_y 0  .

We give in Table 1 the symbolsF ( ˜Sj) of ˜Sjin the 2d case for conductive media for

five different Schwarz algorithms, where the parameters s = p(1 + i), s1= p1(1 + i)

and s2= p2(1 + i) are solutions of some min-max problems, as explained in [11]

(section 5, table 5.1). Note that the Fourier symbols of the operators in algorithms 1, 2 and 4 are constants, therefore they have the same expression as in the physical space. In this case (6) can be written in the 2d situation considered here as

E₁n+1− NnHn+11 + ˜S1(E1n+1+ NnHn+11 ) = E2n− NnHn2+ ˜S1(E2n+ NnHn2),

E₂n+1+ NnHn+1₂ + ˜S2(E2n+1− NnH n+1

2 ) = E1n+ NnHn1+ ˜S2(E1n− NnHn1).

4 El Bouajaji et al. This is not the case for algorithms 3 and 5 which involved second order transmission conditions. Here, the ˜Sjare operators whose Fourier symbols have the form

F ( ˜Sj) =

qj(k)

rj(k)

with qj(k) = −(k2+ iωσ ) and rj(k) = k2− 2ω2+ iωσ + 2iωsj.

where the Fourier variable k corresponds to a transform with respect to the tan-gential direction τ along the interface, assuming a two-subdomain decomposition with a straight interface. In that case,F−1(q_j) andF−1(r_j) are partial differential operators in the τ variable,

F−1_(q

j) = ∂τ τ− iωσ ,F−1(rj) = −∂τ τ− 2ω2+ iωσ + 2iωsj, sj∈ C,

and (7) can be re-written as F−1 _r 1(E1n+1− NnHn+11 )
+F−1 q1(E1n+1+ NnHn+11 )
=F−1(r1(E2n− NnHn2)) +F−1(q1(E2n+ NnHn2)) , F−1 _r 2(E2n+1+ NnHn+12 )
+F−1 q2(E2n+1− NnHn+12 )
=F−1(r2(E1n+ NnHn1)) +F−1(q2(E1n− NnHn1)) .

3 Discontinuous Galerkin approximation

Let Th be a discretization of Ω and Γ0, Γm and Γabe the sets of purely

inter-nal, metallic and absorbing faces of Th. We denote by K an element of Th and

by F = K ∩ ˜Kthe face shared by two neighboring elements K and ˜K. On this face F, we define the average by {W} = 1₂(W_K+ WK˜) and the tangential trace jump

by [[W]] = GnKWK+ GnK˜WK˜. For two vector functions U and V in (L

2_(D))6_{, we}

denote (U, V)D= R

DU · V dx, if D is a domain of R3 and hU, ViF = R

FU · V ds

if F is a face of R2. For sake of simplicity, we will skip some subscripts, that is (·, ·) = (·, ·)_T

h= ∑K∈Th(·, ·)K. On the boundaries we define

MF,K=     ηFNnKN T nK NnK −NT nK 03×3  with ηF6= 0, if F belongs to Γm, |GnK| if F belongs to Γ a_.

Using these notations, the weak formulation of the problem is

(G0W, V) +

∑

l∈{x,y,z} Gl∂lW, V ! −

_∑

F∈Γ0 h[[W]], {V}i_F+

_∑

F∈Γ0  1 2[[W]], {V}  F +

_∑

F∈Γm_∪Γa  1 2(MF,K− GnK)W, V  F =

_∑

F∈Γa  1 2(MF,K− GnK)Winc, V  F .

Note that we have implicitly adopted an upwind scheme for the calculation of the boundary integral over an internal face F ∈ Γ0. An alternative choice is that of a centered scheme. Both of these options are discussed and compared in [8]. Let Pp(D) denote the space of polynomial functions of degree at most p on a domain

D. For any element K ∈T_h, let Dp_{(K) ≡ (P}

p(K))6. The vectors W and V will be

taken in the space D_hp=V ∈ (L2_{(Ω ))}6_{| V|}

K ∈ Dp(K), ∀K ∈Th .

For the discretization of optimized transmission conditions, let ΓΣ be the set of

faces on Σ , Γ₀j be the set of interior faces of Ωjand Γ_bj be the set of faces of Ωj

lying on ∂ Ω . Then the weak form in the two-subdomain case can be written as L (W1, V1) +

_∑

Γ₀1  +

_∑

Γ_b1  +

_∑

F∈ΓΣ  1 2(|GnK| − GnK) (W1− W2), V1  F = 0, L (W2, V2) +

∑

Γ₀2  +

_∑

Γ_b2  +

_∑

F∈ΓΣ  1 2 |GnK˜| − GnK˜
(W2− W1), V2  F = 0, (8)

whereL (Wj, Vj) ≡ (G0Wj, Vj) + (∑lGl∂lWj, Vj) and, for simplicity, we have

replaced some terms on the faces that are not important for the presentation by a . For any face F = K ∩ ˜Kon Σ , if n denotes the normal on Σ directed from Ω1towards

Ω2, and K and ˜Kare elements of Ω1and Ω2, we have nK= n = −nK˜. In order to

simplify the notation, we make use of G−_n =1₂(Gn− |Gn|) and G+n =12(Gn+ |Gn|).

Then, starting from initial guesses W0₁ and W0₂, the classical Schwarz algorithm computes the iterates Wn+1_j from Wn_jby solving on Ω1and Ω2the subproblems

L (Wn+1 1 , V1) +

∑

Γ₀1  +

_∑

Γ_b1  −

_∑

F∈ΓΣ G− n(Wn+11 − Wn2), V1_F= 0, L (Wn+1 2 , V2) +

∑

Γ₀2  +

_∑

Γ_b2  +

_∑

F∈ΓΣ G+ n(Wn+12 − W n 1), V2_F= 0. (9)

In order to introduce optimized transmission conditions (3) into the DG discretiza-tion, we first want to show explicitly what transmission conditions the classical relaxation in (9) corresponds to. To do so, the subdomain problems solved in (9) are not allowed to depend on variables of the other subdomain anymore, since the coupling will be performed with the transmission conditions, and we thus need to introduce additional unknowns, namely Wn+1_2,Ω

1 on Ω1and W

n+1

1,Ω2 on Ω2, in order to

write the classical Schwarz iteration with local variables only, i.e. L (Wn+1 1 , V1) +

∑

Γ₀1  +

_∑

Γ_b1  −

_∑

F∈ΓΣ D G−_n(Wn+1₁ − Wn+1_2,Ω 1), V1 E F= 0, L (Wn+1 2 , V2) +

∑

Γ₀2  +

_∑

Γ_b2  +

_∑

F∈ΓΣ D G+_n(Wn+1₂ − Wn+1_1,Ω 2), V2 E F= 0. (10)

Comparing with the classical Schwarz algorithm (9), we see that in order to ob-tain the same algorithm, the transmission conditions for (10) need to be chosen as G−_nWn+1_2,Ω

1= G

−

nWn2and G+nWn+11,Ω2 = G

+

6 El Bouajaji et al. the algorithm converges, we must verify the coupling conditions

G−_nW_2,Ω1 = G−_nW2, G+nW1,Ω2= G

+

nW1, (11)

where we dropped the iteration index to denote the limit quantities. The Schwarz algorithm (10) can however also be used with optimized transmission conditions (3), which have to be the DG discretization of the strong relations

G−_nWn+1_2,Ω 1+ S1G + nWn+11 = G − nWn2+ S1G + nWn1,Ω2, G+_nWn+1_1,Ω 2+ S2G − nWn+12 = G+nWn1+ S2G−nWn2,Ω1. (12)

Then, we want to show the equivalence between (11) and the DG discretization we adopt for the transmission conditions (12) at convergence in a 2d case. First, from (4) note that relation (11) is equivalent to

NnNnTE2,Ω1− NnH2,Ω1 = NnN T nE2− NnH2, NnNnTE1,Ω2+ NnH1,Ω2 = NnN T nE1+ NnH1. (13)

We translate these relations using auxiliary variables Λ2,Ω1 := E2,Ω1− NnH2,Ω1,

Λ2:= E2− NnH2, Λ1,Ω2 := E1,Ω2+ NnH1,Ω2 and Λ1:= E1+ NnH1 belonging to

the trace space M_hp=

η ∈ L2(Σ ) | η|F∈ Pp(F), ∀F ∈ Σ . Then (13) becomes

Λ2,Ω1= Λ2 and Λ1,Ω2 = Λ1. (14)

From (12) and (14), we have to find for optimized transmission conditions a suitable DG discretization of the relations

Λ2,Ω1+ ˜S1Λ1= Λ2+ ˜S1Λ1,Ω2 and Λ1,Ω2+ ˜S2Λ2= Λ1+ ˜S2Λ2,Ω1. (15)

We focus on the case of second order transmission conditions and (15) becomes (−∂τ2+ iωσ − 2ω 2_{+ 2iωs} 1)(Λ2,Ω1− Λ2) + (−∂ 2 τ+ iωσ )(Λ1,Ω2− Λ1) = 0, (−∂τ2+ iωσ − 2ω 2_{+ 2iωs} 2)(Λ1,Ω2− Λ1) + (−∂ 2 τ+ iωσ )(Λ2,Ω1− Λ2) = 0. (16)

Let (ηj)jbe a basis of M_hp. We define the discrete matrices MΣ and KΣby

(M_Σ)i, j=

_∑

F∈Σ hηi, ηjiF, (KΣ)i, j =

_∑

F∈Σ h∂τηi, ∂τηjiF+

∑

n∈Σ0 αnh−1[[[[ηi]]]]n[[[[ηj]]]]n −

_∑

n∈Σ0 {{∂τηi}}n[[[[ηj]]]]n− [[[[ηi]]]]n  ∂τηj n,

where positiveness is guaranteed for sufficiently large αn, Σ0denotes the set of

in-terior nodes of Σ , [[[[·]]]]nand {{·}}ndenotes the jump and the average at a node

nbetween values of the neighboring segments. The matrix KΣ comes from the

dis-cretization of −∂τ2using a symmetric interior penalty approach [2]. If we denote by

 AΣ− 2(ω2− iωs1)MΣ AΣ AΣ AΣ− 2(ω2− iωs2)MΣ   Λ2,Ω1− Λ2 Λ1,Ω2− Λ1  = 0. (17)

Theorem 1. If s1 and s2 are defined as given in [11] (section 5, table 5.1) then

relations (14) and (17) are equivalent.

The proof is based on the invertibility of the matrix of (17) and can be found in [7].

4 Numerical results

In order to illustrate numerically the proposed discrete versions of the optimized Schwarz algorithms, we consider the propagation of a plane wave in a homoge-neous conductive medium with Ω = [0, 1]2 and σ = 0.5. We use DG with sev-eral orders of polynomial interpolation, denoted by DG-Pk with k = 1, 2, 3, 4,

and impose on ∂ Ω = Γa an incident wave Winc = (k_ωy −k_ωx 1 )Te−ik·x, and k =

( kxky)T = ( ω

q 1 − iσ

ω 0 )

T_{. The domain Ω is decomposed into two subdomains}

Ω1= [0, 0.5] × [0, 1] and Ω2= [0.5, 1] × [0, 1]. The aim is to retrieve numerically the

asymptotic behavior of the convergence factors of the optimized Schwarz methods. It has been proved that these factors behave like 1 − O(hαi_{), i = 2, 3, 4, 5. We show}

here that numerically they behave like 1 − O(hβi_{), i = 2, 3, 4, 5, with β}

i≈ αi. The

performance of these algorithms is summarized in Figure 1.

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8 El Bouajaji et al. 10ï1 102 h Iterations Alg. 2 O(hï1/2₎ Alg. 3 O(hï3/7₎ Alg. 4 O(hï1/4) Alg. 5 O(hï3/13₎ 10ï1 102 h Iterations Alg. 2 O(hï1/2₎ Alg. 3 O(hï3/7₎ Alg. 4 O(hï1/4) Alg. 5 O(hï3/13₎ DG-P1method DG-P2method 10ï1 102 h Iterations Alg. 2 O(hï1/2₎ Alg. 3 O(hï3/7) Alg. 4 O(hï1/4) Alg. 5 O(hï3/13₎ 10ï1 102 h Iterations Alg. 2 O(hï1/2) Alg. 3 O(hï3/7₎ Alg. 4 O(hï1/4₎ Alg. 5 O(hï3/13₎ DG-P3method DG-P4method

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