A Parallel Schwarz Method for Multiple Scattering Problems

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1 A Parallel Schwarz Method for Multiple Scattering Problems Daisuke Koyama The University of Electro-Communications, Chofu, Japan, 1 Introduction Multiple scattering of waves is one of the important research topics in scientific and industrial fields A number of numerical methods have been developed to compute waves scattered by several obstacles, eg, acoustic waves scattered by schools of fish, water waves by ocean structures, and elastic waves by particles in composite materials In this paper, we focus on the acoustic scattering, which is described as solutions of boundary value problems of the Helmholtz equation in an unbounded domain In order to compute such solutions numerically, the following method is well known [2, 6]: one introduces an artificial boundary and imposes an artificial boundary condition on it to reduce the original problem on the unbounded domain to a problem on a bounded domain enclosed by the artificial boundary Recently, [ 4] have developed a new method by extending the method above to the multiple scattering problem In their method, one introduces several disjoint artificial boundaries, each of which surrounds one of the obstacles, and imposes an exact non-reflecting boundary condition on such artificial boundaries This boundary condition is called the multiple DtN (Dirichlet-to-Neumann) boundary condition In this paper, we parallelize their method by a parallel nonoverlapping Schwarz method due to [5] The original unbounded domain is then decomposed into bounded subdomains, each of which is surrounded by one of the artificial boundaries, and the remaining unbounded subdomain A particular feature of this method is including a problem in the unbounded subdomain, imposing Sommerfeld s radiation condition This problem is reduced to a certain problem on the multiple artificial boundaries by the natural boundary reduction due to [2] The remainder of this paper is organized as follows In Sect 2 we introduce the exterior Helmholtz problem, and present a parallel Schwarz algorithm and its convergence theorem, which is proved by the energy method due to [ 1] in Sect 5We introduce the multiple DtN operator associated with the problem on the unbounded subdomain in Sect 3 We describe how to reduce the problem on the unbounded subdomain to the problem on the multiple artificial boundaries in Sect 4

2 352 D Koyama 2 Exterior Helmholtz Problem and Schwarz Method We consider the following exterior Helmholtz problem: Δu k 2 u = f in Ω, lim r + r 1 2 ( u r iku ) u =0 =0, on J O j, (1) where k is a ( positive constant, O j (1 j J) are bounded open sets of R 2, J ) Ω := R 2 \ O j, and f is a given datum Assume that Ω is connected, f has a compact support, and O j (1 j J) are of class C Problem (1) has a unique solution belonging to Hloc 2 (Ω ) for every compactly supported f L 2 (Ω ) (see [7]), where H m loc (Ω ):={u u H m (B) for all bounded open set B Ω (m N) 21 Domain Decomposition Suppose that for each 1 j J, there exists a ball B j with radius a j and center c j such that O j B j, supp f J B j, and B j B l = if j l We introduce, for every 1 j J, artificial boundaries: j := { x R 2 x c j = a j and bounded domains: Ω ( j := B j \ O j We further introduce the following unbounded J ) domain: Ω 0 := R 2 \ B j Then we can decompose Ω into subdomains Ω 0, Ω 1,, Ω J : Ω = J j=0 Ω j, and we have Ω j Ω l = if j l 22 A Parallel Schwarz Method To solve problem (1), we consider the following parallel Schwarz method of Lions: (1) Choose u 0 0 and u0 j (1 j J) (2) For n =1, 2,, solve Δu n 0 k2 u n 0 =0 in Ω 0, un 0 iku n 0 = un 1 j iku n 1 j on j (1 j J), ( ) u n lim 0 r + r1/2 r ikun 0 =0, (2) and for 1 j J, Δu n j k2 u n j = f in Ω j, u n j =0 on O j, u n (3) j iku n j = un 1 0 iku n 1 0 on j

3 A Parallel Schwarz Method for Multiple Scattering Problems 353 Here, for each 1 j J, (r j,θ j ) are polar coordinates with origin c j, and then the normal derivative on j is expressed by / As is well known, problems (2) and (3), respectively, have a unique solution Theorem 1 Let u, u n 0, and u n j (1 j J) be the solutions of problems (1), (2), and (3), respectively If u 0 0 iku 0 0 and u0 j iku 0 j H 1/2 ( j ) (1 j J), (4) j j then we have u n 0 u in L2 ( ) and u n j u in H 1 (Ω j )(1 j J) as n +, where := J j 3 Multiple DtN Operator We introduce the multiple DtN operator S : H 1/2 ( ) J = H 1/2 ( j ) H 1/2 ( ) J = H 1/2 ( j ) defined by p 1 p 2 S : p = p J u r 1 1 u r 2 2 u r J J where p j := p j (1 j J), and u is the solution of the following problem: Δu k 2 u =0 in Ω 0, ( ) u = p on, u lim r + r1/2 r iku =0 To represent S in an implicit form involving some operators which can be analytically represented, we introduce, for each 1 j J, an operator G j : H 1/2 ( j ) Hloc 1 (D j) defined by G j [ϕ j ]:=w j, where w j Hloc 1 (D j) is the solution of the following problem: Δw j k 2 w j =0 in D j, ( w) j = ϕ j on j, (5) lim r 1 2 wj r j ikw j =0, j + where D j := { x R 2 x c j >a j,

4 354 D Koyama Theorem 2 If u Hloc 1 (Ω 0) satisfies ( Δu k 2 ) u =0 in Ω 0, lim r 1 u 2 r + r iku =0, then there uniquely exists a ϕ H 1/2 ( ) such that u = J G j [ϕ j ] in Ω 0 (6) Proof See [4] Taking the traces and the normal derivatives / of both sides of (6) on j (1 j J), weget u = ϕ j + P jl [ϕ l ] on j (7) l j and u = S j [ϕ j ] T jl [ϕ l ] on j, (8) l j respectively, where P jl [ϕ l ]:= G l [ϕ l ] j, T jl [ϕ l ]:= G l [ϕ l ] S j [ϕ j ]:= G j [ϕ j ] j j (1 j J) (1 j l J), We here remark that S j is the single DtN operator associated with (5) Deleting ϕ j from (7) and (8), we can see that the multiple DtN operator S can be represented as follows: S = TC 1, (9) where I P 12 P 1J S 1 T 12 T 1J P 21 I P 2J C :=, T := T 21 S 2 T 2J P J1 P J2 I T J1 T J2 S J As we all know, we can obtain an analytical representation of G j by separation of variables, and hence, from this representation, we can derive analytical representations of the operators S j, T jl, and P jl Note that we have C Isom(H 1/2 ( ), H 1/2 ( )), the proof of which will appear elsewhere

5 4 How to Solve Problem (2) A Parallel Schwarz Method for Multiple Scattering Problems 355 If u n 0 is the solution of problem (2), then we have un 0 = Su n 0 (1 j J) j Hence we can reduce problem (2) to the following problem on : find p H 1/2 ( ) such that Sp ikp = λ, (10) where λ H 1/2 ( ) and λ = u n 1 j / iku n 1 j on j This process is often called the natural boundary reduction (cf [2]) The solution p of this problem gives the trace on of the solution u n 0 of problem (2) By (9), we have (S iki)p = λ (T ikc)c 1 p = λ Hence, we can solve (10) by executing the following two processes: Solve (T ikc)ϕ = λ Compute p = Cϕ Equation (T ikc)ϕ = λ can be written in the following variational form: find ϕ H 1/2 ( ) such that S j ϕ j,ψ j j + T jl [ϕ l ] ψ j d j l j j ik ϕ j ψ j d j + P jl [ϕ l ] ψ j d j j l j j = λ j,ψ j j ψ j H 1/2 ( j ), 1 j J, (11) where, j is the duality pairing between H 1/2 ( j ) and H 1/2 ( j ) We can practically compute (11) by using the analytical representations of the operators S j, T jl, and P jl Discretizing (11) by a FEM, we need to solve a linear system whose matrix is full and of order the number of nodes on 5 Proof of Theorem 1 We can prove Theorem 1 by the energy technique due to [1] Let u, u n 0, and un j (1 j J) be the solutions of problems (1), (2), and (3), respectively Put e n j := u un j (0 j J) We should keep in mind that if (4) is satisfied, then e n 0 Hloc 2 (Ω 0) and e n j H2 (Ω j )(1 j J) for all n N We now define a pseudo energy E n by J E n := Se n 0 ike n 0 2 e d + n 2 j ike n j d j j

6 356 D Koyama Lemma 1 {E n n=1 is a decreasing sequence Proof Because e n j satisfies the homogeneous equation in Ω j and the homogeneous boundary condition on O j,wehave { Im en j e n j r d j =0 (12) j j for every n N and for each 1 j J Hence, for every n N, we have the following energy equality: { E n+1 = E n +4kIm Se n 0 en 0 d (13) Since e n 0 also satisfies the homogeneous equation in Ω 0,wehave { Im Se n 0 e n 0 d = kr μ= Im {H (1) μ (kr) H μ (1) (kr) e n 0,μ(R) 2 (14) for an arbitrary positive number R satisfying {x R 2 x <R, where H μ (1) is the first kind Hankel function of order μ, and e n 0,μ(R) is the μth Fourier coefficient of e n 0 on R := {x R 2 x = R As we all know, we have Im {H (1) μ (kr) > 0 (15) H μ (1) (kr) for all μ Z Therefore, combining (13), (14) and (15) completes the proof of Lemma 1 Proposition 1 We have S iki Isom(H 1/2 ( ), H 1/2 ( )) Proof We can prove from the two facts that S iki is a bounded linear operator from H 1/2 ( ) onto H 1/2 ( ), and that the following problem: Δu k 2 u =0 in Ω 0, u iku = λ on j (1 j J), ( ) u lim r + r1/2 r iku =0 has a unique solution belonging to Hloc 1 (Ω 0) for every λ H 1/2 ( ) Proof of Theorem 1 Proposition 1 assures us that there exists a positive constant C such that e n+1 0 H 1/2 ( ) C λ H 1/2 ( ), (16) where λ j = e n j / ike n j Using (12) and Lemma 1,wehave

7 A Parallel Schwarz Method for Multiple Scattering Problems 357 J λ 2 L 2 ( ) = j e n j ike n j 2 d j E 1 (17) From (16) and (17), {e n 0 is a bounded sequence in H 1/2 ( ), and hence {e n 0 has a subsequence {e n l 0 such that en l 0 e 0 in H 1/2 ( ) weakly This indicates that for all q H 1/2 ( ), Se n l 0 qd = en l 0,S q e 0,S q = Se 0,q, (18) where S is the DtN operator corresponding to the incoming radiation condition, and, is the duality pairing between H 1/2 ( ) and H 1/2 ( ) On the other hand, from Lemma 1,(14), and (15), we have E 1 Se n 0 iken 0 2 d { = Se n k 2 e n 0 2 { d 2k Im Se n 0 e n 0 d Se n 0 2 d This implies that there exists a subsequence of {Se n l 0, still denoted by {Sen l 0, which converges in L 2 ( ) weakly Thus, we can see from (18) that Se 0 L 2 ( ) and Se n l 0 Se 0 in L 2 ( ) weakly Further, by the compact imbedding of H 1/2 ( ) in L 2 ( ), e n l 0 e 0 in L 2 ( ) strongly, and hence we have Se n l 0 en l 0 d Se 0 e 0 d Now, from (13), we get E n+1 = E 1 +4k n m=1 { Im Se m 0 e m 0 d, and hence Im { Sem 0 e m 0 d 0 Thereby we have Im { Se 0e 0 d =0 This implies e 0 =0 Therefore, we can conclude that u n 0 u in L2 ( ) We can show that u n j u in H 1 (Ω j )(1 j J) in the same way as in the proof of Theorem 26 in [1] 6 Concluding Remarks We demonstrated the convergence of the parallel Schwarz method of Lions for multiple scattering problems Many techniques of acceleration of the convergence of the Schwarz method have been developed (see [3, 8]) The investigation of acceleration techniques is yet to be done for the Schwarz method presented in this paper

8 358 D Koyama References 1 B Després Domain decomposition method and the Helmholtz problem In Mathematical and Numerical Aspects of Wave Propagation Phenomena (Strasbourg, 1991), pp SIAM, Philadelphia, PA, K Feng Finite element method and natural boundary reduction In Proceeding of the International Congress of Mathematicians, Warsaw, Poland, MJ Gander, F Magoulès, and F Nataf Optimized Schwarz methods without overlap for the Helmholtz equation SIAM J Sci Comput, 24(1):38 60, MJ Grote and C Kirsch Dirichlet-to-Neumann boundary conditions for multiple scattering problems J Comput Phys, 201(2): , P-L Lions On the Schwarz alternating method III: a variant for nonoverlapping subdomains In TF Chan, R Glowinski, J Périaux, and O Widlund, editors, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20 22, 1989, SIAM, Philadelphia, PA, M Masmoudi Numerical solution for exterior problems Numer Math, 51:87 101, RS Phillips On the exterior problem for the reduced wave equation In Partial Differential Equations (Proc Sympos Pure Math, Vol XXIII, Univ California, Berkeley, CA, 1971), pp , AMS, Providence, RI, D Tromeur-Dervout Aitken-Schwarz method: acceleration of the convergence of the Schwarz method In F Magoulès and T Kako, editors, Domain Decomposition Methods: Theory and Applications, chapter 2, pp 37 64, Gakkotosho, Tokyo, 2006