A robust domain decomposition method for the Helmholtz equation with high wave number
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 3, pp. 921-944.

In this paper we present a robust Robin−Robin domain decomposition (DD) method for the Helmholtz equation with high wave number. Through choosing suitable Robin parameters on different subdomains and introducing a new relaxation parameter, we prove that the new DD method is robust, which means the convergence rate is independent of the wave number k for kh=constant and the mesh size h for fixed k. To the best of our knowledge, from the theoretical point of view, this is a first attempt to design a robust DD method for the Helmholtz equation with high wave number in the literature. Numerical results which confirm our theory are given.

Received:
DOI: 10.1051/m2an/2015058
Classification: 65N55
Keywords: Robin−Robin domain decomposition method, Helmholtz equation, optimal convergence rate
Chen, Wenbin 1; Liu, Yongxiang 2; Xu, Xuejun 2, 3

1 School of Mathematical Sciences, Fudan University, Shanghai 200437, China
2 LSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, China
3 Department of Mathematics, Tongji University, Shanghai 200092, China
@article{M2AN_2016__50_3_921_0,
     author = {Chen, Wenbin and Liu, Yongxiang and Xu, Xuejun},
     title = {A robust domain decomposition method for the {Helmholtz} equation with high wave number},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {921--944},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {3},
     year = {2016},
     doi = {10.1051/m2an/2015058},
     zbl = {1361.65093},
     mrnumber = {3507279},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015058/}
}
TY  - JOUR
AU  - Chen, Wenbin
AU  - Liu, Yongxiang
AU  - Xu, Xuejun
TI  - A robust domain decomposition method for the Helmholtz equation with high wave number
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 921
EP  - 944
VL  - 50
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015058/
DO  - 10.1051/m2an/2015058
LA  - en
ID  - M2AN_2016__50_3_921_0
ER  - 
%0 Journal Article
%A Chen, Wenbin
%A Liu, Yongxiang
%A Xu, Xuejun
%T A robust domain decomposition method for the Helmholtz equation with high wave number
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 921-944
%V 50
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015058/
%R 10.1051/m2an/2015058
%G en
%F M2AN_2016__50_3_921_0
Chen, Wenbin; Liu, Yongxiang; Xu, Xuejun. A robust domain decomposition method for the Helmholtz equation with high wave number. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 3, pp. 921-944. doi : 10.1051/m2an/2015058. http://www.numdam.org/articles/10.1051/m2an/2015058/

I.M. Babuška, and S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451–484. | MR | Zbl

A. Brandt and I. Livshits, Wave-ray multigrid method for standing wave equations, Electron. Trans. Numer. Anal. 6 (1997) 162–181. | MR | Zbl

W. Chen, X. Xu and S. Zhang, On a Robin−Robin domain decomposition method with optimal convergence rate. J. Comput. Math. 32 (2014) 456–475. | DOI | MR | Zbl

Q. Deng, A nonoverlapping domain decomposition method for nonconforming finite element problems. Commun. Pure Appl. Anal. 2 (2003) 295–306. | MR | Zbl

B. Despres, Domain Decomposition Method and Helmholtz Problem. Mathematical and Numerical Aspects of Wave Propagation Phenomena, edited by G. Cohen, L. Halpern and P. Joly. Philadelphia, SIAM (1991) 44–52. | MR

J. Douglas and C.S. Huang, An accelerated domain decomposition procedures based on Robin transmission conditions. BIT 37 (1997) 678-686. | DOI | MR | Zbl

J. Douglas and C.S. Huang, Accelerated domain decomposition iterative procedures for mixed methods based on Robin transmission conditions. Calcolo 35 (1998) 131–147. | DOI | MR | Zbl

O.G. Ernst and M.J. Gander, Why is Difficult to Solve Helmholtz Problems with Classical Iterative Methods. Numerical Analysis of Multiscale Problems, edited by I. Graham, T. Hou, O. Lakkis and R. Scheichl. Springer-Verlag, New York (2011) 325–363. | MR | Zbl

C. Farhat, A. Macedo and R. Tezaur, FETI-H: A Scalable Domain Decomposition Method for High Frequency Exterior Helmholtz Problem. In 11th International Conference on Domain Decomposition Method, edited by P. Bjørstad, M. Cross and O. Widlund. Choi-Hong Lai, DDM.ORG (1999) 231–241. | MR

C. Farhat, P. Avery, R. Tezaur and J. Li, FETI-DPH: a dual-primal domain decomposition method for accoustic scattering. J. Comput. Acoustics 13 (2005) 499–524. | DOI | MR | Zbl

M.J. Gander, L. Halpern and F. Nataf, Optimized Schwarz Methods. In 12th International Conference on Domain Decomposition Methods, edited by T. Chan, T. Kako, H. Kawarada and O. Pironneau. Chiba, Japan, Domain Decomposition Press (2001) 15–18. | MR

M.J. Gander, L. Halpern and F. Magoules, An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation. Int. J. Numer. Meth. Fluids 55 (2007) 163–175. | DOI | MR | Zbl

M.J. Gander, F. Magoules and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38–60. | DOI | MR | Zbl

W. Guo and L.S. Hou, Generalization and accelerations of Lions’ nonoverlapping domain decomposition method for linear elliptic PDE. SIAM J. Numer. Anal. 41 (2003) 2056–2080. | DOI | MR | Zbl

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering. Vol. 132 of Appl. Math. Sci. Springer-Verlag, New York (1998). | MR | Zbl

J. Li and X. Tu, Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems. Numer. Linear Algebra Appl. 16 (2009) 745–773. | DOI | MR | Zbl

L. Qin and X. Xu, On a parallel Robin-type nonoverlapping domain decomposition method. SIAM J. Numer. Anal. 44 (2006) 2539–2558. | DOI | MR | Zbl

L. Qin, Z. Shi and X. Xu, On the convergence rate of a parallel nonoverlapping domain decomposition method. Sci. China, Ser. A: Math. 51 (2008) 1461–1478. | DOI | MR | Zbl

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory. Vol. 34 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (2005). | MR | Zbl

M.B. Van Gijzen, Y.A. Erlangga and C. Vuik, Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29 (2007) 1942–1958. | DOI | MR | Zbl

X. Xu and L. Qin, Spectral analysis of DN operators and optimized Schwarz methods with Robin transmission conditions. SIAM J. Numer. Anal. 47 (2010) 4540–4568. | DOI | MR | Zbl

Cited by Sources: