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 for and the mesh size for fixed . 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.
DOI: 10.1051/m2an/2015058
Keywords: Robin−Robin domain decomposition method, Helmholtz equation, optimal convergence rate
@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/
Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451–484. | MR | Zbl
, and ,Wave-ray multigrid method for standing wave equations, Electron. Trans. Numer. Anal. 6 (1997) 162–181. | MR | Zbl
and ,On a Robin−Robin domain decomposition method with optimal convergence rate. J. Comput. Math. 32 (2014) 456–475. | DOI | MR | Zbl
, and ,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
An accelerated domain decomposition procedures based on Robin transmission conditions. BIT 37 (1997) 678-686. | DOI | MR | Zbl
and ,Accelerated domain decomposition iterative procedures for mixed methods based on Robin transmission conditions. Calcolo 35 (1998) 131–147. | DOI | MR | Zbl
and ,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
FETI-DPH: a dual-primal domain decomposition method for accoustic scattering. J. Comput. Acoustics 13 (2005) 499–524. | DOI | MR | Zbl
, , and ,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
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
, and ,Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38–60. | DOI | MR | Zbl
, and ,Generalization and accelerations of Lions’ nonoverlapping domain decomposition method for linear elliptic PDE. SIAM J. Numer. Anal. 41 (2003) 2056–2080. | DOI | MR | Zbl
and ,F. Ihlenburg, Finite Element Analysis of Acoustic Scattering. Vol. 132 of Appl. Math. Sci. Springer-Verlag, New York (1998). | MR | Zbl
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
and ,On a parallel Robin-type nonoverlapping domain decomposition method. SIAM J. Numer. Anal. 44 (2006) 2539–2558. | DOI | MR | Zbl
and ,On the convergence rate of a parallel nonoverlapping domain decomposition method. Sci. China, Ser. A: Math. 51 (2008) 1461–1478. | DOI | MR | Zbl
, and ,A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory. Vol. 34 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (2005). | MR | Zbl
Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29 (2007) 1942–1958. | DOI | MR | Zbl
, and ,Spectral analysis of DN operators and optimized Schwarz methods with Robin transmission conditions. SIAM J. Numer. Anal. 47 (2010) 4540–4568. | DOI | MR | Zbl
and ,Cited by Sources: