×

A Schwarz alternating algorithm for elliptic boundary value problems in an infinite domain with a concave angle. (English) Zbl 1071.65171

The authors study a Schwarz iterative algorithm used to solve elliptic boundary value problems formulated upon an infinite domain with a concave angle. The introduction of two artificial boundaries allows to solve the original problem in a bounded domain by a standard finite element method and in an unbounded domain by the natural boundary element method. The convergence of the resulting algorithm is carefully analyzed and some numerical experiments prove the effectiveness of the method.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Givoli, D., Numerical Methods for Problems in Infinite Domains (1992), Elsevier: Elsevier Amsterdam · Zbl 0788.76001
[2] Yu, Dehao, Harmonic canonical integral equations with cracked and concave angle domains, Chinese Journal of Numerical Mathematics and Applications, 4, 3, 193-198 (1983)
[3] Yu, Dehao, Coupling canonical boundary element methods with FEM to solve harmonic problem over cracked domain, Journal of Computational Mathematics, 1, 3, 195-202 (1983) · Zbl 0557.65070
[4] Du, Qikui; Yu, Dehao, On the natural integral equation for initial boundary value problems of two dimensional hyperbolic equations, Acta Mathematica Applied Sinica, 24, 1, 17-26 (2001), (in Chinese) · Zbl 0974.35071
[5] Givoli, D.; Rivkin, L.; Keller, J. B., A finite element method for domains with corners, International Journal for Numerical Methods in Engineering, 35, 1329-1345 (1992) · Zbl 0768.73072
[6] Wu, Xiaonan; Han, Houde, A finite-element method for Laplace and Helmholtz-type boundary value problems with singularities, SIAM Journal of Numerical Analysis, 34, 3, 1037-1050 (1997) · Zbl 0873.65100
[7] Bao, Weizhu; Han, Houde, High-order local artificial boundary conditions for problems in unbounded domains, Computer Methods in Applied Mechanics and Engineering, 188, 1/3, 455-471 (2000) · Zbl 0957.65096
[8] Chinese Journal of Numerical Mathematics and Applications, 17, 1, 95-105 (1995)
[9] Dehao Yu, Domain decomposition method for unbounded domains, in: 8th International Conference on Domain Decomposition Methods (Beijing, 1997), 1997, pp. 125-132; Dehao Yu, Domain decomposition method for unbounded domains, in: 8th International Conference on Domain Decomposition Methods (Beijing, 1997), 1997, pp. 125-132
[10] Chinese Journal of Numerical Mathematics and Applications, 18, 4, 93-102 (1996) · Zbl 0928.65146
[11] Chinese Journal of Numerical Mathematics and Applications, 20, 1, 89-101 (1998)
[12] Chinese Journal of Numerical Mathematics and Applications, 22, 3, 55-72 (2000) · Zbl 0960.65133
[13] Lu, Tao; Shih, T. M.; Lie, C. B., Domain Decomposition Method-New Numerical Techniques for PDE (1992), Science Press: Science Press Beijing, (in Chinese)
[14] Givoli, D.; Patlashenko, I., Finite-element solution of nonlinear time-dependent exterior wave problems, Journal of Computational Physics, 143, 1, 241-258 (1999) · Zbl 0923.65066
[15] Keller, J. B.; Givoli, D., Exact non-reflecting boundary conditions, Journal of Computational Physics, 82, 1, 172-192 (1989) · Zbl 0671.65094
[16] Yu, Dehao, Mathematical Theory of Natural Boundary Element Methods (1993), Science Press: Science Press Beijing, (in Chinese)
[17] Du, Qikui; Yu, Dehao, A domain decomposition method based on natural boundary reduction for nonlinear time-dependent exterior wave problems, Computing, 68, 2, 111-129 (2002) · Zbl 1004.65098
[18] Yu, Dehao, Natural Boundary Integral Method and Its Applications (2002), Science Press/Kluwer Academic Publishers · Zbl 1115.65379
[19] Harari, I.; Huges, T. J.R., Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Computer Methods Applied Mechanics and Engineering, 97, 1, 103-124 (1992) · Zbl 0769.76063
[20] Givoli, D.; Keller, J. B., A finite element method for large domains, Computer Methods Applied Mechanics and Engineering, 76, 1, 41-66 (1989) · Zbl 0687.73065
[21] Givoli, D., Non-reflecting boundary conditions, Journal of Computational Physics, 94, 1-24 (1991) · Zbl 0731.65109
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.