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Keywords:
method of fictitious right-hand sides; fast algorithm; standard domain; convergence; numerical example; Poisson equation
method of fictitious right-hand sides; fast algorithm; standard domain; convergence; numerical example; Poisson equation
Summary:
The paper deals with the application of a fast algorithm for the solution of finite-difference systems for boundary-value problems on a standard domain (e.g. on a rectangle) to the solution of a boundary-value problem on a domain of general shape contained in the standard domain. A simple iterative procedure is suggested for the determination of fictitious right-hand sides for the system on the standard domain so that its solution is the desired one. Under the assumptions that are usual for matrices obtained by discretization of elliptic boundary-value problems, the convergence of this procedure for all sufficiently small positive values of a parameter is proved. The method is illustrated by a simple numerical example (solution of the Poisson equation on an $L$-shaped domain).
References:
[1] B. L. Buzbee F. W. Dorr J. A. George G. H. Golub: The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Anal. 8 (1971), 722 -736. DOI 10.1137/0708066 | MR 0292316
[2] W. Proskurowski O. Widlund: On the numerical solution of Helmholtz's equation by the capacitance matrix method. Math. Comput. 30 (1976), 433 - 468. MR 0421102
[3] A. S. L. Shieh: Fast Poisson solves on general two dimensional region for the Dirichlet problem. Numer. Math., 31 (1979), 405-429. DOI 10.1007/BF01404568 | MR 0516582
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