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The artificial boundary method – numerical solutions of partial differential equations on unbounded domains. (English) Zbl 1097.65115
Li, Tatsien (ed.) et al., Frontiers and prospects of contemporary applied mathematics. Selected papers based on the presentation at the symposium on frontiers and prospects of contemporary applied mathematics, Xiangtan, China, August 24–30, 2004. Beijing: Higher Education Press; Hackensack, NJ: World Scientific (ISBN 7-04-018575-X/hbk). Ser. Contemp. Appl. Math. CAM 6, 33-58 (2005).

Summary: The artificial boundary method has been established for computing the numerical solutions of partial differential equations on unbounded domains. The key points of this method are to find the exact boundary conditions or the approximate artificial boundary conditions on the given artificial boundary for various problems arising in many fields of science and engineering. In general, the artificial boundary conditions can be classified into implicit boundary conditions and explicit boundary conditions including global artificial boundary conditions, local artificial boundary conditions and discrete artificial boundary condition.

The explicit artificial boundary conditions are more convenient in applications, but the implicit artificial conditions can handle the artificial boundary with any shape. This method has attained successful applications in many fields in science and engineering and has shown wider and wider application prospect. In this field there are still many open problems, for example, how to solve the nonlinear partial difference equations on unbounded domains numerically? It is an interesting and important problem waiting for solving.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation