Numerically derived boundary conditions on artificial boundaries. (English) Zbl 0832.65129
The article is concerned with boundary value problems for elliptic differential equations (in particular, for Laplace’s and Helmholtz’ equation) on unbounded domains. An artificial boundary, and a certain nonlocal boundary condition on it, are introduced to replace the problem appropriately by a boundary value problem on a bounded domain, so that “usual” numerical approximation methods (e.g., finite elements) can be applied. The artificial boundary is chosen such that the exterior domain has a simple geometry, so that separation of variables (generalized polar coordinates) is possible on it. Representation of the solution by the “free space” Green’s function or by a series expansion provides a nonlocal boundary condition on the artificial boundary, which can easily be approximated by a finite element approach.
Reviewer: M.Plum (Clausthal-Zellerfeld)
MSC:
| 65N45 | Method of contraction of the boundary for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
Laplace equation; Helmholtz’ equation; unbounded domains; artificial boundary; nonlocal boundary condition; finite elements; series expansionReferences:
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