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Solution of unbounded domain problems using elliptic artificial boundaries. (English) Zbl 0851.65074

In order to solve the two-dimensional Laplace and Helmholtz equations in an unbounded domain, the authors propose the introduction of an artificial boundary with exact conditions on it. While in previous papers this problem has been dealt with by choosing circular boundaries, in the present one ellipses are used instead, because they can enclose slender obstacles in a more adapted way. Accordingly, the introduction of elliptic coordinates produces appropriate eigenfunctions for each of the equations; for the explicit finite element calculation the authors refer to the circular boundary case.

The performance of the method is demonstrated with the two-dimensional potential flow around an aerfoil and with the time harmonic waves radiated from a rigid circular obstacle; in the first example, comparison with the numerical solution obtained by a far-field flow condition imposed on a rectangular boundary shows the much better accuracy that can be obtained with the approach proposed in the paper.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
76B10Jets and cavities, cavitation, free-streamline theory, water-entry problems, etc.
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation