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A weakly singular boundary integral formulation of the external Helmholtz problem valid for all wavenumbers. (English) Zbl 1099.65120

Constanda, Christian (ed.) et al., Integral methods in science and engineering. Theoretical and practical aspects. Selected papers based on the presentation at the 8th international conference, Orlando, FL, USA, August 2–5, 2004. Boston: Birkhäuser (ISBN 0-8176-4377-X/hbk). 79-87 (2006).
From the introduction: Over the last fourty or so years the boundary integral method has become established as one of the most widely used methods for solving the exterior Helmholtz problem. The underlying differential equation can be reformulated as a boundary integral equation either by applying Green’s theorem directly to the solution or by representing the solution in terms of a layer potential function. It is well known that the integral equation formulation arising from either of these methods does not have a unique solution for all real and positive values of the wavenumber.
We consider the formulation of A. J. Burton and G. F. Miller [Proc. R. Soc. Lond. Ser. A 323, 201–210 (1971; Zbl 0235.65080)] for overcoming the nonuniqueness problem. This method has the advantage that it is guaranteed to have a unique solution for all real and positive wavenumbers, but has the disadvantage that it introduces an integral operator with a hypersingular kernel. We present a method for reformulating the hypersingular integral operator in terms of integral operators that have kernel functions which are at worst weakly singular and hence relatively straightforward to approximate by standard numerical methods. Further, we show that the numerical results obtained using the methods described here are considerably more accurate than those obtained by the most widely used existing methods.
For the entire collection see [Zbl 1084.65003].

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Citations:

Zbl 0235.65080
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