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Asymptotically derived boundary elements for the Helmholtz equation in high frequencies. (English) Zbl 1102.65120
Summary: We present an asymptotically derived boundary element method for the Helmholtz equation in exterior domains. Each basis function is the product of a smooth amplitude and an oscillatory phase factor, like the asymptotic solution. The phase factor is determined a priori by using arguments from geometrical optics and the geometrical theory of diffraction, while the smooth amplitude is represented by high-order splines. This yields a high-order method in which the number of unknowns is virtually independent of the wavenumber \(k\).
Two types of diffracted basis functions are presented: the first accounts for the dominant oscillatory behavior in the shadow region while the second also accounts for the decay of the amplitude there. We show that the matrix \(A\), associated with the discrete problem, has only \(O(N)\) significant entries as \(k\to\infty\), where \(N\) is the number of basis functions. Hence it can be approximated with a matrix \(\widehat A\) having \(O(N)\) terms, and the relative error between \(A\) and \(\widehat A\) rapidly converges to zero as \(k\to\infty\). Although the method is applicable to a variety of scatterers, we focus our attention here on scattering from smooth closed convex bodies in two dimensions. Computations on a circular cylinder illustrate our results.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A05 Geometric optics
78A45 Diffraction, scattering
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
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