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An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

On the localization principle of high frequency scattering problems in integral equation discretizations

Author: Daan Huybrechs (K.U. Leuven)

Abstract

The numerical simulation of scattering problems at moderate to high frequencies is a challenging problem. The discretization of any suitable mathematical model usually requires resolving the oscillations, which naturally leads to large and, in the context of integral equations, dense discretization matrices. Yet, high frequency scattering problems have a very local nature. The localization principle states that the reflection, refraction or diffraction of a wave that hits an obstacle is governed mainly by the geometry of the scattering object which is local to the point of contact. This principle is exploited by asymptotic methods, such as geometrical optics and the geometrical theory of diffraction. Suddenly, higher frequencies are desirable and lower frequencies become problematic.

In this talk we examine how the localization principle can be exploited numerically in a more classical finite element setting. In particular, we may arrive at a sparse discretization matrix for integral equations by the use of Filon-type quadrature rules for oscillatory integrals. We discuss the advantages and limitations of this approach and we examine the asymptotic nature of this sparse representation.

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