A Galerkin Bem For High-Frequency Scattering Problems Based On Frequency-Dependent Changes Of Variables

In this paper we develop a new class of semidiscrete Galerkin boundary element methods for the solution of two-dimensional exterior single-scattering problems. Our approach is based upon construction of Galerkin approximation spaces confined to the asymptotic behavior of the solution through a certain direct sum of appropriate function spaces weighted by the oscillations in the incident field of radiation. Specifically, the function spaces in the illuminated/shadow regions and the shadow boundaries are simply algebraic polynomials, whereas those in the transition regions are generated utilizing novel, yet simple, frequency-dependent changes of variables perfectly matched with the boundary layers of the amplitude in these regions. We rigorously verify for compact, smooth and strictly convex obstacles that, with increasing wavenumber k, these methods require, for any epsilon > 0, only an O (k(epsilon)) increase in the number of degrees of freedom (DoF) to maintain any given accuracy independent on frequency. These theoretical results are confirmed by numerical tests that show that, for sufficiently large DoF, the error tends to decrease with increasing wavenumber k.