Abstract

Introduction

There is huge mathematical and engineering interest in acoustic and electromagnetic wave scattering problems, driven by many applications such as modelling radar, sonar, acoustic noise barriers, atmospheric particle scattering, ultrasound and VLSI. For time harmonic problems in infinite domains and media which are predominantly homogeneous, the boundary element method is a very popular solver, used in a number of large commercial codes, see e.g. [CSCVHH04]. In many practical applications the characteristic length scale L of the domain is large compared to the wavelength λ. Then the small dimensionless wavelength λ/L induces oscillatory solutions, and the application of conventional (piece-wise polynomial) boundary elements for this multiscale problem yields full matrices of dimension at least N = (L/λ)d-1 (in ℝd). (Domain finite elements lead to sparse matrices but require even larger N.) Since this “loss of robustness” as L/λ→∞ puts high frequency problems outside the reach of many standard algorithms, much recent research has been devoted to finding more robust methods.