Abstract

Green’s integral representations for two-dimensional PDEs arising in scattering can be expressed as smooth modifications of logarithmic and Cauchy kernels. An efficient way to numerically solve these is to find exact expressions for the logarithmic and Cauchy kernels on the underlying geometries. This talk explores how this can be accomplished using orthogonal polynomials on an interval, and by mapping the interval to a curve or wedge by a polynomial, rational, or algebraic map. The Cauchy transform over the mapped domain can be expressed exactly in terms of the unmapped Cauchy transform using new change-of-variable formulae, and multiple wedges can be combined to solve PDEs on polygons, that is, by breaking the polygon into elements but where the elements have a corner in the middle. These maps exactly capture the singularities that arise in corners for solutions ot PDEs leading to spectral convergence.