Fully discrete spectral boundary integral methods on smooth closed surfaces in IR³ and applications to acoustic scattering

In this talk we describe and analyse a class of spectral methods, based on spherical harmonic approximation, for second-kind weakly singular boundary integral equations arising from the Helmholtz equation on smooth closed 3D surfaces which are diffeomorphic to the sphere. Our methods are fully discrete Galerkin methods, based on the application of special quadrature rules for computing both the outer and inner integrals arising in the Galerkin matrix elements. For the inner integrals, a variant of the classical product integration procedure is employed to remove the singularity arising in the kernel. With the aid of recent results of Sloan and Womersley on the norm of discrete orthogonal projection operators on the sphere, we prove that our methods are stable and superalgebraically convergent for smooth data.

Our method, although suitable only for a restricted range of 3D surfaces, has applications in the iterative solution of certain inverse problems and when artificial boundaries are employed in the handling of inhomogeneous exterior problems. Our theory includes as a special case a method closely related to one of those proposed by Wienert (1990), which is one of the key fast spectral methods for direct and inverse acoustic scattering in 3D.