Fully discrete spectral methods for boundary integral equations on slender spheroids

We develop fully discrete spectral boundary integral algorithms to solve interior and exterior potential boundary value problems on three-dimensional slender spheroids. Our approach is based on identifying and using analytical solutions of associated (generalised) eigenvalue problems of boundary integral operators. We generalise approximation theory results on the sphere using spheroidally appropriate eigenfunctions and solve an open problem of stability of spectral methods for first kind boundary integral equations on spheroids.