A New Spectral Boundary Integral Collocation Method for Three-Dimensional Potential Problems

Abstract

In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces $\Gamma \subset {\Bbb R}^3$. Under the assumption that $\Gamma$ is diffeomorphic to the unit sphere $\partial B$, the original equation is transferred to an equivalent one on $\partial B$ which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.