Rapid parallel evaluation of integrals in potential theory on general three-dimensional regions

We present a new, high order accurate method for the rapid, parallel evaluation of certain integrals in potential theory on general three-dimensional regions. These methods use fast methods for solving the differential equation which the kernel satisfies, and the number of operations needed to evaluate the integrals is essentially equal to the number of operations needed to solve the differential equation on a regular rectangular grid. In particular, one can evaluate integrals whose kernels are the Greens function for Poissons equation by using Fourier methods on a rectangular grid, or, a fast Poisson solver. Thus, these methods avoid the problems associated with using quadrature methods to evaluate an integral with a singular kernel. Numerical results are presented for experiments on a variety of geometries. (C) 1998 Academic Press.