Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions

We present two methods for the rapid, high order accurate evaluation of integrals in potential theory on general, unbounded 3D regions. Our methods allow for direct calculation of derivatives of the integrals as well. One of the methods uses a fourth order compact stencil, and the other uses a nonstandard variant of Richardson extrapolation. Both methods involve calculation of discontinuities in high order derivatives of the integrals across the boundary of the integration region. The extrapolation method, in addition, involves correction for the discontinuities in truncation error. The number of operations required for the methods is essentially equal to twice the number of operations needed to solve Poisson’s equation on a regular grid. Both methods avoid problems associated with using quadrature methods to evaluate integrals with singular kernels. Numerical results are presented for experiments on a variety of geometries in free space.