The rapid evaluation of volume integrals of potential theory on general regions

We present a new method for the rapid, high order accurate evaluation of certain volume integrals in potential theory on general irregular regions. The kernels of the integrals are either a fundamental solution, or a linear combination of the derivatives of a fundamental solution of a second-order linear elliptic differential equation. Instead of using a standard quadrature formula or the exact evaluation of any integral, the methods rely on rapid methods of solving the differential equation which the kernel is the solution of. Therefore, the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid, and the method requires no evaluation of the kernel. © 1992.