Rapid evaluation of Newtonian potentials on planar domains

The accurate and efficient evaluation of Newtonian potentials over general 2-D domains is a subject of great importance for the numerical solution of Poisson's equation and volume integral equations. Complicated domains typically require discretization by unstructured meshes, over which the direct evaluation of the potential by quadrature becomes costly. In this paper, we present a simple and effective algorithm for computing Newtonian potentials, based on the use of Green's third identity for transforming the volume integral into a collection of boundary integrals, which can then be easily handled by the Helsing-Ojala method. As a result, the time cost of the classically expensive near field and self-interaction computations over an unstructured mesh becomes roughly the same as the time cost of the FMM-based far field interaction computation. One of the key components of our algorithm is the high-order 2-D monomial expansion approximation of a function over a mesh element, which is often regarded as an ill-conditioned problem, since it involves the solution of a Vandermonde linear system. In fact, it has long been observed that, when the function is sufficiently smooth, and when an appropriate linear system solver is used, the resulting monomial expansion can approximate the function uniformly to high accuracy. We rigorously formalize this observation in this paper. The performance of our algorithm is illustrated through several numerical experiments.