Newtonian potentials of Legendre polynomials on rectangles have displacement structure

Particular solutions of the Poisson equation can be constructed via Newtonian potentials, integrals involving the corresponding Green's function which in two-dimensions has a logarithmic singularity. The singularity represents a significant challenge for computing the integrals, which is typically overcome via specially designed quadrature methods involving a large number of evaluations of the function and kernel. We present an attractive alternative: we show that Newtonian potentials (and their gradient) applied to (tensor products of) Legendre polynomials can be expressed in terms of complex integrals which satisfy simple and explicit recurrences that can be utilised to exactly compute singular integrals, i.e., singular integral quadrature is completely avoided. The inhomogeneous part of the recurrence has low rank structure (its rank is at most three for the Newtonian potential) and hence these recurrences have displacement structure. Using the recurrence directly is a fast approach for evaluation on or near the integration domain that remains accurate for low degree polynomial approximations, while high-precision arithmetic allows accurate use of the approach for moderate degree polynomials.