Sheehan Olver: Newtonian potentials have displacement structure

Abstract

We show that Newtonian potentials (and their gradient) applied to tensor products of Legendre polynomials can be expressed in terms of a matrix which has displacement structure, a simple recurrence relationship that lends itself to exact computation which avoids the need for singular integral quadrature. 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. The results can also be extended to trapeziums.We show that Newtonian potentials (and their gradient) applied to tensor products of Legendre polynomials can be expressed in terms of a matrix which has displacement structure, a simple recurrence relationship that lends itself to exact computation which avoids the need for singular integral quadrature. 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. The results can also be extended to trapeziums.