Newtonian potentials of Legendre polynomials on rectangles have displacement structure
File(s)
Published version
Author(s)
Olver, Sheehan
Type
Journal Article
Abstract
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.
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.
Date Issued
2026-08-14
Date Acceptance
2026-06-17
Citation
Advances in Computational Mathematics, 2026, 52
ISSN
1019-7168
Publisher
Springer
Journal / Book Title
Advances in Computational Mathematics
Volume
52
Copyright Statement
©TheAuthor(s) 2026 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Publication Status
Published
Article Number
66
Date Publish Online
2026-08-14