On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

TL;DR

This paper presents a rational-function fitting method for efficiently solving 2D Laplace boundary-value problems, achieving high accuracy with fewer basis functions and addressing singularity issues.

Contribution

It introduces a desingularized rational-function approach with pole placement and least-squares boundary enforcement, improving accuracy and efficiency over existing methods.

Findings

Errors near machine epsilon for complex boundary features

Fewer basis functions needed compared to Nyström method

Effective for large-scale problems

Abstract

A computational scheme for solving 2D Laplace boundary-value problems using rational functions as the basis functions is described. The scheme belongs to the class of desingularized methods, for which the location of singularities and testing points is a major issue that is addressed by the proposed scheme, in the context of the 2D Laplace equation. Well-established rational-function fitting techniques are used to set the poles, while residues are determined by enforcing the boundary conditions in the least-squares sense at the nodes of rational Gauss-Chebyshev quadrature rules. Numerical results show that errors approaching the machine epsilon can be obtained for sharp and almost sharp corners, nearly-touching boundaries, and almost-singular boundary data. We show various examples of these cases in which the method yields compact solutions, requiring fewer basis functions than the Nystr\"{o}m method, for the same accuracy. A scheme for solving fairly large-scale problems is also presented.