# Solving Laplace problems with corner singularities via rational functions

## TL;DR

This paper introduces a rational function approximation method for efficiently solving 2D Laplace problems with corner singularities, achieving rapid convergence and high accuracy with a simple computational approach.

## Contribution

It extends approximation theory to corner singularities and develops a practical numerical method using rational functions with fixed poles for fast, accurate solutions.

## Key findings

- Achieves root-exponential convergence near corners
- Solves typical problems in less than 1 second on a laptop
- Provides solutions with 8-digit accuracy and rapid evaluation

## Abstract

A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities. We then develop a numerical method to compute approximations via linear least-squares fitting on the boundary. Typical problems are solved in < 1s on a laptop to 8-digit accuracy, with the accuracy guaranteed in the interior by the maximum principle. The computed solution is represented globally by a single formula, which can be evaluated in tens of microseconds at each point.