Solving Laplace problems with corner singularities via rational functions
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.
Keywords
Cite
@article{arxiv.1905.02960,
title = {Solving Laplace problems with corner singularities via rational functions},
author = {Abinand Gopal and Lloyd N. Trefethen},
journal= {arXiv preprint arXiv:1905.02960},
year = {2019}
}