Reciprocal-log approximation and planar PDE solvers

TL;DR

This paper introduces reciprocal-log approximation for analytic functions with branch points, proving exponential convergence, and applies it to develop a near-exponential convergence method for solving 2D PDEs with corner singularities.

Contribution

It develops a new reciprocal-log approximation framework and a corresponding numerical PDE solver with near-exponential convergence.

Findings

Reciprocal-log approximations converge exponentially with respect to the number of poles.

The log-lightning method achieves near-exponential convergence for PDEs with corner singularities.

The approach outperforms traditional lightning methods based on rational functions.

Abstract

This article is about both approximation theory and the numerical solution of partial differential equations (PDEs). First we introduce the notion of {\em reciprocal-log} or {\em log-lightning approximation} of analytic functions with branch point singularities at points $\{z_k\}$ by functions of the form $g(z) = \sum_k c_k /(\log(z-z_k) - s_k)$, which have $N$ poles potentially distributed along a Riemann surface. We prove that the errors of best reciprocal-log approximations decrease exponentially with respect to $N$ and that exponential or near-exponential convergence (i.e., at a rate $O(\exp(-C N / \log N))$) also holds for near-best approximations with preassigned singularities constructed by linear least-squares fitting on the boundary. We then apply these results to derive a "log-lightning method" for numerical solution of Laplace and related PDEs in two-dimensional domains with corner singularities. The convergence is near-exponential, in contrast to the root-exponential convergence for the original lightning methods based on rational functions.