Reciprocal-log approximation and planar PDE solvers

This article is about both approximation theory and the numerical solution of partial differential equations (PDEs). First we introduce the notion of reciprocal-log or 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 on different sheets of a Riemann surface. We prove that the errors of minimax 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 constructed by linear least-squares fitting on the boundary with suitably chosen preassigned singularities. We then apply these results to derive a “log-lightning method” for the 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.