Paper Info
Title
RECIPROCAL-LOG APPROXIMATION AND PLANAR PDE SOLVERS
Abstract
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(kappa)} by functions of the form g(z) = Sigma(kappa) c(kappa)/(log(z - z(kappa)) - s(kappa)), 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(-CN/ log N))) also holds for near-best approximations constructed by linear leastsquares 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.
Year
DOI
Venue
2021
10.1137/20M1369555
SIAM JOURNAL ON NUMERICAL ANALYSIS
Keywords
DocType
Volume
reciprocal-log approximation, rational approximation, lightning solver
Journal
59
Issue
ISSN
Citations 
6
0036-1429
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Yuji Nakatsukasa19717.74
Lloyd N. Trefethen21024203.66