Reciprocal-log approximation and planar PDE solvers
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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) = ∑_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((-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.
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