Computation of harmonic functions on higher genus surfaces

TL;DR

This paper extends classical approximation results for harmonic functions to Riemann surfaces, providing a method to approximate local harmonic functions globally with prescribed poles, and introduces an efficient numerical approach.

Contribution

It generalizes Bernstein and Walsch's approximation results to higher genus Riemann surfaces and develops a numerical method for solving boundary value problems on these surfaces.

Findings

Provides an exact characterization of approximation rates.

Develops an efficient basis computation method with arbitrary precision.

Demonstrates the effectiveness of the approach in solving Laplace problems.

Abstract

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a harmonic function in a subdomain of a compact Riemann surface may be approached by globally defined harmonic functions with prescribed poles. We illustrate the effectiveness and the impact of the method solving general boundary value Laplace problems in subdomains of the surface; we lay the groundwork for this numerical method in Riemann surfaces represented by a gluing of hyperbolic polygons. In particular, we give a general approximation procedure that computes this basis efficiently with arbitrary precision.