Harmonic functions on finitely-connected tori

TL;DR

This paper establishes a logarithmic conjugation theorem for harmonic functions on finitely-connected tori, enabling precise approximate solutions to Laplace and Steklov eigenvalue problems with extremely low error margins.

Contribution

It introduces a new conjugation theorem for harmonic functions on finitely-connected tori and demonstrates its application in high-precision solutions to classical boundary value problems.

Findings

Achieved error less than 10^{-100} in Laplace problem solutions.

Provided high-accuracy Steklov eigenvalue computations.

Validated the method with arbitrary precision implementation.

Abstract

In this paper, we prove a Logarithmic Conjugation Theorem on finitely-connected tori. The theorem states that a harmonic function can be written as the real part of a function whose derivative is analytic and a finite sum of terms involving the logarithm of the modulus of a modified Weierstrass sigma function. We implement the method using arbitrary precision and use the result to find approximate solutions to the Laplace problem and Steklov eigenvalue problem. Using a posteriori estimation, we show that the solution of the Laplace problem on a torus with a few circular holes has error less than $10^{-100}$ using a few hundred degrees of freedom and the Steklov eigenvalues have similar error.