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arXiv · 2309.12459

Harmonic functions on finitely-connected tori

Chiu-Yen Kao·Braxton Osting·Édouard Oudet
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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.

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Chiu-Yen Kao, Braxton Osting, Édouard Oudet. 2023-09-21. Harmonic functions on finitely-connected tori. https://arxiv.org/abs/2309.12459

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