SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2795-2812, December 2023.
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 the Steklov eigenvalue problem. Using a posteriori estimation, we show that the solution of the Laplace problem on a torus with a few holes has error less than [math] using a few hundred degrees of freedom and the Steklov eigenvalues have similar error.

中文翻译：

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有限连通圆环上的调和函数

《SIAM 数值分析杂志》，第 61 卷，第 6 期，第 2795-2812 页，2023 年 12 月。
摘要。在本文中，我们证明了有限连通圆环上的对数共轭定理。该定理指出，调和函数可以写成函数的实部，该函数的导数是解析的，并且是涉及修正的 Weierstrass sigma 函数模数对数的有限项和。我们使用任意精度实现该方法，并使用结果来找到拉普拉斯问题和 Steklov 特征值问题的近似解。使用后验估计，我们表明，使用几百个自由度，在具有几个孔的环面上解决拉普拉斯问题的误差小于[数学]，并且 Steklov 特征值具有类似的误差。