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.

翻译：本文证明了有限连通环面上的对数共轭定理。该定理指出，调和函数可表示为导数解析的函数的实部与有限个包含修正Weierstrass sigma函数模的对数项之和。我们采用任意精度实现了该方法，并利用结果求解拉普拉斯问题与Steklov特征值问题的近似解。通过后验估计，我们证明在带有少量圆孔的环面上，拉普拉斯问题的解在仅用几百个自由度时误差小于$10^{-100}$，而Steklov特征值也具有相近的误差精度。