We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere \(\hat {\mathbb {C}}\). Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

中文翻译：

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黎曼球分枝覆盖的离散周期矩阵和离散全纯积分的收敛性

我们考虑一类紧凑的黎曼曲面，它们是黎曼球体\(\hat {\mathbb {C}}\) 的分支覆盖。基于此覆盖的三角剖分，我们定义了离散（多值）谐波和全纯函数。我们证明相应的离散周期矩阵收敛到它们的连续对应矩阵。为了实现在三角形的最大边长中线性的误差估计，我们在每个分支点的邻域中适当地调整三角剖分。最后，我们还证明了离散全纯积分的收敛结果，用于我们适应的分枝覆盖三角剖分。