The boundary value problem for discrete analytic functions
The boundary value problem for discrete analytic functions
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DOI：
10.1016/j.aim.2013.03.002
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发表时间：
2011-10
影响因子：
1.7
通讯作者：
M. Skopenkov
中科院分区：
文献类型：
--
作者：
M. Skopenkov
本文是对R. Isaacs、J. Ferrand、R. Duffin和C. Mercat提出的离散复分析的进一步发展。考虑复平面上一个具有四边形面的图.一个在顶点上的函数叫做离散解析函数，如果对于每个面，沿着两条对角线的差导数相等。证明了离散解析函数的真实的部分的狄利克雷边值问题有唯一解。当每个面有正交对角线的情况下，我们证明了这个解决方案一致收敛到一个调和函数的比例限制。这解决了S. Smirnov在2010年提出的一个问题。这一点在较早的时候由R. Courant-K. Friedrichs-H. Lewy和L. Lusternik针对正方形格、D. Chelkak-S. Smirnov以及P.G. Ciarlet-P. A. Raviart的菱形晶格。特别地，我们的结果暗示了有限元方法在Delaunay三角剖分上的一致收敛性。这解决了A. Bobenko在2011年提出的一个问题。该方法是基于交流网络理论启发的能量估计。
This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal. We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant–K. Friedrichs–H. Lewy and L. Lusternik for square lattices, by D. Chelkak–S. Smirnov and implicitly by P.G. Ciarlet–P.-A. Raviart for rhombic lattices. In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.