Boundary value problem for discrete analytic functions

Details Boundary value problem for discrete analytic functions Boundary value problem for discrete analytic functions

2011-10-31

arxiv.org/abs/1110.6737v1

Mathematics

Combinatorics

20 pages, 6 figures

Scientific paper

This paper is on further development of discrete complex analysis introduced by R. Isaacs, 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 converges to a harmonic function in the scaling limit (under certain regularity assumptions). This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant - K. Friedrichs - H. Lewy 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 finite element method on Delauney triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current networks theory.

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