The boundary value problem for discrete analytic functions

TL;DR

This paper advances discrete complex analysis by proving boundary value problem solutions for discrete analytic functions on quadrilateral graphs, with convergence results applicable to finite element methods and solving longstanding open problems.

Contribution

It proves the uniqueness of solutions for the Dirichlet problem for discrete analytic functions and their convergence to harmonic functions in orthogonal cases, extending prior results to general quadrilateral lattices.

Findings

Unique solution for Dirichlet boundary value problem.

Convergence to harmonic functions in orthogonal diagonals.

Implication for finite element methods on Delaunay triangulations.

Abstract

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.