Green function for a two-dimensional discrete Laplace-Beltrami operator

TL;DR

This paper develops a discrete Laplace-Beltrami operator on a 2D grid that preserves geometric structure, proves its mathematical properties, and derives an explicit Green function formula.

Contribution

It introduces a discrete model of the Laplace-Beltrami operator that maintains geometric and Riemannian structures, with proofs of self-adjointness and boundedness, and provides an explicit Green function.

Findings

Proved self-adjointness and boundedness of the discrete Laplacian.

Derived an explicit formula for the Green function.

Established a discrete analog of differential forms preserving geometric structure.

Abstract

We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential operators by difference ones but also a discrete analog of the Riemannian structure. We consider this structure on the appropriate combinatorial analog of differential forms. Self-adjointness and boundness for a discrete Laplacian are proved. We define the Green function for this operator and also derive an explicit formula of the one.