Convergence of finite volume scheme for three dimensional Poisson's equation

TL;DR

This paper extends a finite volume scheme to three dimensions for solving Poisson's equation, deriving optimal convergence rates and validating results through numerical examples.

Contribution

It develops a 3D finite volume scheme for Poisson's equation, providing new convergence analysis and fixing a gap in previous proofs.

Findings

Optimal convergence in discrete H^1 norm

Sub-optimal convergence in maximum norm

Validated theoretical results with 3D numerical examples

Abstract

We construct and analyze a finite volume scheme for numerical solution of a three-dimensional Poisson equation. This is an extension of a two-dimensional approach by Suli 1991. Here we derive optimal convergence rates in the discrete H^1 norm and sub-optimal convergence in the maximum norm, where we use the maximal available regularity of the exact solution and minimal smoothness requirement on the source term. We also find a gap in the proof of a key estimate in a reference in Suli 1991 for which we present a modified and completed proof. Finally, the theoretical results derived in the paper are justified through implementing some canonical examples in 3D.