Finite volume schemes of any order on rectangular meshes

TL;DR

This paper provides a comprehensive analysis of high-order vertex-centered finite volume methods on rectangular meshes, establishing convergence rates, superconvergence properties, and validating results through numerical experiments.

Contribution

It offers a unified proof of the inf-sup condition for any order FVM on rectangular meshes, demonstrating optimal convergence and superconvergence properties.

Findings

Optimal convergence in energy norm

Superconvergence in L2 norm

Numerical validation of theoretical results

Abstract

In this paper, we analyze vertex-centered finite volume method (FVM) of any order for elliptic equations on rectangular meshes. The novelty is a unified proof of the inf-sup condition, based on which, we show that the FVM approximation converges to the exact solution with the optimal rate in the energy norm. Furthermore, we discuss superconvergence property of the FVM solution. With the help of this superconvergence result, we find that the FVM solution also converges to the exact solution with the optimal rate in the $L^2$-norm. Finally, we validate our theory with several numerical experiments.