Convergence of a cell-centered finite volume discretization for linear elasticity

TL;DR

This paper proves the convergence of a cell-centered finite volume method, called MPSA, for linear elasticity, providing the first rigorous justification for its use in elasticity problems.

Contribution

It offers the first rigorous convergence analysis of the MPSA finite volume method for linear elasticity, including stability and robustness results.

Findings

Convergence of the MPSA method is established.

Stability is proven with respect to the Poisson ratio.

A discrete Korn's inequality is developed for the analysis.

Abstract

We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Secondly, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.