Two-point stress approximation: A simple and robust finite volume method for linearized (poro-)mechanics and Stokes flow

TL;DR

This paper introduces a simple, robust two-point finite volume method for linearized elasticity, Stokes flow, and poroelasticity, emphasizing stability, convergence, and practical effectiveness through theoretical analysis and 3D numerical experiments.

Contribution

It presents a novel two-point finite volume discretization that is simple, stable, and applicable to a range of mechanics and flow problems, including incompressible limits.

Findings

Proves stability and convergence of the discretization.

Demonstrates robustness with respect to material parameters.

Validates the method's effectiveness through 3D numerical experiments.

Abstract

In this paper, we construct a simple and robust two-point finite volume discretization applicable to isotropic linearized elasticity, valid in also in the incompressible Stokes' limit. The discretization is based only on co-located, cell-centered variables, and has a minimal discretization stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. The discretization naturally couples to finite volume discretizations of flow, providing a stable discretization of poroelasticity. We show well-posedness of a weak statement of the continuous formulation in appropriate Hilbert spaces, and identify the appropriate weighted norms for the problem. For the discrete approximations, we prove stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments in 3D support the theoretical results, and provide additional insight into the practical performance of the discretization.