Optimal error bounds for the two point flux approximation finite volume scheme

TL;DR

This paper derives optimal error bounds for the two-point flux approximation finite volume scheme applied to Laplace problems with minimal regularity, including extensions to evolution problems with implicit Euler discretization.

Contribution

It establishes the first optimal error bounds for TPFA finite volume schemes with low regularity solutions, including gradient approximation and evolution problems.

Findings

Error bounds match the sum of interpolation and conformity errors.

Numerical example confirms theoretical error estimates.

Extension to evolution problems with implicit Euler scheme.

Abstract

We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to $H^1_0(\Omega)$. We establish in this case some error bounds for both the solution and the approximation of the gradient component orthogonal to the mesh faces. This estimate is optimal, in the sense that the approximation error has the same order as that of the sum of the interpolation error and a conformity error. A numerical example illustrates the error estimate in the context of a solution with minimal regularity. This result is extended to evolution problems discretized via the implicit Euler scheme in an appendix.