Some error estimates for the finite volume element method for a parabolic problem

TL;DR

This paper provides error estimates for finite volume element methods applied to the heat equation, highlighting conditions for optimal convergence and demonstrating the impact of triangulation symmetry on accuracy.

Contribution

It extends previous error analysis results to the finite volume element method for parabolic problems, emphasizing the role of triangulation symmetry for optimal error bounds.

Findings

Optimal second order error estimates require symmetric triangulations.

Without symmetry, only first order convergence is guaranteed.

Almost symmetric triangulations yield improved error estimates.

Abstract

We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work \cite{clt11} for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in $L_2$ to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric.