Some error estimates for the finite volume element method for a parabolic problem
Provides theoretical error bounds for finite volume element methods, extending prior work on lumped mass methods to this context.
The paper derives error estimates for finite volume element methods applied to the heat equation, showing that optimal second-order convergence for nonsmooth initial data requires symmetric triangulations; otherwise only first-order convergence is achieved.
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.