  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.
