We consider piecewise linear finite element discretizations
of the model initialboundary value problem for the homogeneous
heat equation, and discuss the validity of the associated discrete
maximumprinciples. We demonstrate that for the spatially semidiscrete
standard Galerkin approximation, the maximumprinciple is not valid in
general. However, as was shown by Fujii in 1973, the maximumprinciple holds
for the lumped mass modification, when the triangulation is of Delaunay
type,
and this condition on the triangulation is essentially sharp. We also present
some results for the simplest time stepping analogues of these
approximations.
