On the sharpness of an error bound for a Galerkin method to solve parabolic differential equations

This paper discusses the sharpness of an error bound for the standard Galerkin method for the approximate solution of a parabolic differential equation. A backward difference is used for discretization in time, and a variational method like the finite element method is considered for discretization in space. The error bound is written in terms of an averaged modulus of continuity. Whereas the direct estimate follows by standard methods, the sharpness of the bound is established by an application of a quantitative extension of the uniform boundedness principle as proposed in Dickmeis et al. (1984) [4].