The $L^2$-Projection and Quasi-Optimality of Galerkin Methods for Parabolic Equations

Abstract

We consider linear parabolic initial-boundary value problems and analyze Galerkin approximation in space. With the help of the inf-sup theory, we derive quasi-optimality results with respect to norms that arise from the standard weak formulation and from a formulation requiring only integrability in time. Moreover, we reveal that the $H^1$-stability of the $L^2$-projection is not only sufficient but also necessary for these results. As application, we consider conforming finite element approximation in space and derive a priori error bounds in terms of the local meshsize and piecewise regularity. The regularity is the minimal one indicated by approximation theory and matches regularity results for linear parabolic problems.