On the Smoothing Property of the Galerkin Method for Parabolic Equations

We analyze the Galerkin approximation of the general second-order parabolic initial-boundary value problem. For the second-order continuous time method with initial data only in L² we prove an L² error estimate of order O ({h² / t}). Analogous results are shown to hold for the error in negative Sobolev norms and for the time derivative of the error. Our analysis uses only elementary energy techniques and the same techniques are shown to give a simple analysis of the backward Euler time discretization.