Optimal $H^{p,{p/2}} $ Error Estimates for a Parabolic Galerkin Method

Abstract

Optimal error estimates are proved for the continuous and discrete Galerkin solution (implicit and Crank-Nicolson scheme) of a general linear parabolic equation subject to nonhomogeneous mixed boundary conditions. The norms used correspond to the nonisotropic Sobolev spaces $H^{{{(p - \alpha )}/2}} ((0,T),H^\alpha (\Omega )) \cap L^2 ((0,T),H^p (\Omega ))$ for $p \geqq \alpha \geqq - 1$. Also, estimates with respect to the norm of $\mathcal{H}^{ - 1} = $ dual space of $H^1 (\Omega )$ are mentioned.