Some Convergence Estimates for Semidiscrete Galerkin Type Approximations for Parabolic Equations

Abstract

In this paper we derive convergence estimates for certain semidiscrete methods used in the approximation of solutions of initial boundary value problems with homogeneous Dirichlet boundary conditions for parabolic equations. These methods contain the ordinary Galerkin method based on approximating subspaces with functions vanishing on the boundary of the basic domain, and also some methods without such restrictions. The results include $L_2 $ estimates, maximum norm estimates, interior estimates for difference quotients and superconvergence estimates. Some proofs depend on known results for the associated elliptic problem. Several of these estimates are derived for positive time under weak assumptions on the initial data.