A Priori $L_2 $ Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations

Abstract

$L_2 $ error estimates for the continuous time and several discrete time Galerkin approximations to solutions of some second order nonlinear parabolic partial differential equations are derived. Both Neumann and Dirichlet boundary conditions are considered. The estimates obtained are the best possible in an $L_2 $ sense. These error estimates are derived by relating the error for the nonlinear parabolic problem to known $L_2 $ error estimates for a linear elliptic problem. With additional restrictions on basis functions and region $L_\infty $ error estimates are derived. Possible extensions to other discrete time Galerkin procedures and to higher order parabolic equations and systems of parabolic equations are suggested.