An Analysis of Some Galerkin Schemes for the Solution of Nonlinear Time-Dependent Problems

Abstract

A priori error estimates are derived for several Galerkin procedures. For a nonlinear parabolic initial-boundary value problem, optimal rate L2-error estimates are derived for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods. For a nonlinear second order hyperbolic initial-boundary value problem, optimal rate L2-error estimates are derived for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method. 1. Introduction. In this paper we derive a priori error estimates for several Galerkin procedures. In ?3 we consider a nonlinear parabolic initial-boundary value problem, and we derive error estimates for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods of (5); these methods were formulated in (5), but in the central difference methods proofs were given only for the case in which the elliptic part was linear. In ?4 we consider a nonlinear second order hyperbolic initial-boundary value problem, and we derive error estimates for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method of (5); these methods are the obvious extensions of the methods of (5) to the nonlinear case. In ?2 we present certain notation, assumptions, and known results which are common to the development of ? 3 and 4. Section 5 contains some computational results.