### A Priori L_{2} Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations

Rice University, 1971

Introduction

In recent years much attention has been devoted to the formulation and analysis of Galerkin methods for approximating solutions of parabolic problems. Douglas and Dupont obtained H^{1} error estimates for the continuous time and for several discrete time Galerkin procedures. Price and Varga obtained L_{2} error estimates for the continuous time Galerkin procedure for linear parabolic problems. It appears however that their proofs are restricted to linear problems and the use of L-splines. Fix and Strang have also obtained L_{2} estimates for linear initial value problems.

This thesis is an extension of the work of Douglas and Dupont. In this thesis L_{2} error estimates for continuous time and several discrete time Galerkin approximations of some second order nonlinear parabolic boundary value problems are derived. It appears that this analysis carries over to higher order parabolic problems and to systems of parabolic equations.

This thesis is divided into three main chapters. In Chapter II the variational problem and the Galerkin procedure are described. The basic techniques of this thesis are developed in Chapter III. There L_{2} error estimates for Galerkin approximations of linear elliptic problems are used to derive a priori L_{2} error estimates for continuous time Galerkin approximations of nonlinear parabolic problems. These estimates are independent of the choice of basis functions used in the Galerkin procedure. However, they do depend on an L_{∞} estimate of the derivative of a function which is the Galerkin solution to a certain linear elliptic problem. In Chapter III we also derive L_{2} and L_{∞} error estimates for Galerkin approximations and derivatives of Galerkin approximations where the region under consideration is a rectangular parallelepiped and the tensor products of piecewise Hermite polynomials of degree 2m-1, m __>__ 1, are used as a basis. In Chapter IV we use the techniques of Chapter III to obtain L_{2} error estimates for several discrete time Galerkin procedures.