Semidiscrete Perturbed Variational Methods for the Numerical Solution of Parabolic Boundary Value Problems

Abstract

The purpose of this paper is to propose semidiscrete methods for approximating the solution of second order parabolic initial boundary value problems based on the penalty method of Babuška. The methods given here consist in discretizing, with respect to time, the parabolic equation and applying the penalty method to the resulting elliptic boundary value problems. It is shown, for a particular method, that the error is asymptotically of second order when measured in the $L_2 $-norm and is of order ${3 / 2}$ with respect to the Sobolev $H^1 $-norm. In contrast to semidiscrete Galerkin methods it is not required that the admissable approximating functions satisfy boundary conditions.