Variational principles and the finite-element method in partial differential equations

Abstract

In recent years, there has been an increasing interest in the numerical solution of elliptic boundary-value problems by finite-element methods based on variational principles. This is now spreading to time-dependent problems and recent papers by Swartz & Wendroff (1969), J. Douglas & T. Dupont (1970, unpublished report), and Wait & Mitchell (1970) have applied Galerkin methods, using local basis functions, to the numerical solution of parabolic and hyperbolic equations. The purpose of the present paper is to describe some of the recent work in the field with special reference to the basis functions which are fundamental to the Galerkin procedure. Particular emphasis is placed on elliptic boundary-value problems because in evolutionary problems, the dependence of the solution on the time is either treated separately or accounted for by discretization of the time derivatives.