A LEAST SQUARES FINITE ELEMENT SOLUTION OF NONLINEAR OPERATORS

Abstract

In order that the finite element method may become a more general analysis tool, mathematical approaches must be formulated for arriving at finite element models of differential equations when the equivalent functional is not known or does not exist. The method of weighted residuals provides such an alternate solution. The method of weighted residuals encompasses the collocation method, Galerkin criterion, least squares method, Trefftz's method, etc. To date, the Galerkin criterion has become the most popular of these methods. This chapter presents the formulation of a least squares finite element solution of nonlinear partial differential equations. From the engineering point of view, the method can be simply thought of as minimizing the square of the residual error in the differential equation. The formulation is restricted to a single differential equation as the extension of the weighted residual procedures to systems of equations, which has been previously outlined by Crandal and Leonard.