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Finite element method - a Galerkin approach Hutton, Stanley George

Abstract

This study is concerned with defining the mathematical framework in which the finite element procedure can most advantageously be considered. It is established that the finite element method generates an approximate solution to a given equation which is defined in terms of assumed co-ordinate functions and unknown parameters. The advantages of determining the parameters by Galerkin's method are discussed and the convergence characteristics of this method are reviewed using functional analysis principles. Comparisons are made between the Galerkin and Rayleigh-Ritz procedures and the connection between virtual work and Galerkin's method is illustrated. The convergence results presented for the Galerkin procedure are used to provide sufficient conditions that ensure the convergence of a finite element solution of a general system of time independent linear differential equations. Application of the principles developed is illustrated with a convergence proof for a finite element solution of a non-symmetric eigenvalue problem and by developing a computer program for the finite element analysis of the two-dimensional steady state flow of an incompressible viscous fluid.

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