# These are being used to verify the accuracy of the

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Unformatted text preview: ese are being used to verify the accuracy of the computation, and also to control an adaptive process whereby meshes are automatically re ned and coarsened and/or the degrees of polynomial approximations are varied so as to compute solutions to desired accuracies in an optimal fashion 1, 2, 3, 4, 5, 7, 14]. 1.2 Weighted Residual Methods Our goal, in this introductory chapter, is to introduce the basic principles and tools of the nite element method using a linear two-point boundary value problem of the form L u] := ; d (p(x) du ) + q(x)u = f (x) dx dx u(0) = u(1) = 0: 0<x<1 (1.2.1a) (1.2.1b) The nite element method is primarily used to address partial di erential equations and is hardly used for two-point boundary value problems. By focusing on this problem, we hope to introduce the fundamental concepts without the geometric complexities encountered in two and three dimensions. Problems like (1.2.1) arise in many situations including the longitudinal deformation of an elastic rod, steady heat conduction, and the transverse de ection of a supported 1.2....
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