TABLE OF CONTENTS The finite element method (FEM) presents to all those who were weaned on finite differences an entirely new outlook on the computation of a numerical solution for differential equations. Although it is often encapsulated in a few buzzwords weak solution , Galerkin , finite element functions an understanding of the FEM calls not just for a different frame of mind but also for the comprehension of several principles. Each principle is important but it is their combination that makes the FEM into such an effective computational tool.
Instead of commencing our exposition from the deep end, let us first examine in detail a simple example, the Poisson equation in just one space variable. In principle, such an equation is u ?? = f, but this is clearly too trivial for our purposes since it can be readily solved by integration. Instead, we adopt a more ambitious goal and examine linear two-point boundary value problems
where a, b and f are given functions, a is differentiable and a( x) > 0, b( x) ? 0, 0 < x < 1. Any equation of the form (9.1) must be specified in tandem with proper initial or boundary data. For the time being, we assume Dirichlet boundary conditions
Two-point boundary problems (9.1) abound in applications, e.g. in mechanics, and their numerical solution is of independent interest. However, in the context of this section, our main...
