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**Unformatted text preview: **Chapter 11 Numerical Methods: Finite Elements In Chapter 5, we introduced the first, the oldest, and in many ways the simplest class of numerical algorithms for approximating the solutions to partial differential equations: finite differences. In the present chapter, we study the second of the two major numerical paradigms: the finite element method. Finite elements are of more recent vintage, having first appeared soon after the Second World War; historical details can be found in [ 134 ]. As a consequence of their flexibility and ability to readily handle complicated geometries, finite elements have, in many situations, become the basic method of choice for solving equilibrium boundary value problems governed by elliptic partial differential equations. Extensions to dynamical problems are also extensively utilized, but these will not be treated here due to lack of space. Finite elements rely on a more sophisticated understanding of the partial differen- tial equation, in that, unlike finite differences, they are not obtained by simply replacing derivatives by their numerical approximations. Rather, they can be based on an associated minimization principle which, as we learned in Chapter 10, can be used to characterize the unique solution to a positive definite boundary value problem. Restricting the minimizing functional to an appropriately chosen finite-dimensional subspace of functions reduces it to a finite-dimensional minimization problem, that can then be solved by numerical linear algebra. When properly formulated, the resulting finite-dimensional minimization problem has a solution that well approximates the true minimizer. After setting up the basic framework, we illustrate the basic constructions in the context of boundary value problems for ordinary differential equations. The following section then extends finite element analysis to boundary value problems associated with the two-dimensional Laplace and Poisson equations. A rigorous justification and proof of convergence of the finite element approximations requires further analysis, and we refer the interested reader to [ 134 , 151 ]. In this chapter, we shall concentrate on understanding how to formulate and implement the finite element method in practical situations. 11.1. Minimization and Finite Elements. To explain the key ideas behind the finite element method, we return to the abstract framework that was developed in Chapter 10. Recall that we can characterize the solution to a positive definite boundary value problem as the function u U , belonging to a cer- tain infinite-dimensional function space, that minimizes an associated quadratic functional Q : U R . This sets the stage for the first key idea of the finite element method. Instead of trying to minimize the functional Q over the entire infinite-dimensional function space, 1/19/12 394 c circlecopyrt 2012 Peter J. Olver we will seek to minimize it over a finite-dimensional subspace W U . The effect is to reduce a problem in analysis a boundary value problem for a differential equation...

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