The Finite Element Method--Linear and Nonlinear Applications

L Numerical analysis is a crazy mixture of pure and applied mathematics. It asks us lo do two things at once, and on the surface they do appear complementary: (i) to propose a good algorithm, and (ii) to analyze it. In principle, the analysis should reveal what makes the algorithm good, and suggest how to make it better. For some problems—computing the eigenvalues of a large matrix, for example, which used to be a hopeless mess—this combination of invention and analysis has actually succeeded. But for partial differential equations, which come to us in such terrible variety, there seems to be a long way to go. We want to speak about an algorithm which, at least in its rapidly developing extensions to nonlinear problems, is still new and flexible enough to be improved by analysis. It is known as the finite element method, and was created to solve the equations of elasticity and plasticity. In this instance, the "numerical analysts" were all engineers. They needed a better technique than finite differences, especially for complicated systems on irregular domains, and they found one. Their method falls into the framework of the Ritz-Galerkin technique, which operates with problems in "variational form"—starting either from an extremum principle, or from the weak form of the differential equation, which is the engineer's equation of virtual work. The key idea which has made this classical approach a success is to use piecewise polynomials as trial functions in the variational problem. We plan to begin by describing the method as it applies to linear problems. Because the basic idea is mathematically sound, convergence can be proved and the error can be estimated. This theory has been developed by a great many numerical