10 - VARIATIONAL METHODS

Abstract

There is an equivalence between problems in the calculus of variations and problems involving partial differential equations (PDE). In particular, any PDE problem can be phrased in variational form. The variational form permits a useful alternative approach to the solution of the PDE problem. One of the simplest and most natural approximate methods based on the calculus of variations is that of Rayleigh and Ritz. It can be used in any problem for which a variational principle exists. The Rayleigh–Ritz method can also be used in situations in which the variational integral is merely stationary. The idea of the Rayleigh–Ritz method can also be extended to the construction of functions that depend nonlinearly on the parameters. The widely used finite-element method is a special case of the Rayleigh–Ritz and Galerkin methods. It is based on the subdivision of the region of interest into a number of subregions and on the use of a different analytic expression for the approximate solution function within each such sub-region.