Saddle Points and Ritz–Galerkin Approximations

Abstract

This paper establishes a theorem for Ritz–Galerkin approximations to a unique saddle point of a function defined on a reflexive Banach space. The functional value at the saddle point is given in terms of an infimum and supremum over two closed, possibly infinite dimensional subspaces whose direct sum determines the original Banach space. Also, an application is given to a nonlinear boundary value problem involving the biharmonic operator.