L² Error Bounds for the Rayleigh–Ritz–Galerkin Method

In this paper we adapt a technique of J. Nitsche to obtain new sharp a priori L²-bounds for the error involved in approximating the solutions to a wide variety of boundary value problems for nonlinear elliptic partial differential equations by the Rayleigh–Ritz–Galerkin method. In particular, we consider linear and semilinear elliptic problems, linear and semilinear forced vibration problems, and eigenvalue problems.