Estimates for the errors in the Rayleigh-Ritz method

Abstract

with homogeneous linear boundary conditions, denoted by (C). It is assumed that the parameter λ does not appear in the boundary conditions. The region R of the problem may be of any number of dimensions; the symbol ff(P)dP will mean an integral over /?, dP standing for the Euclidean volume element; (/, g) will stand for ff(P)g{P)dP; and ||/||, for (f, f)ι/ The symbol ffF (P, Q)dPdQ will mean an integral over the Cartesian product of R with itself. We assume that the problem (1)+(C) is positive definite and selfadjoint; that is, {φ, Lφ) > 0 and (φ, Lφ)= {Lφ, ψ) for any admissible functions φ and ψ, an admissible function being a real-valued function, not identically zero, which satisfies the boundary conditions (C) and is continuously differentiate up to derivatives of the order of the operator L. The class of admissible functions will be denoted by G.