A numerical investigation of the Rayleigh-Ritz method for the solution of variational problems

Abstract

The results of a numerical investigation of the Haylaigh·-H:l:tz method for the approxi.mate solution of two-point boundary value problems in ordinary differential equations are presented. Theoretical results are developed which indicate that the observed behaviour ie typical of the method in more general applications. In particular9 a number of choices of co-ord:i.natefunctions for certain second order equations are considerede A new algorit~~ for the efficient evaluation of an established sequence of functions related to the Legendre polynomials is desoribed, and the sequence is compared in use with a similar sequence related to the Chebyshev polynomials. Algebraic properties of the Rayleigh ••R1tz equations tor these and other co-ordinate systems are discussede The Chebyshev system is shown to lead to equations with oonvenient computational and theoretical properties, and the latter are used to characterize the asymptotic convergence of the approximations for linear equationse These results are subsequently extended to a certain type of non-linear equatione An orthonormalization approach to the solution ot the R~leigh- Ritz equations which has been suggested in the literature is compa.red in practice with more usual methods, and it is shown that the properties of the resulting approximations are not improvedo Since it is knoWli.that the method requires more work than established ones it cannot be recommendedo Quadrature approximations of elements of the ~leigh-Ritz matrices a.re investigated, and known results for a restricted class ot quadra·t';.re approximation are extended towards the more general case. In a final chapter extensions of the material of earlier chapters to partial differential equations are described, and new forms of the 'finite element' and 'extended Kantorovich' methods are proposed. A summary of the conclusions discerned from the investigation is given.