Computable error bounds for approximate solutions of ordinary differential equations

Abstract

This thesis is concerned with an error analysis of approximate methods for second order linear two point boundary value problems, in particular for the method of collocation using piecewise polynomial approximations. As in previous related work on strict error bounds an operator theoretic approach is taken. We consider operators acting between two spaces Xl and X2 with uniformly equivalent metrics. The concept of a "collectively compact sequence of operators" is examined in relation to "pointwise convergence" - relevant to many approximate numerical methods. The introduction of a finite dimensional projection operator permits considerable theoretical development which enables us to relate various inverse approximate operators directly to a certain inverse matrix. The application of this theory to the approximate solution of linear two point boundary value problems is then considered. It is demonstrated how the method of collocation can be expressed in terms of a projection method applied to a certain operator equation. The conditions required by the theory are expressed in terms of continuity requirements on the coefficients of the differential equation and in terms of the distribution of the collocation pOints. Various estimates of bounds on the inverse differential operator are presented and it is demonstrated that the "residual" can be a very useful error estimate. The use of a "weighted infinity norm" is shown to improve the applicability of the theory for "stiff" problems. Some real problems are then examined and a selection of numerical results illustrating the theory and application are presented. The thesis concludes with a brief review, outlining some of the deficiencies in the work and possible improvements and extensions of the analysis.