Error analysis of collocation methods for the numerical solution of ordinary differential equations

Abstract

This thesis is concerned with an error analysis of numerical methods for two point boundary value problems and much of the investigation is concentrated on collocation methods from an 'a posteriori' point of view. Most of the previous work on error bounds for boundary value problems has been of an 'a priori' nature, requiring knowledge of the inverse of the differential operator under consideration and furnishing convergence proofs and theoretical bounds on the error. There are however a few results of the converse nature and in this thesis means of determining error bounds in practice are developed, much of the analysis also applying to Fredholm integral equations of the second kind. In more detail, having firstly considered certain preliminaries the setting for the theory and the principal results for later use are presented. It is demonstrated how the approximate solution by collocation of linear differential equations fits into this background and different 'a priori' approaches are examined by example and shown to be rather unsatisfactory. The 'a posteriori' outlook is then considered and to achieve practical results the inverse of the approxi- mating operator is related to the inverse of the collocation matrix. However the problem of obtaining a suitable bound on the norm of this inverse operator is encountered and after examination of the most obvious approach which proves unsatisfactory a convenient bound is developed. Certain interesting computational properties of matrices involved in the process are discussed and a brief examination of condition numbers is given. A different theoretical analysis using the concept of a 'collectively compact sequence of operators' is considered and it is demonstrated that the approximate solution by collocation of linear differential equations can be 'extended' to satisfy the conditions for this theory. Again the error bounds are reduced to a more practical level and subsequently a generalisation of the notion of this extension is suggested. The implementation of the various practical error bounds which have been deduced is then considered in detail and formulae for their evaluation are presented. The numerical results of examples of this application are then given followed by a discussion of certain relevent points concerning the experiments. In the final chapter certain possible extensions of the analysis herein are briefly examined and lastly a review of the work of this thesis with appropriate conclusions is given.