Collocation algorithms and error analysis for approximate solutions of ordinary differential equations

Abstract

This thesis is mainly concerned with an error analysis of numerical methods for two point boundary value problems. in particular for the method of collocation using polynomial and certain piecewise polynomial bases. As in previous work on strict error bounds an operator theoretical approach is taken. The setting for the theory and the principal results for later use are firstly considered. Then two types of 'a posteriori' error bounds are developed. These"bounds are made computable by relating the inverse of the approximating operator to the inverse of certain matrices formed in the actual application of the approximation method. The application of this theory to the numerical solution of linear two point boundary value problems is then considered. It is demonstrated how the differential equation can be split to fit into the setting required by the theory. It is also demonstrated how the global and the piecewise collocation method can be expressed in terms of a projection method applied to the 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. In examining these bounds on a variety of problems. it is noticed that with some problems the conditions for applicability may not hold except for more points than one actually required to obtain a satisfactory solution. To improve the applicability. the theory is reconsidered with a different splitting of the differential equation. The method of collocation is expressed accordingly in terms of a new projection operator which is proved to have some nice properties in practice. This new approach is then compared with the original one and it is shown to be superior on various problems. By examining the inverse differential operator and the residual improved error bounds and estimates are shown to be obtainable. These estimates are tested in a large variety of examples and some graphs are presented to describe their behaviour in more detail. Finally these estimates are used to develop various adaptive mesh selection algorithms for solving two point boundary value problems. These strategies are tested and compared in several representative examples and some conclusions are drawn. The thesis concludes with a brief review of the work with an indication of possible improvements and extensions.