An investigation of mesh selection algorithms in the numerical solution of boundary value problems by piecewise polynomial collocation method

Abstract

This thesis is concerned with the investigation and evaluation of adaptive mesh selection strategies for solving two points boundary value problems using a piecewise collocation method. General definitions and descriptions for adaptive strategies and piecewise collocation methods are given at the beginning. A description of a data structure which is suitable for implementing adaptive algorithms is also given. A preliminary investigation of four adaptive strategies is introduced and evaluated on a set of test problems. From this evaluation it is found that a strategy called the Q-matrix has done generally well, but its cost is high compared with the rest. An improvement to one of the cheap strategies is introduced by An improving the initial mesh from which the cheap strategy starts. algorithm is designed to build a good initial mesh which is fairly dense in the layer regions. This algorithm is based on the behaviour of the asymptotic solution of the problem. A definition and a short description for such solution are also introduced. A further improvement to the Q-matrix strategy, is then introduced. In this, we used an error estimate instead of the error bound used originally in the strategy. This estimate is obtained by multiplying two ,polynomials, one representing the residual and the other representing the kernel (using the elements of Q ). The effect of having a singular point in the middle of an interval on these representations is also investigated. A final evaluation of three strategies, the Q-matrix and the two new strategies is introduced. This evaluation shows the improvement in the new modified strategies in terms of cost and accuracy. The thesis concludes with comment on the strategies and some suggestions for further research and improvement.