Hybrid Mesh Selection Algorithms Based on Conditioning for Two-Point Boundary Value Problems

In this paper we demonstrate how hybrid mesh selection strategies based on conditioning can be used in codes designed for the numerical solution of singularly perturbed boundary value problems. The new mesh selection strategies are based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on standard estimates of the local discretization error. The code TOM was the first code where this kind of mesh selection was implemented. Subsequently, a different version of mesh selection, based on the same principles, was implemented in the well known code TWPBVP. We have now implemented a similar strategy in a deferred correction code based on Lobatto formulae and this has proved to be very efficient for stiff problems. The results obtained show that the modified code is often much more efficient than the original one which uses a standard mesh selection strategy. Furthermore the estimate of the conditioning of the problem, which is automatically provided by the code, gives information about the complexity of the differential equation being solved. This can be useful especially for singularly perturbed boundary value problems for which the complexity of the problem is related to a parameter which appears in the differential equation. Finally some numerical results are given to illustrate the greatly improved performance of the new algorithm. c © 2006 European Society of Computational Methods in Sciences and Engineering