A new mesh selection algorithm, based on conditioning, for two-point boundary value codes

We present a hybrid mesh selection strategy for use in codes for the numerical solution of two-point boundary value problems. This new mesh strategy is based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on a standard estimate of the local discretisation error. We have implemented this algorithm in the well known code TWPBVP and have found that the modified code is often considerably more efficient than the original. Another strong advantage of using the new mesh selection algorithm is that it automatically provides an estimate of the conditioning of the discrete problem. This is very valuable (arguably indispensible) for use either in an a posteriori error estimate or, in situations where the conditioning constants are large, as a warning that the accuracy obtained in the solution may be worse than anticipated.