On optimally scaled systems for second-order scalar singularly perturbed problems

The conditioning of the singularly perturbed scalar Dirichlet problem is considered. It is shown how this is related to the conditioning of an appropriate associated first-order system. Through this the dichotomy of the solution space (a concept that only makes sense in a vectorial setting) can be investigated. Two typical equations are studied in more detail, one with possible boundary layers on both sides of the interval and one with an internal layer (i.e., the turning point case). The results are applied to obtain estimates of global discretization errors for difference methods. Several examples illustrate the analysis