Error control for approximate solutions of a class of singularly perturbed boundary value problems

Reaction–convection–diffusion equations with a small parameter at the highest derivative are considered. The question as to how the accuracy of approximate solutions of such problems can be effectively controlled with the help of a posteriori estimates is studied. The resulting estimates do not depend on the method used to construct the approximate solution and perform well in a wide range of parameter values. The estimates are derived relying on special (a posteriori) identities whose left-hand side represents a measure of the deviation of the approximate solution from the exact one and the right-hand side involves data of the problem and a known approximate solution. It is shown on a series of examples that the errors of both rough and accurate approximations of problems can be efficiently computed for various values of the small parameter by applying these identities and estimates following from them.