Reliable and robust a posteriori error estimation for singularly perturbed reaction-diffusion problems

Problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior. Such features lend themselves to adaptive solution methods. The quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control. An estimator that has proved to be one of the most reliable is the equilibrated residual method. The main property of the estimator is that it bounds the true error from above. However, the method is not robust in the singularly perturbed limit. The current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit. It is shown that the new method is robust in the singularly perturbed limit while maintaining reliability, yielding a guaranteed upper bound on the true error. Finally, the application of the estimator to the problem of controlling the spatial error in Rothe's method for the time discretization of a simple parabolic problem is included.