Robust solution of singularly perturbed problems using multigrid methods (Q2870642)

The authors are interested in the solution of systems of linear equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction-diffusion type. The classical finite difference schemes on the layer adapted meshes of Shishkin and Bakhvalov are considered. The solution of the resulting linear systems is considered and it is shown that standard direct solvers exhibit a poor scaling behavior with respect to the perturbation parameter. A new block-structured preconditioning approach is proposed and the optimality of that is shown. The method is robust for small values of the perturbation parameter. Stopping criteria are derived ensuring that the potential accuracy of the layer-resolving meshes is achieved.