Robust Solution of Singularly Perturbed Problems Using Multigrid Methods

Abstract

We consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction-diffusion type. Standard discretization techniques are not suitable for such problems and, so, specially tailored methods are required, usually involving adapted or fitted meshes that resolve important features such as boundary and/or interior layers. In this study, we consider classical finite difference schemes on the layer adapted meshes of Shishkin and Bakhvalov. We show that standard direct solvers exhibit poor scaling behavior, with respect to the perturbation parameter, when solving the resulting linear systems. We propose and prove optimality of a new block-structured preconditioning approach that is robust for small values of the perturbation parameter, and compares favorably with standard robust multigrid preconditioners for these linear systems. We also derive stopping criteria which ensure that the potential accuracy of the layer-resolving meshes is achieved.