Robust solution of singularly perturbed problems using multigrid methods: analysis and results in one and two dimensions

We consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial di #64256;erential equations of reaction-di #64256;usion type. Standard discretization techniques are not suitable for such problems and, so, specially tailored methods are required, usually involving adapted or #64257;tted meshes that resolve important features such as boundary and or interior layers. In this study, we consider classical #64257;nite di #64256;erence schemes on the layer adapted meshes of Shishkin and Bakhvalov. We show that standard direct solvers exhibit poor scaling behaviour when solving the resulting linear systems. We investigate the use of standard robust multigrid preconditioners for these linearsystems, and we propose and prove optimality of a new block-structured preconditioning approach.