Discretization of ordinary and partial differential equations by finite difference or finite element methods (for example) lead to large systems of linear equations that must be solved numerically. In this talk, we will focus on the linear systems that arise when solving singularly perturbed problems. These problems feature a small (perturbation) parameter multiplying the highest derivative, while their solutions typically feature boundary and/or interior layers. Numerous specialized methods have been proposed for computing solutions to these problems, most often involving specially designed, highly anisotropic layer-adapted meshes.

A discretization technique is called parameter robust if its accuracy is independent of the perturbation parameter, for a given number of degrees of freedom. Simply put, the computational cost of a robust method achieving a fixed accuracy should be independent of the perturbation parameter. Most of this cost is incurred when solving associated linear system, so a truly robust discretization needs to be implemented with a linear solver whose efficiency is independent of the perturbation parameter.

In this talk, I will demonstrate that, surprisingly, some standard direct solvers are not parameter robust in the above sense. This leads us to consider iterative techniques. Since the problems are discretized on highly non-uniform meshes, the linear systems tend to be ill-conditioned. We consider several approaches, giving particular attention to a recently proposed "boundary layer preconditioner", that uses multigrid techniques, for a finite element method.

Dr. Niall Madden is a Senior Lecturer in Mathematics at the National University of Ireland Galway. He holds a B.Sc. in Mathematical and Computer Sciences from University College Galway (1994), and a Masters (1996) and PhD in Mathematics (2000) from University College Cork. His research interests are in the design and analysis of algorithms for solving singularly perturbed differential equations, particularly highly efficient techniques for partial differential equations, and in the application of novel methods in numerical modelling.