In recent years, substantial effort has been focused on developing methods capable of solving the large linear systems that arise from the discretization of partial differential equations. Multilevel methods, such as multigrid, provide optimal or near-optimal order solution techniques for many of these systems, based on the construction of coarse-scale problems that complement the fine-scale relaxation. Classical geometric and algebraic multigrid methods rely on (implicit) assumptions about the performance of relaxation in order to develop appropriately complementary coarse-grid correction processes. These assumptions lead to optimal (linear-scaling) methods, at the expense of robustness; the true potential of the multigrid approach has only been realized for a small class of problems.

In this talk, I will introduce the adaptive multigrid framework and discuss its application to problems in heterogeneous media. The aim of this approach is to reduce the restrictions imposed by the assumptions on relaxation, thus allowing for efficient black-box multigrid solution of a wider class of problems. Relaxation is used directly, to expose the components that slow convergence. Having identified these troublesome components, we construct complementary processes to avoid such failings. In this way, the multigrid solver itself is adapted to improve performance. The robustness of this approach is demonstrated, including its application to systems from linear elasticity and quantum chromodynamics.