In many application areas the diffusive component offers a significant challenge because it is characterized by a discontinuous diffusion coefficient with fine-scale anisotropic spatial structure; moreover, the underlying grid may be severely distorted. Thus, increasingly Mixed and Mixed-Hybrid discretizations (e.g., Support Operator Methods, and Mixed-Hybrid finite element methods) are employed because they explicitly enforce important physical properties of the problem, such as mass balance. However, these discretizations are based on the first order form, and hence, naturally lead to an indefinite linear system. Thus, the primary hurdle in the widespread adoption of these methods has been the robust and efficient solution of this linear system.
In the mixed-hybrid case it is possible to eliminate the flux (i.e., the vector unknowns) locally to obtain a sparse symmetric positive definite system. Unfortunately, the nonstandard sparsity structure of this reduced system has posed a significant challenge for the design of efficient multigrid preconditioners. In this presentation we compare two different approaches to preconditioning these systems. First, we consider a mimetic approach to developing alternative or approximate discretizations that generate reduced systems that have a sparsity structure amenable to existing robust multigrid solvers. In contrast, we consider the direct application of variational coarsening, the robust but potentially costly technique that lies at the heart of most robust multigrid algorithms. Both of these techniques have been used to generate a number of preconditioners for orthogonal grids, as well as distorted logically rectangular grids. We will discuss the varying degrees of robustness associated with these preconditioners, as well as their potential extension to unstructured grids.