Block preconditioners for coupled physics problems

The finite element discretization and, if necessary, linearization of systems of partial differential equations (PDEs) yields large, sparse linear systems of equations.
These systems are usually too large for direct methods and so iterative methods must be used instead.
Iterative methods require preconditioning in order to be effective. We choose to use block-style preconditioning in the spirit of \cite{MurphyGolubWathen2000}. The specific preconditioners we use are block diagonal and block upper-triangular. The block diagonal is a standard choice, and the block triangular preconditioner uses a very simple approximation to the Schur complement that in some cases leads to a compact perturbation of the identity. We apply these preconditioners to a variety of coupled physics problems.
In order to analyze the effectiveness of these preconditioners, we conduct an eigenvalue analysis and a rudimentary field of values analysis of the preconditioned systems. The field of values analysis is done in order to obtain rigorous bounds on the convergence rate of GMRES, which is information an eigenvalue analysis cannot provide.
We implement each of these systems of PDEs in Sundance, a part of the Trilinos library, and provide numerical results corresponding to a wide range of parameter choices.