Block preconditioners for coupled physics problems

Finite element discretizations of multiphysics problems frequently give rise to blockstructured linear algebra problems that require effective preconditioners. We build two classes of preconditioners in the spirit of well-known block factorizations [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput., 21 (2000), pp. 1969-1972; I. C. F. Ipsen, SIAM J. Sci. Comput., 23 (2001), pp. 1050-1051] and apply these to the diffusive portion of the bidomain equations and the Bénard convection problem. An abstract generalized eigenvalue problem allows us to give application-specific bounds for the real parts of eigenvalues for these two problems. This analysis is accompanied by numerical calculations with several interesting features. One of our preconditioners for the bidomain equations converges in five iterations for a range of problem sizes. For Bénard convection, we observe mesh-independent convergence with reasonable robustness with respect to physical parameters, and offer some preliminary parallel scaling results on a multicore processor via message passing interface (MPI).