Mimetic Finite Difference (MFD) discretizations of the diffusion equation, which are based on the first-order form, have gained significant popularity in recent years. The mimetic methodology has similarities with finite volume methods and mixed finite element methods (MFEM), but are the first to be extended to general polygonal and polyhedral elements while maintaining symmetry. In all cases, since these discretizations are based on the first-order form, they naturally lead to an indefinite linear system. Block elimination of the flux (i.e., vector unknowns), destroys the overall sparsity of the reduced system for the scalar variable. Hybridization, is one approach which improves the sparsity of this reduced system. However, for many existing codes, hybridization is not practical, hence we focus on treating the full indefinite system.

Motivated by the more localized sparsity structure of hybridized discretizations, we increase the degrees of freedom in a manner that leads to a more localized sparsity structure. Specifically, for the support operator MFD we consider augmentation of the flux (i.e., vector unknowns) such that an appropriate ordering of the unknowns leads to a new block diagonal system for this component. In contrast to the block diagonal structure of the hybrid system this system has blocks centered about vertices, and block elimination of the flux (i.e., formation of the Schur complement) leads to a symmetric positive definite scalar problem with a standard cell-based 9-point structure (in two dimensions). In previous work, we demonstrated that this reduced system is readily solved with existing robust multigrid methods, and that it is an effective preconditioner for conjugate gradient iterations on the reduced system. But we are really interested in efficiently solving the full indefinite system. Here we investigate two approaches to this problem. First we consider using the augmented flux system as a block preconditioner for GMRES iterations of the full system. Alternatively, we use inner and outer preconditioned conjugate gradient iterations to mitigate the lack of sparsity in the reduced system, the Schur complement, for the scalar variable. The inner iteration inverts the mass matrix for the vector flux and the outer iteration inverts the Schur complement for the scalar variable. Although, this nested approach is commonly thought to be too costly, here we find that it is faster, and scales slightly better than the preconditioned GMRES.