The diffusive component in many important application areas is characterized by a discontinuous diffusion coefficient with fine-scale anisotropic spatial structure; moreover, the underlying grid may be severely distorted. Thus, increasingly mixed discretizations, which include mixed finite element methods (FEM) and support operator methods (SOM), are employed because they explicitly enforce important physical physical properties of the problem. Unfortunately, these discretizations are based on the first order form, and hence naturally lead to an indefinite linear system. In the case of orthogonal grids and a diagonal diffusion tensor optimal preconditioners have been developed. However, the performance of these methods degrades with increasing full tensor anisotropy or grid distortion. Moreover, the development of alternative preconditioners that are robust with respect to general forms of anisotropy has been thwarted by a significant loss of sparsity.
In this work we are motivated by one specific advantage that the hybrid or local forms of mixed discretizations exhibit, namely, their more localized sparsity structure. Specifically, for the SOM we consider augmentation of the flux (i.e., vector unknowns) such that an appropriate ordering of the augmented flux 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). This reduced system is readily solved with existing robust multigrid methods, such as Dendy's black box multigrid. An analogous approach is used to augment the hybrid system and derive the equivalent preconditioner for this case. We demonstrate the effectiveness of this preconditioner for both GMRES iterations of the full indefinite system as well as a CG iterations of the reduced scalar problem.