Dual Variable Methods for Mixed-hybrid Finite Element Approximation of the Potential Fluid Flow Problem in Porous Media

Mixed-hybrid finite element discretization of Darcy’s law and the continuity equation that describe the potential fluid flow problem in porous media leads to symmetric indefinite saddlepoint problems. In this paper we consider solution techniques based on the computation of a nullspace basis of the whole or of a part of the left lower off-diagonal block in the system matrix and on the subsequent iterative solution of a projected system. This approach is mainly motivated by the need to solve a sequence of such systems with the same mesh but different material properties. A fundamental cycle null-space basis of the whole off-diagonal block is constructed using the spanning tree of an associated graph. It is shown that such a basis may be theoretically rather ill-conditioned. Alternatively, the orthogonal null-space basis of the sub-block used to enforce continuity over faces can be easily constructed. In the former case, the resulting projected system is symmetric positive definite and so the conjugate gradient method can be applied. The projected system in the latter case remains indefinite and the preconditioned minimal residual method (or the smoothed conjugate gradient method) should be used. The theoretical rate of convergence for both algorithms is discussed and their efficiency is compared in numerical experiments.