Schur complement systems in the mixed-hybrid finite element approximation of the potential fluid flow problem

The mixed-hybrid finite element discretization of Darcy's law and continuit y equation describing the potential fluid ow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vecto r components. As a method of solution the reduction to three Schur compleme nt systems based on successive block elimination is considered. The rst and second Schur complement matrices are formed eliminating the velocity and p ressure variables, respectively, and the third Schur complement matrix is o btained by elimination of a part of Lagrange multipliers that come from the hybridization of a mixed method. The structural properties of these consec utive Schur complement matrices in terms of the discretization parameters a re studied in detail. Based on these results the computational complexity o f a direct solution method is estimated and compared to the computational c ost of the iterative conjugate gradient method applied to Schur complement systems. It is shown that due to special block structure the spectral prope rties of successive Schur complement matrices do not deteriorate and the ap proach based on the block elimination and subsequent iterative solution is well justified. Theoretical results are illustrated by numerical experiment s.