Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem

The mixed hybrid nite element discretization of Darcy's law and continuity equation describing the potential uid ow problem in porous media leads to a symmetric indeenite linear system for the pressure and velocity vector components. As a method of solution the reduction to Schur complement systems based on successive block elimination is considered. The computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugate gradient or conjugate residual method applied to certain Schur complement systems. It is shown that due to special block structure the spectral properties of successive Schur complement matrices do not deteriorate and the approach based on the block elimination and subsequent iterative solution is well justiied. Theoretical results are illustrated by numerical experiments.