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

Abstract

The mixed-hybrid finite element discretization of Darcy's law and continuity equation describing the potential fluid flow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vector components. As a method of solution the reduction to three Schur complement systems based on successive block elimination is considered. The first and second Schur complement matrices are formed eliminating the velocity and pressure variables, respectively, and the third Schur complement matrix is obtained by elimination of a part of Lagrange multipliers that come from the hybridization of a mixed method. The structural properties of these consecutive Schur complement matrices in terms of the discretization parameters are studied in detail. Based on these results the computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugate gradient method applied to 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 justified. Theoretical results are illustrated by numerical experiments.