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

*(English)*Zbl 0978.76052Authors’ summary: 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 pressure and velocity components. As a method of solution, we consider the reduction to three Schur complement systems based on successive block elimination. 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 structured properties of these consecutive Schur complement matrices are studied in detail in terms of discretization parameters. Based on these results, the computational complexity of a direct solution method is estimated and compared to the computational cost of iterative conjugate gradient method applied to Schur complement systems. It is shown that, due to the 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.

Reviewer: A.Meister (Hamburg)

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76S05 | Flows in porous media; filtration; seepage |

76B10 | Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing |

65F10 | Iterative numerical methods for linear systems |

65F50 | Computational methods for sparse matrices |