Schur complement reduction in the mixed-hybrid approximation of Darcyʹs law: rounding error analysis

Mixed-hybrid finite element approximation of the potential fluid flow problem leads to the solution of a large symmetric indefinite system for the velocity and potential head vector components. Such discretization gives rise to a very accurate approximation of the continuity equation in every element, and for low-order discretizations, the structural properties of the discrete matrix blocks allow cheap block elimination of the positive-definite diagonal block and subsequent reduction to the Schur complement system for the pressure and Lagrangian vector components. This system is then frequently solved by the iterative conjugate gradient-type method. Whereas this approach is well known, considerably less attention has been paid to the numerical stability aspects of such transformation. It was shown in [5] that block LU factorization can be unstable even when the system matrix is symmetric positive definite. In this paper we examine this type of conditional stability for a particular application in the underground water flow modelling. We show that the actual error of the computed approximate solution depends not only on the user-defined tolerance in the conjugate gradient process but also on the spectral properties of the corresponding matrix blocks eliminated during the Schur complement reduction. It is often observed that although the backward error of the approximate solution in the iterative part is reduced to the level of machine accuracy, the total residual norm after the back-substitution process remains at certain accuracy level. We give a bound for this maximal attainable accuracy and illustrate our theoretical results on a model example.