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.

Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing

“…When the saddle point problem (8.1) originates from certain classes of PDEs, it can be shown that σ min (B T S −1 B) = O(h α ) for some known integer α, where h is a mesh-dependent parameter. Therefore, rough estimates of such quantity can be easily obtained; see, e.g., [20], [25], [31] and references therein. When running the same example with ε inner = ε/ r k−1 , that is, m = 1, we did not notice any change in the convergence behavior (the curves are the same as those in Figure 8.1), showing that our bound is very conservative on this problem.…”

“…For example, when operating with A implies a solution of a linear system, as is the case in Schur complement computations (see, e.g., [19], [20], [37]), and in certain eigenvalue algorithms [16], [32], or when the matrix is very large (and/or dense), and a reasonable approximation can be used [2], [11]. Several authors have studied different aspects of the use of inexact matrix-vector multiplication in iterative methods, sometimes in the context of small perturbations, and in some other instances allowing for large tolerances (though not letting them grow); see, e.g., [13], [14], [15], [38].…”

A Preconditioner for Generalized Saddle Point Problems

“…Besides specialized sparse direct solvers [16,17] we mention, among others, Uzawa-type schemes [11,21,24,27,62], block and approximate Schur complement preconditioners [4,15,20,22,41,45,46,48,51], splitting methods [18,30,31,49,57], indefinite preconditioning [23,35,39,43,48], iterative projection methods [5], iterative null space methods [1,32,54], and preconditioning methods based on approximate factorization of the coefficient matrix [25,50]. Several of these algorithms are based on some form of reduction to a smaller system, for example, by projecting the problem onto the null space of B, while others work with the original (augmented) matrix in (1.1).…”

BDDC for mixed‐hybrid formulation of flow in porous media with combined mesh dimensions

“…the first term in (11). It was shown in [28] that the system of equations (14) can be reduced (twice) to the Schur complement corresponding to the Lagrange multipliers λ and solved efficiently by a direct or iterative solver. Here, we will look for an efficient solution of a slightly modified, and in general also block dense, system which is introduced in the next section.…”

“…In addition, the presentation of the BDDC algorithm is driven more by an efficient implementation, while it is more oriented towards underlying theory in [6]. We take advantage of the special structure of the blocks in matrix (1) studied in detail in [26,28,29]. In particular, the nonzero structure of block C resulting from a combination of meshes with different spatial dimensions is considered in [30].…”