Block preconditioning for saddle point systems with indefinite (1, 1) block

Abstract

We investigate the solution of linear systems of saddle point type with an indefinite (1, 1) block by preconditioned iterative methods. Our main focus is on block matrices arising from eigenvalue problems in incompressible fluid dynamics. A block triangular preconditioner based on an augmented Lagrangian formulation is shown to result in fast convergence of the GMRES iteration for a wide range of problem and algorithm parameters. Some theoretical estimates for the eigenvalues of the preconditioned matrices are given. Inexact variants of the preconditioner are also considered.

An Augmented Lagrangian Approach to Linearized Problems in Hydrodynamic Stability

“…The results in [1] show the good performance of the AL-based approach, especially in terms of robustness with respect to problem and algorithmic parameters. In that paper, however, the crucial question of how to efficiently approximate the action of (A α +γB T W −1 B) −1 was left open.…”

“…We note that some preliminary experiments with a block triangular preconditioner (2.2) for systems of the form (1.7) arising from marker-and-cell (MAC) discretizations of flow problems can be found in [1]. The results in [1] show the good performance of the AL-based approach, especially in terms of robustness with respect to problem and algorithmic parameters.…”

“…Indeed, it can be easily shown that the standard ellipticity, continuity, and stability conditions for corresponding finite element bilinear forms imply c 1 …”

Block triangular preconditioner for static Maxwell equations

“…The form of (1.2) frequently occurs in a large number of applications, such as the (linearized) Navier-Stokes equations [21], the time-harmonic Maxwell equations [7,8,10], the linear programming (LP) problem and the quadratic programming (QP) problem [17,20]. At present, there usually exist four kinds of preconditioners for the saddle point linear systems (1.2): block diagonal preconditioner [22,23,24,25], block triangular preconditioner [15,16,26,27,28,37], constraint preconditioner [29,30,31,32,33] and Hermitian and skew-Hermitian splitting (HSS) preconditioner [34]. One can [12] for a general discussion.…”

Sparse block factorization of saddle point matrices

“…Numerous solution methods for the saddle point systems of the form (1) can be found in the literature and many of them have focused on preconditioning techniques for Krylov subspace iterative solvers [1,2,3,7,12,18,23,24,26,28,31]. As a direct method against iterative solvers, various techniques on symmetric indefinite factorization P TÅ P = LDL T can be found in [9,14,21,34,35,38], where P is a permutation matrix, L is unit lower triangular matrix, D is block-diagonal matrix with blocks of order 1 or 2.…”