An Augmented Lagrangian‐Based Approach to the Oseen Problem

Abstract

We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schöberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2-P0 and isoP2-P1 finite elements in support of our conclusions. We also show results of a comparison with two state-of-the-art preconditioners, showing the competitiveness of our approach.

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

“…Moreover, the quality of the preconditioner was not significantly affected when linear solves with the (1, 1) block of (4) were performed inexactly via a single W-cycle of a specially developed multigrid method. It was also shown in [5] that γ ≈ 1 gave sufficiently good results in many cases, although in a few situations the best overall results were obtained using smaller values of γ (up to about γ ≈ 0.02). We note that in the Oseen problem, the matrix A is nonsymmetric.…”

“…We show that for this "ideal" version of the preconditioner, under some fairly mild assumptions the eigenvalues of the preconditioned matrix become tightly clustered around 1 as γ → ∞. Our analysis makes use of the following simple Lemma, which is a straightforward consequence of [10,Exercise 12.12]; see also [5,14].…”

“…Note that for β ≤ 0 or β > 0 and sufficiently small, the spectrum of the coefficient matrix of (3) is entirely contained in the right half-plane; see, e.g., [3,6]. For the case β = 0, it was shown in [5] that a block triangular preconditioner of the type…”

“…For the definite case β = 0, an effective multigrid methods has been developed in [5]. The applicability of such method in the indefinite case β > 0 is questionable, unless β is sufficiently small.…”

“…For the Oseen problem this rescaling is not useful, and the viscosity parameter ν controls the difficulty of the problem: the smaller ν, the harder the problem. Discretization of problem (9)-(11) by MAC or any LBB-stable finite element method leads to the solution of linear systems in saddle point form; see, e.g., [3,5,9,11,13,16]. The diagonal blocks of the n × n matrix A consist of d discrete Laplace operators.…”