A New Analysis of Block Preconditioners for Saddle Point Problems

Abstract

We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these latter based on symmetric block Gauss-Seidel type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow to localize accurately both real and non real eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory.

CPU and GPU Performance of Large Scale Numerical Simulations in Geophysics

“…On the other hand, approximating S by S has less profound effect, compared with that of M . This is confirmed by the rigorous eigenvalue analysis provided in [25] (see in particular Corollary 4.5). For the purpose of this study we compute S using the so-called element-by-element (EBE) approach, see for instance, [16,5,22,21] and the references therein.…”

A new algebraic multigrid approach for Stokes problems

“…The stated results (4.13), (4.14) are straightforward corollaries of Theorem 4.1 of [29] if we can show that the eigenvalues of (4.12) are also that of a matrix…”

“…The first lemma extends, for saddle point matrices in the form (4.10) (i.e., nonsymmetric but positive definite in R n+m ), the eigenvalue analysis developed in [3, Proposition 2.12] and [29,Theorem 4.1] to a form (4.12) of generalized eigenvalue problem where the right hand side matrix is an orthogonal projector. Observe that the matrices (3.3) resulting from the transformation suggested in Section 3 satisfy the assumptions of the lemma.…”

“…It goes without saying that the mentioned difficulties do not concern the use of AMG techniques within the framework of block preconditioning methods. These (see, e.g., [2,11,12,29]) require approximations of the inverse of certain matrix blocks, and are in fact most effective when multigrid is used for this purpose. Here, one may apply AMG without particular difficulty because these matrix blocks are related to scalar Poissonlike problems.…”

Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

“…The literature on this class of preconditioners is huge. We refer for more details to the articles [2,3,4,18,23], the surveys [5,8,10,30] and the books [12,28], with numerous references therein. In general, the exact factorization of a two-by-two block matrix is…”