On Block Preconditioners for Nonsymmetric Saddle Point Problems

Krzyżanowski, Piotr

doi:10.1137/s1064827599360406

Cited by 24 publications

(21 citation statements)

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“…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.…”

Section: Block Triangular Preconditioner For Static Maxwell Equationsmentioning

confidence: 99%

Block triangular preconditioner for static Maxwell equations

Wu¹,

Huang²,

Li³

2011

Comput. Appl. Math.

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Abstract. In this paper, we explore the block triangular preconditioning techniques applied to the iterative solution of the saddle point linear systems arising from the discretized Maxwell equations. Theoretical analysis shows that all the eigenvalues of the preconditioned matrix are strongly clustered. Numerical experiments are given to demonstrate the efficiency of the presented preconditioner.Mathematical subject classification: 65F10.

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Section: Block Triangular Preconditioner For Static Maxwell Equationsmentioning

confidence: 99%

Block triangular preconditioner for static Maxwell equations

Wu¹,

Huang²,

Li³

2011

Comput. Appl. Math.

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“…The techniques for solving systems like (1.1) are so numerous that it is almost impossible to give an overview. In addition to the methods developed specifically for Navier-Stokes problems, existing techniques also include Uzawa-type algorithms [9], splitting schemes [6,12], constraint preconditioning [12,17,21,24,25,26], and (preconditioned) Krylov subspace methods based on (approximations to) the Schur complement [1,18,23].…”

Section: Introduction We Study Preconditioners For General Nonsingulmentioning

confidence: 99%

Block-Diagonal and Constraint Preconditioners for Nonsymmetric Indefinite Linear Systems. Part I: Theory

Sturler

Liesen²

2005

SIAM J. Sci. Comput.

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Abstract. We study block-diagonal preconditioners and an efficient variant of constraint preconditioners for general two-by-two block linear systems with zero (2,2)-block. We derive block-diagonal preconditioners from a splitting of the (1,1)-block of the matrix. From the resulting preconditioned system we derive a smaller, so-called related system that yields the solution of the original problem. Solving the related system corresponds to an efficient implementation of constraint preconditioning. We analyze the properties of both classes of preconditioned matrices, in particular their spectra. Using analytical results, we show that the related system matrix has the more favorable spectrum, which in many applications translates into faster convergence for Krylov subspace methods. We show that fast convergence depends mainly on the quality of the splitting, a topic for which a substantial body of theory exists. Our analysis also provides a number of new relations between block-diagonal preconditioners and constraint preconditioners. For constrained problems, solving the related system produces iterates that satisfy the constraints exactly, just as for systems with a constraint preconditioner. Finally, for the Lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a Newton iteration. Our theoretical results are confirmed by numerical experiments on a constrained optimization problem.We consider the general, nonsymmetric, nonsingular case. Our only additional requirement is the nonsingularity of the Schur-complement-type matrix derived from the splitting that defines the preconditioners. In particular, the (1,2)-block need not equal the transposed (2,1)-block, and the (1,1)-block might be indefinite or even singular. This is the first paper in a two-part sequence. In the second paper we will study the use of our preconditioners in a variety of applications.

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“…We follow the general idea of Krzyżanowski [2001] (see also Klawonn [1998b] for the symmetric case analysis). Using the natural formulation of the variational discrete problem (5) in the operator form,…”

Section: Nonoverlapping Domain Decomposition Block Preconditionermentioning

confidence: 99%

“…We use the technique of Krzyżanowski [2001]. Due to stability results for A h (·, ·), it is sufficient to check only the influence of the quality of the velocity preconditioner.…”

Section: Cond(p)mentioning

confidence: 99%

“…First we use the approach of block preconditioning, see Krzyżanowski [2001], Klawonn [1998b], Klawonn [1998a]. Using available results related to nonoverlapping Additive Schwarz preconditioning DG discretizations of the second order equations, we prove the convergence rate bounds for the corresponding block preconditioner for the DG Stokes discretization.…”

Section: Introductionmentioning

confidence: 99%

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