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Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 322 (25 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Optimization Of The Hermitian And SkewHermitian Splitting Iteration For SaddlePoint Problems
, 2003
"... We study the asymptotic rate of convergence of the alternating Hermitian/skewHermitian iteration for solving saddlepoint problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal con ..."
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Cited by 22 (13 self)
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We study the asymptotic rate of convergence of the alternating Hermitian/skewHermitian iteration for solving saddlepoint problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal convergence parameters for the model problem of the Poisson equation written in divgrad form. We show that the optimized convergence rate for small mesh parameter h is asymptotically 1  O(h ). Furthermore we show that when the splitting is used as a preconditioner for a Krylov method, a different optimization leading to two clusters in the spectrum gives an optimal, hindependent, convergence rate. The theoretical analysis is supported by numerical experiments.
PRESSURE SCHUR COMPLEMENT PRECONDITIONERS FOR THE DISCRETE OSEEN PROBLEM
, 2007
"... We consider several preconditioners for the pressure Schur complement of the discrete steady Oseen problem. Two of the preconditioners are well known from the literature and the other is new. Supplemented with an appropriate approximate solve for an auxiliary velocity subproblem, these approaches g ..."
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Cited by 20 (3 self)
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We consider several preconditioners for the pressure Schur complement of the discrete steady Oseen problem. Two of the preconditioners are well known from the literature and the other is new. Supplemented with an appropriate approximate solve for an auxiliary velocity subproblem, these approaches give rise to a family of the block preconditioners for the matrix of the discrete Oseen system. In the paper we critically review possible advantages and difficulties of using various Schur complement preconditioners. We recall existing eigenvalue bounds for the preconditioned Schur complement and prove such for the newly proposed preconditioner. These bounds hold both for LBB stable and stabilized finite elements. Results of numerical experiments for several model twodimensional and threedimensional problems are presented. In the experiments we use LBB stable finite element methods on uniform triangular and tetrahedral meshes. One particular conclusion is that in spite of essential improvement in comparison with “simple” scaled massmatrix preconditioners in certain cases, none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex (e.g., circulating) advection vector field.
Block preconditioning of realvalued iterative algorithms for complex linear systems
, 2008
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CONVERGENCE PROPERTIES OF PRECONDITIONED HERMITIAN AND SKEWHERMITIAN SPLITTING METHODS FOR NONHERMITIAN POSITIVE SEMIDEFINITE MATRICES
"... Abstract. For the nonHermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skewHermitian splitting iteration methods. We then apply these results to block tri ..."
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Cited by 13 (7 self)
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Abstract. For the nonHermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skewHermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skewHermitian splitting iteration methods. 1.
Spectral properties of the hermitian and skewhermitian splitting preconditioner for saddle point problems
 SIAM J. Matrix Anal. Appl
, 2004
"... Abstract. In this paper we derive bounds on the eigenvalues of the preconditioned matrix that arises in the solution of saddle point problems when the Hermitian and skewHermitian splitting preconditioner is employed. We also give sufficient conditions for the eigenvalues to be real. A few ..."
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Cited by 13 (7 self)
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Abstract. In this paper we derive bounds on the eigenvalues of the preconditioned matrix that arises in the solution of saddle point problems when the Hermitian and skewHermitian splitting preconditioner is employed. We also give sufficient conditions for the eigenvalues to be real. A few
A Relaxed Dimensional Factorization Preconditioner for the Incompressible NavierStokes Equations
, 2010
"... In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo in [8]. Nu ..."
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Cited by 9 (3 self)
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In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo in [8]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.
A Dimensional Split Preconditioner for Stokes and Linearized Navier–Stokes Equations
, 2010
"... In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier– Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixedpoi ..."
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Cited by 9 (2 self)
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In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier– Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixedpoint iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included. Key words. saddle point problems, matrix splittings, iterative methods, preconditioning, Stokes problem, Oseen problem, stretched grids