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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.
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
 SIAM J. Numer. Anal
, 1997
"... . In this paper, we consider the socalled "inexact Uzawa" algorithm for iteratively solving block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity and mixed finite ..."
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Cited by 93 (4 self)
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. In this paper, we consider the socalled "inexact Uzawa" algorithm for iteratively solving block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity and mixed finite element discretization of second order problems. We consider both the linear and nonlinear variants of the inexact Uzawa algorithm. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block. In the nonlinear case, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in t...
Approximate Inverse Techniques for BlockPartitioned Matrices
 SIAM J. Sci. Comput
, 1995
"... This paper proposes some preconditioning options when the system matrix is in blockpartitioned form. This form may arise naturally, for example from the incompressible NavierStokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to g ..."
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Cited by 44 (12 self)
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This paper proposes some preconditioning options when the system matrix is in blockpartitioned form. This form may arise naturally, for example from the incompressible NavierStokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner. The storage requirements for these preconditioners may be much less than for ILU preconditioners for tough, largescale CFD problems. The numerical experiments reported show that these preconditioners can help us solve difficult linear systems whose coefficient matrices are highly indefinite. 1 Introduction Consider the block partitioning of a matrix A, in the form A = ` B F E C ' (1) where the blocking naturally occurs due the ordering of the equations and the variables. Matrices of this form arise in many applications, such as in the incompressible NavierStokes equations, where the sc...
Iterative Methods for Problems in Computational Fluid Dynamics
 ITERATIVE METHODS IN SCIENTIFIC COMPUTING
, 1996
"... We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible NavierStokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equ ..."
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Cited by 28 (6 self)
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We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible NavierStokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equations at each time step, and discretization in space then produces a series of linear algebraic systems. We give an overview of commonly used time and space discretization techniques, and we discuss a variety of algorithmic strategies for solving the resulting systems of equations. The emphasis is on preconditioning techniques, which can be combined with Krylov subspace iterative methods. In many cases the solution of subsidiary problems such as the discrete convectiondiffusion equation and the discrete Stokes equations plays a crucial role. We examine iterative techniques for these problems and show how they can be integrated into effective solution algorithms for the NavierStokes equa...
Fast iterative solvers for discrete Stokes equations
 SIAM J. Sci. Comput
"... Abstract. We consider saddle point problems that result from the finite element discretization of stationary and instationary Stokes equations. Three efficient iterative solvers for these problems are treated, namely the preconditioned CG method introduced by Bramble and Pasciak, the preconditioned ..."
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Cited by 19 (5 self)
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Abstract. We consider saddle point problems that result from the finite element discretization of stationary and instationary Stokes equations. Three efficient iterative solvers for these problems are treated, namely the preconditioned CG method introduced by Bramble and Pasciak, the preconditioned MINRES method and a method due to Bank et al. We give a detailed overview of algorithmic aspects and theoretical convergence results. For the method of Bank et al a new convergence analysis is presented. A comparative study of the three methods for a 3D Stokes problem discretized by the HoodTaylor P2 − P1 finite element pair is given. AMS subject classifications. 65N30, 65F10 Key words. Stokes equations, inexact Uzawa methods, preconditioned MINRES, multigrid
A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a . . .
, 2002
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Preconditioners For Saddle Point Problems Arising In Computational Fluid Dynamics
 Appl. Numer. Math
, 2002
"... Discretization and linearization of the incompressible NavierStokes equations leads to linear algebraic systems in which the coefficient matrix has the form of a saddle point problem F B T B 0 u p = f g : (0.1) In this paper, we describe the development of efficient and general iterative solut ..."
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Cited by 16 (1 self)
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Discretization and linearization of the incompressible NavierStokes equations leads to linear algebraic systems in which the coefficient matrix has the form of a saddle point problem F B T B 0 u p = f g : (0.1) In this paper, we describe the development of efficient and general iterative solution algorithms for this class of problems. We review the case where (0.1) arises from the steadystate Stokes equations and show that solution methods such as the Uzawa algorithm lead naturally to a focus on the Schur complement operator BF 1 B T together with efficient strategies of applying the action of F 1 to a vector. We then discuss the advantages of explicitly working with the coupled form of the block system (0.1). Using this point of view, we describe some new algorithms derived by developing efficient methods for the Schur complement systems arising from the NavierStokes equations, and we demonstrate their effectiveness for solving both steadystate and evolutionary problems.
The Convergence Of Iterative Solution Methods For Symmetric And Indefinite Linear Systems
, 1997
"... this paper we concentrate on convergence estimates for miminum residual iteration applied to a linear system Ax = b, (so that A represents the preconditioned coefficient matrix if preconditioning is employed). In particular we generalise the results of [25] to establish rigorous convergence estimate ..."
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Cited by 15 (5 self)
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this paper we concentrate on convergence estimates for miminum residual iteration applied to a linear system Ax = b, (so that A represents the preconditioned coefficient matrix if preconditioning is employed). In particular we generalise the results of [25] to establish rigorous convergence estimates for families of matrices which depend on an asymptotically small parameter ff (in applications ff is typically a positive power of the mesh size parameter h). These results prove the superiority of the minimum residual approach over the solution of normal equations for all except one very special type of symmetric and indefinite matrix. More background and an easy introduction to this problem can be found in [8], pp. 310315
Block Preconditioners for Nonsymmetric Saddle Point Problems
, 2001
"... this paper is independent of the parameter h. ..."
Analysis of preconditioned Picard iterations for the NavierStokes equations
, 2001
"... this paper was introduced in [18] and is given by p ; (1.8) with M p ; A p de ned as above and F p the projection onto the pressure nite element space of the velocity operator of equation (1.2a), + b r + . Note that when b = 0 we recover the preconditioner (1.6). The numerical results presen ..."
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Cited by 9 (3 self)
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this paper was introduced in [18] and is given by p ; (1.8) with M p ; A p de ned as above and F p the projection onto the pressure nite element space of the velocity operator of equation (1.2a), + b r + . Note that when b = 0 we recover the preconditioner (1.6). The numerical results presented in [18] together with those in [10] show no mesh dependence and a dependence on of order or less. Our analysis con rms and re nes these results. We provide meshindependent bounds on the eigenvalues of the preconditioned system for the general case of timedependent, stabilized problems. Moreover, our bounds are shown numerically to be descriptive with respect to the parameters in the problem. We show in particular that the modulus of the eigenvalues of the preconditioned system grows like R = kbk L =, which for problems with length scale O(1) is the Reynolds number, and demonstrate that this leads to the number of iterations growing linearly with . As a byproduct of our analysis, we con rm the bounds in [9] for the steadystate case with b S = M p = and extend them to the case of stabilized problems. The paper is structured as follows. In Section 2 we derive the preconditioner for the timedependent problem (1.1) with the general set of boundary conditions (1.1c, 1.1d). In Section 3 we present the weak formulation of problem (1.2) and consider the stabilized formulation and some useful related bilinear forms. Moreover, we derive results for the discrete operators arising from our mixed stabilized nite element formulation. Section 4 presents the main results, Theorem 8 and Theorem 9 which establish eigenvalue bounds for preconditioners (1.7), (1.8). Finally Section 5 investigates the issue of spectral description of convergence for iterative solvers with ...