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93
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.
Preconditioning in H(div) and Applications
 Math. Comp
, 1998
"... . We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I \Gamma grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational fo ..."
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Cited by 47 (5 self)
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. We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I \Gamma grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems. 1. Introduction The Hilbert space H(div) consists of squareintegr...
Adaptive Wavelet Methods For Saddle Point Problems  Optimal Convergence Rates
 IGPM report, RWTH Aachen
, 2001
"... In this paper an adaptive wavelet scheme for saddle point problems is developed and analysed. Under the assumption that the underlying continuous problem satisfies the infsup condition it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity. Thi ..."
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Cited by 45 (18 self)
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In this paper an adaptive wavelet scheme for saddle point problems is developed and analysed. Under the assumption that the underlying continuous problem satisfies the infsup condition it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity. This means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the best wavelet Nterm approximation of the solution with respect to the relevant norms. Moreover, the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom. It is remarkable that compatibility constraints on the trial spaces such as the LadyshenskajaBabuskaBrezzi (LBB) condition do not arise. In the second part the general results are applied to the Stokes problem. Aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon, the regularity of the solutions in certain Besov scales is analyzed. These results reveal under which circumstances the work/accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements. This in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the Stokes problem.
A preconditioner for generalized saddle point problems
 SIAM J. Matrix Anal. Appl
, 2004
"... Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skewsymmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matri ..."
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Cited by 44 (23 self)
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Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skewsymmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical
Preconditioning discrete approximations of the ReissnerMindlin plate model
 Math. Modelling Numer. Anal
, 1997
"... Abstract. We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner–Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear sys ..."
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Cited by 38 (10 self)
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Abstract. We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner–Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear system has spectral condition number independent of both the mesh size h and the plate thickness t. We further discuss how this preconditioner may be implemented and then apply it to efficiently solve this indefinite linear system. Although the mixed formulation of the Reissner–Mindlin problem has a saddlepoint structure common to other mixed variational problems, the presence of the small parameter t and the fact that the matrix in the upper left corner of the partition is only positive semidefinite introduces new complications.
Analysis of iterative methods for saddle point problems: A unified approach
 Math. Comp
, 1998
"... Abstract. In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but esse ..."
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Cited by 36 (4 self)
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Abstract. In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.
On solving blockstructured indefinite linear systems
, 2003
"... We consider 2 × 2 block indefinite linear systems whose (2, 2) block is zero. Such systems arise in many applications. We discuss two techniques that are based on modifying the (1, 1) block in a way that makes the system easier to solve. The main part of the paper focuses on an augmented Lagrangian ..."
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Cited by 35 (6 self)
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We consider 2 × 2 block indefinite linear systems whose (2, 2) block is zero. Such systems arise in many applications. We discuss two techniques that are based on modifying the (1, 1) block in a way that makes the system easier to solve. The main part of the paper focuses on an augmented Lagrangian approach: a technique that modifies the (1,1) block without changing the system size. The choice of the parameter involved, the spectrum of the linear system, and its condition number are discussed, and some analytical observations are provided. A technique of deflating the (1,1) block is then introduced. Finally, numerical experiments that validate the analysis are presented.
A ParticlePartition Of Unity Method For The Solution Of Elliptic, Parabolic And Hyperbolic PDEs
 SIAM J. SCI. COMP
"... In this paper, we present a meshless discretization technique for instationary convectiondiffusion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used a ..."
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Cited by 35 (6 self)
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In this paper, we present a meshless discretization technique for instationary convectiondiffusion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h or pversion. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convectiondiffusion, instationary diffusion, linear advection and elliptic problems.
A General Mixed Covolume Framework For Constructing Conservative Schemes For Elliptic Problems
 Math. Comp
, 1999
"... We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a onetoone transfer operator between trial and test space ..."
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Cited by 26 (9 self)
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We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a onetoone transfer operator between trial and test spaces. In the nonsymmetric case (convectiondiffusion equation) we show onehalf order convergence rate for the flux variable which is approximated either by the lowest order RaviartThomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.
Interior Penalty Preconditioners For Mixed Finite Element Approximations Of Elliptic Problems
 Math. Comp
, 1996
"... It is established that an interior penalty method applied to secondorder elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method. This relation can be utilized in order to construct preconditi ..."
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Cited by 26 (6 self)
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It is established that an interior penalty method applied to secondorder elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method. This relation can be utilized in order to construct preconditioners for the discrete mixed system. As an example, a family of additive Schwarz preconditioners for these systems is constructed. Numerical examples which confirm the theoretical results are also presented. 1.