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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...
A Multigrid Algorithm For The Mortar Finite Element Method
 SIAM J. NUMER. ANAL
"... The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar finite element discretization of second order elliptic boundary value problems. In order to establish the infsup condition for the saddle point formulation and to motivate ..."
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Cited by 59 (10 self)
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The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar finite element discretization of second order elliptic boundary value problems. In order to establish the infsup condition for the saddle point formulation and to motivate the subsequent treatment of the discretizations we revisit first briefly the theoretical concept of the mortar finite element method. Employing suitable meshdependent norms we verify the validity of the LBB condition for the resulting mixed method and prove an L 2 error estimate. This is the key for establishing a suitable approximation property for our multigrid convergence proof via a duality argument. In fact, we are able to verify optimal multigrid efficiency based on a smoother which is applied to the whole coupled system of equations. We conclude with several numerical tests of the proposed scheme which confirm the theoretical results and show the efficiency and the robustness of the method even in situations not covered by the theory.
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
Multigrid And Krylov Subspace Methods For The Discrete Stokes Equations
 INT. J. NUMER. METH. FLUIDS
, 1994
"... Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations, a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper, we compare the perfo ..."
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Cited by 40 (3 self)
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Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations, a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper, we compare the performance of four such methods: variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual, and multigrid methods, for solving several twodimensional model problems. The results indicate that where it is applicable, multigrid with smoothing based on incomplete factorizaton is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantages of being both independent of iteration parameters and widely applicable.
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 Posteriori Error Estimates For The Stokes Problem
 SIAM J. Numer. Anal
, 1991
"... . We derive and analyze an a posteriori error estimate for the minielement discretization of the Stokes equations. The estimate is based on the solution of a local Stokes problem in each element of the finite element mesh, using spaces of quadratic bump functions for both velocity and pressure erro ..."
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Cited by 29 (2 self)
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. We derive and analyze an a posteriori error estimate for the minielement discretization of the Stokes equations. The estimate is based on the solution of a local Stokes problem in each element of the finite element mesh, using spaces of quadratic bump functions for both velocity and pressure errors. This results in solving a 9 \Theta 9 system which reduces to two 3 \Theta 3 systems easily invertible. Comparisons with other estimates based on a PetrovGalerkin solution are used in our analysis, which shows that it provides a reasonable approximation of the actual discretization error. Numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures. Key words. Mixed finite element methods, Stokes equations, a posteriori error estimates, mesh adaptation, minielement formulation, PetrovGalerkin formulation. AMS subject classifications. 65F10, 65N20, 65N30. 1. Introduction. The need for accurate solutions of large scal...
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.
A finite element based level set method for twophase incompressible flows
 Comput. Vis. Sci
, 2004
"... Abstract. We present a method that has been developed for the efficient numerical simulation of twophase incompressible flows. For capturing the interface between the flows the level set technique is applied. The continuous model consists of the incompressible NavierStokes equations coupled with a ..."
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Cited by 24 (6 self)
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Abstract. We present a method that has been developed for the efficient numerical simulation of twophase incompressible flows. For capturing the interface between the flows the level set technique is applied. The continuous model consists of the incompressible NavierStokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (socalled continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the LaplaceBeltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θscheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the Fast Marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, LaplaceBeltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented. AMS subject classifications. 65M60, 65T10, 76D05, 76D45, 65N22 1. Introduction. In
Adaptive Multilevel Techniques for Mixed Finite Element Discretizations of Elliptic Boundary Value Problems
 SIAM J. NUMER. ANAL
, 1994
"... We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. By a well known postprocessing technique the discrete problem is equivalent to a modifie ..."
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Cited by 21 (7 self)
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We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. By a well known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned cgiterations using a multilevel BPXtype preconditioner designed for standard nonconforming approximations. Local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results. The performance of the preconditioner and the error estimator is illustrated by several numerical examples.