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Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations
, 1002
"... We study different variants of the augmented Lagrangianbased block triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied ..."
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Cited by 41 (10 self)
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We study different variants of the augmented Lagrangianbased block triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied to various finite element and MAC discretizations of the Oseen problem in two and three space dimensions. Both steady and unsteady problems are considered. Numerical experiments show the effectiveness of the proposed preconditioners for a wide range of problem parameters. Implementation on parallel architectures is also considered. The augmented Lagrangianbased approach is further generalized to deal with linear systems from stabilized finite element discretizations. Copyright c ○ 2000 John Wiley & Sons, Ltd. key words: preconditioning; saddle point problems; Oseen problem; augmented Lagrangian method; Krylov subspace methods; parallel computing 1.
PRECONDITIONING DISCRETIZATIONS OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
, 2009
"... This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be c ..."
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Cited by 26 (4 self)
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This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be considered. In particular, parameter dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several examples and models which have been discussed in the literature previously. However, here each example is discussed with reference to a more unified abstract approach.
An overlapping Schwarz algorithm for almost incompressible elasticity
 SIAM J. Numer. Anal
"... Abstract. Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a lowdimensional space previously developed for scalar elliptic problems and a domain decomp ..."
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Cited by 25 (7 self)
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Abstract. Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a lowdimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type, i.e., a method based on nonoverlapping decompositions of the domain, while the local components of the preconditioner are based on solvers on a set of overlapping subdomains. A bound is established for the condition number of the algorithm which grows in proportion to the square of the logarithm of the number of degrees of freedom in individual subdomains and the third power of the relative overlap between the overlapping subdomains, and which is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. A positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward. Numerical results, which include a comparison with problems of compressible elasticity, illustrate the findings.
A comparison of preconditioners for incompressible Navier–Stokes solvers
 International Journal for Numerical Methods in Fluids 2008; 57:1731–1751. DOI: 10.1002/fld.1684
"... We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the socalled saddle point type, which means that there is a large block of zeros on the main diagonal. To solve th ..."
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Cited by 22 (10 self)
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We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the socalled saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For mediumsized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright q 2007
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.
ANALYSIS OF AUGMENTED LAGRANGIANBASED PRECONDITIONERS FOR THE STEADY INCOMPRESSIBLE NAVIERSTOKES EQUATIONS
"... Abstract. We analyze a class of modified augmented Lagrangianbased preconditioners for both stable and stabilized finite element discretizations of the steady incompressible Navier–Stokes equations. We study the eigenvalues of the preconditioned matrices obtained from Picard linearization, and we d ..."
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Cited by 9 (3 self)
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Abstract. We analyze a class of modified augmented Lagrangianbased preconditioners for both stable and stabilized finite element discretizations of the steady incompressible Navier–Stokes equations. We study the eigenvalues of the preconditioned matrices obtained from Picard linearization, and we devise a simple and effective method for the choice of the augmentation parameter γ based on Fourier analysis. Numerical experiments on a wide range of model problems demonstrate the robustness of these preconditioners, yielding fast convergence independent of mesh size and only mildly dependent on viscosity on both uniform and stretched grids. Good results are also obtained on linear systems arising from Newton linearization. We also show that performing inexact preconditioner solves with an algebraic multigrid algorithm results in excellent scalability. Comparisons of the modified augmented Lagrangian preconditioners with other stateoftheart techniques show the competitiveness of our approach.
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
AN AUGMENTED LAGRANGIAN APPROACH TO LINEARIZED PROBLEMS IN HYDRODYNAMIC STABILITY
"... Abstract. The solution of linear systems arising from the linear stability analysis of solutions of the Navier–Stokes equations is considered. Due to indefiniteness of the submatrix corresponding to the velocities, these systems pose a serious challenge for iterative solution methods. In this paper, ..."
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Cited by 8 (4 self)
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Abstract. The solution of linear systems arising from the linear stability analysis of solutions of the Navier–Stokes equations is considered. Due to indefiniteness of the submatrix corresponding to the velocities, these systems pose a serious challenge for iterative solution methods. In this paper, the augmented Lagrangianbased block triangular preconditioner introduced by the authors in [2] is extended to this class of problems. We prove eigenvalue estimates for the velocity submatrix and deduce several representations of the Schur complement operator which are relevant to numerical properties of the augmented system. Numerical experiments on several model problems demonstrate the effectiveness and robustness of the preconditioner over a wide range of problem parameters. Key words. Navier–Stokes equations, incompressible flow, linear stability analysis, eigenvalues, finite elements, preconditioning, iterative methods, multigrid AMS subject classifications. 65F10, 65N22, 65F50. 1. Introduction. In
FIELDOFVALUES CONVERGENCE ANALYSIS OF AUGMENTED LAGRANGIAN PRECONDITIONERS FOR THE LINEARIZED NAVIERSTOKES PROBLEM
"... We study a block triangular preconditioner for finite element approximations of the linearized Navier–Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [2]. In this paper we prove fieldofvalues type estimates ..."
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Cited by 8 (4 self)
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We study a block triangular preconditioner for finite element approximations of the linearized Navier–Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [2]. In this paper we prove fieldofvalues type estimates for the preconditioned system which lead to optimal convergence bounds for the GMRES algorithm applied to solve the system. Two variants of the preconditioner are considered: an ideal one based on exact solves for the velocity submatrix, and a more practical one based on block triangular approximations of the velocity submatrix.