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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 180 (30 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.
An overview of the trilinos project
 ACM Transactions on Mathematical Software
"... The Trilinos Project is an effort to facilitate the design, development, integration and ongoing support of mathematical software libraries within an objectoriented framework for the solution of largescale, complex multiphysics engineering and scientific problems. Trilinos addresses two fundament ..."
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Cited by 72 (9 self)
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The Trilinos Project is an effort to facilitate the design, development, integration and ongoing support of mathematical software libraries within an objectoriented framework for the solution of largescale, complex multiphysics engineering and scientific problems. Trilinos addresses two fundamental issues of developing software for these problems: (i) Providing a streamlined process and set of tools for development of new algorithmic implementations and (ii) promoting interoperability of independently developed software. Trilinos uses a twolevel software structure designed around collections of packages. A Trilinos package is an integral unit usually developed by a small team of experts in a particular algorithms area such as algebraic preconditioners, nonlinear solvers, etc. Packages exist underneath the Trilinos top level, which provides a common lookandfeel, including configuration, documentation, licensing, and bugtracking. Here we present the overall Trilinos design, describing our use of abstract interfaces and default concrete implementations. We discuss the services that Trilinos provides to a prospective package and how these services are used by various packages. We also illustrate how packages can be combined to rapidly develop new algorithms. Finally, we discuss how Trilinos facilitates highquality software engineering practices that are increasingly required from simulation software. Sandia is a multiprogram laboratory operated by Sandia Corporation, a LockheedMartin Company, for the United States Department of Energy under Contract DEAC0494AL85000. Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. Sci. Comput
, 2006
"... Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a no ..."
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Cited by 52 (23 self)
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Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schöberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2P0 and isoP2P1 finite elements in support of our conclusions. We also show results of a comparison with two stateoftheart preconditioners, showing the competitiveness of our approach. Key words. Navier–Stokes equations, finite element, iterative methods, multigrid, preconditioning AMS subject classifications. 65F10, 65N22, 65F50 DOI. 10.1137/050646421 1. Introduction. We consider the numerical solution of the steady Navier– Stokes equations governing the flow of a Newtonian, incompressible viscous fluid. Let Ω ⊂ R d (d =2,3) be a bounded, connected domain with a piecewise smooth
Block Preconditioners Based on Approximate Commutators
 SIAM J. SCI. COMPUT
, 2006
"... This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible NavierStokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [16]. Numerous t ..."
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Cited by 24 (9 self)
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This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible NavierStokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [16]. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall e#cacy of this methodology. A drawback, however, is that it requires the construction of a convectiondiffusion operator (denoted Fp ) projected onto the discrete pressure space. This means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues, something often held only by the developers of the model. As an alternative, we consider automatic ways of computing Fp based on purely algebraic considerations. The new methods are closely related to the "BFBt preconditioner" of Elman [6]. We use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convectiondiffusion operators, together with methods for computing sparse approximate inverses, to generate the required matrix Fp automatically. We demonstrate that with this strategy, the favorable convergence properties of the preconditioning methodology are retained.
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 18 (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
Least squares preconditioners for stabilized discretizations of the NavierStokes equations
 University of Maryland, College Park
, 2006
"... Abstract. This paper introduces two stabilization schemes for the least squares commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth, and Tuminaro [SIAM J. Sci. Comput., 27 (2006), pp. 1651–1668] for the incompressible Navier–Stokes equations. This preconditioning methodol ..."
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Cited by 15 (5 self)
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Abstract. This paper introduces two stabilization schemes for the least squares commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth, and Tuminaro [SIAM J. Sci. Comput., 27 (2006), pp. 1651–1668] for the incompressible Navier–Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier–Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to divstable discretizations of the Navier–Stokes equations. This paper shows how to extend the same methodology to stabilized loworder mixed finite element approximation methods. Key words. preconditioning, Navier–Stokes, iterative algorithms
Moving mesh finite element methods for the incompressible Navier–Stokes equations
 SIAM J. Sci. Comput
, 2005
"... Abstract. This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergencefree for the velocity field at each ti ..."
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Cited by 15 (5 self)
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Abstract. This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergencefree for the velocity field at each time level. The proposed numerical scheme extends a recent moving grid method based on harmonic mapping [R. Li, T. Tang, and P. W. Zhang, J. Comput. Phys., 170 (2001), pp. 562–588], which decouples the PDE solver and the meshmoving algorithm. This approach requires interpolating the solution on the newly generated mesh. Designing a divergencefreepreserving interpolation algorithm is the first goal of this work. Selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations, which is the second goal of this study. The performance of the moving mesh scheme is tested on the standard periodic double shear layer problem. No spurious vorticity patterns appear when even fairly coarse grids are used.
Preconditioning techniques for Newton’s method for the incompressible Navier–Stokes equations
, 2003
"... Newton’s method for the incompressible Navier–Stokes equations gives rise to large sparse nonsymmetric indefinite matrices with a socalled saddlepoint structure for which Schur complement preconditioners have proven to be effective when coupled with iterative methods of Krylov type. In this work ..."
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Cited by 14 (4 self)
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Newton’s method for the incompressible Navier–Stokes equations gives rise to large sparse nonsymmetric indefinite matrices with a socalled saddlepoint structure for which Schur complement preconditioners have proven to be effective when coupled with iterative methods of Krylov type. In this work we investigate the performance of two preconditioning techniques introduced originally for the Picard method for which both proved significantly superior to other approaches such as the Uzawa method. The first is a block preconditioner which is based on the algebraic structure of the system matrix. The other approach uses also a block preconditioner which is derived by considering the underlying partial differential operator matrix. Analysis and numerical comparison of the methods are presented.
A parallel block multilevel preconditioner for the 3d incompressible navierstokes equations
 J. Comput. Phys
, 2003
"... Abstract. The development of robust and efficient algorithms for both steadystate simulations and fullyimplicit time integration of the Navier–Stokes equations is an active research topic. To be effective, the linear subproblems generated by these methods require solution techniques that exhibit r ..."
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Cited by 13 (2 self)
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Abstract. The development of robust and efficient algorithms for both steadystate simulations and fullyimplicit time integration of the Navier–Stokes equations is an active research topic. To be effective, the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence. In particular, they should be insensitive to parameters in the problem such as mesh size, time step, and Reynolds number. In this context, we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an Oseen nonlinear iteration for the primitive variable formulation of the system. The key to this preconditioner is the approximation of a certain Schur complement operator by a technique first proposed by Kay, Loghin, and Wathen [26] and Silvester, Elman, Kay, and Wathen [46]. The resulting operator entails subsidiary computations (solutions of pressure Poisson and convection–diffusion subproblems) that are similar to those required for decoupled solution methods; however, in this case these solutions are applied as preconditioners to the coupled Oseen system. One important aspect of this approach is that the convection–diffusion and Poisson subproblems are significantly easier to solve than the entire coupled system, and a solver can be built using tools developed for the subproblems. In this paper, we apply smoothed aggregation algebraic multigrid to both subproblems. Previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial, twodimensional, steadystate computations employing geometric multigrid methods; we focus on extending these methods to largescale, parallel, threedimensional, transient and steadystate simulations employing algebraic multigrid (AMG) methods. Our results display nearly optimal convergence rates for steadystate solutions as well as for transient solutions over a wide range of CFL numbers on the twodimensional and threedimensional liddriven cavity problem. 1. Introduction. Recently
Algorithm 866: IFISS, a Matlab toolbox for modellingincompressible flow
 2007.CODEN ACMSCU. ISSN
, 2007
"... IFISS is a graphical Matlab package for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions. The package can also be used as a computational laboratory f ..."
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Cited by 12 (2 self)
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IFISS is a graphical Matlab package for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions. The package can also be used as a computational laboratory for experimenting with stateoftheart preconditioned iterative solvers for the discrete linear equation systems that arise in incompressible flow modelling. A unique feature of the package is its comprehensive nature; for each problem addressed, it enables the study of both discretization and iterative solution algorithms as well as the interaction between the two and the resulting effect on overall efficiency. Categories and Subject Descriptors: G.1.3 [Numerical Analysis]: Numerical Linear Algebra— Linear systems (direct and iterative methods); G.1.8 [Numerical Analysis]: Partial Differential