A parallel block multi-level preconditioner for the 3D incompressible Navier–Stokes equations

Elman, Howard C.; Howle, Victoria E.; Shadid, John N.; Tuminaro, Raymond S.

doi:10.1016/s0021-9991(03)00121-9

Cited by 49 publications

(40 citation statements)

References 43 publications

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“…The Meros package in Trilinos is designed to provide scalable preconditioners for the incompressible Navier-Stokes equations and similarly structured problems [Elman et al 2003]. It is based on and extends the work of Kay, Loghin and Wathen [Kay et al 2002] and Silvester, Elman, Kay and Wathen [Silvester et al 2001].…”

Section: An Illustration Of Trilinos Interoperabilitymentioning

confidence: 99%

“…Since these tasks are too expensive, we instead use approximations to S −1 and F −1 . A variety of approximations to S −1 and F −1 have been developed [Elman et al 2003]. In general, the strength of this preconditioning approach is that wellestablished preconditioning methods can be applied on the subblock operators, in turn building up a preconditioner for fully-coupled problem.…”

Section: An Illustration Of Trilinos Interoperabilitymentioning

confidence: 99%

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An overview of the Trilinos project

Heroux

Bartlett

Howle

et al. 2005

ACM Trans. Math. Softw.

959

671

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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 object-oriented framework for the solution of large-scale, complex multi-physics 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 two-level 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 look-and-feel, including configuration, documentation, licensing, and bug-tracking.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.

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Section: An Illustration Of Trilinos Interoperabilitymentioning

confidence: 99%

Section: An Illustration Of Trilinos Interoperabilitymentioning

confidence: 99%

An overview of the Trilinos project

Heroux

Bartlett

Howle

et al. 2005

ACM Trans. Math. Softw.

959

671

View full text Add to dashboard Cite

show abstract

“…A disadvantage with this approach is that the constraint preconditioner must be applied exactly if subsequent iterates are to lie in the null space. This limits the ability to perform approximate solves with the preconditioner, as is often required when the matrix A has a PDE-like structure that also must be handled using an iterative solver (see, e.g., Saad [41], Notay [37], Simoncini and Szyld [43], and Elman et al [14]). In section 3 we consider preconditioners that do not require the assumption that b 2 = 0, and hence do not require an accurate solve with the preconditioner.…”

Section: Properties Of the Kkt Systemmentioning

confidence: 99%

Iterative Solution of Augmented Systems Arising in Interior Methods

2007

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Abstract. Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization. Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton's method for a point satisfying the perturbed optimality conditions. These equations involve both the primal and dual variables and become increasingly ill-conditioned as the optimization proceeds. In this context, an iterative linear solver must not only handle the ill-conditioning but also detect the occurrence of KKT matrices with the wrong matrix inertia. A one-parameter family of equivalent linear equations is formulated that includes the KKT system as a special case. The discussion focuses on a particular system from this family, known as the "doubly augmented system," that is positive definite with respect to both the primal and dual variables. This property means that a standard preconditioned conjugate-gradient method involving both primal and dual variables will either terminate successfully or detect if the KKT matrix has the wrong inertia. Constraint preconditioning is a well-known technique for preconditioning the conjugate-gradient method on augmented systems. A family of constraint preconditioners is proposed that provably eliminates the inherent ill-conditioning in the augmented system. A considerable benefit of combining constraint preconditioning with the doubly augmented system is that the preconditioner need not be applied exactly. Two particular "active-set" constraint preconditioners are formulated that involve only a subset of the rows of the augmented system and thereby may be applied with considerably less work. Finally, some numerical experiments illustrate the numerical performance of the proposed preconditioners and highlight some theoretical properties of the preconditioned matrices.

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“…The well-posedness of the boundary conditions (11) and (12) is analyzed in [19], [25], and [34]. The pressure is well defined by the last conditions in (11) and (12) and their discretization is easily incorporated in our iterative method in the next section. Furthermore, the system matrix of the discretized equations is non-singular and the equations can be solved to machine precision.…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…An overview of iterative methods and preconditioners for (17) is found in [10] and the convergence of a particular preconditioning operator is analyzed in [12] and applied in [11]. The multigrid method is applied to the solution of the incompressible Navier-Stokes equations in [2], [15], [31], [36].…”

Section: Comparison With Other Approachesmentioning

confidence: 99%

High order accurate solution of the incompressible Navier–Stokes equations

Brüger

Gustafsson

Lötstedt

et al. 2005

Journal of Computational Physics

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High order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation. With this goal in mind, the incompressible Navier-Stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid. The equations are integrated in time by a second order semi-implicit method. Stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions. In every time step, a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning. The convergence properties of the iterative method are analyzed. The order of accuracy of the method is demonstrated in numerical experiments. The method is used to compute the flow in a channel, the driven cavity and a constricted channel.

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A parallel block multi-level preconditioner for the 3D incompressible Navier–Stokes equations

Cited by 49 publications

References 43 publications

An overview of the Trilinos project

An overview of the Trilinos project

Iterative Solution of Augmented Systems Arising in Interior Methods

High order accurate solution of the incompressible Navier–Stokes equations

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