A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations

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

doi:10.1016/j.jcp.2007.09.026

Cited by 141 publications

(108 citation statements)

References 30 publications

Supporting

Mentioning

101

Contrasting

Unclassified

Order By: Relevance

“…Note that the SIMPLE preconditioner (e.g., [35]) is a particular case of the block approximate factorization preconditioner as defined above; see also [13] for further related variants. This framework also describes some multigrid smoothers based on "distributive relaxation"; see [4,Section 11.1] for a discussion and further references.…”

Section: 2)mentioning

confidence: 99%

A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these latter based on symmetric block Gauss-Seidel type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow to localize accurately both real and non real eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory.

show abstract

Section: 2)mentioning

confidence: 99%

A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…It goes without saying that the mentioned difficulties do not concern the use of AMG techniques within the framework of block preconditioning methods. These (see, e.g., [2,11,12,29]) require approximations of the inverse of certain matrix blocks, and are in fact most effective when multigrid is used for this purpose. Here, one may apply AMG without particular difficulty because these matrix blocks are related to scalar Poissonlike problems.…”

Section: Introductionmentioning

confidence: 99%

A new algebraic multigrid approach for Stokes problems

Notay

2015

Numer. Math.

View full text Add to dashboard Cite

Standard discretizations of Stokes problems lead to linear systems of equations in saddle point form, making difficult the application of algebraic multigrid methods. In this paper, a new approach is proposed. It consists in first transforming the system by pre-and post-multiplication with simple, algebraic, sparse block triangular matrices. This is a form of pre-conditioning in the literal sense, designed to make sure that the transformed matrix is well adapted to multigrid. In particular, after transformation, all the diagonal blocks are symmetric and positive definite, and correspond to, or resemble, a discrete Laplace operator. Then, to each of these diagonal blocks is associated a prolongation that works well for it, using any relevant algebraic or geometric multigrid method. Next, a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a Galerkin coarse grid matrix. For this approach combined with damped Jacobi-smoothing, a uniform two-grid convergence bound is derived for the global system under the assumption that the two-grid schemes for the different diagonal blocks are themselves uniformly convergent. This result is illustrated by a few examples, showing further that time-dependent problems and variable viscosity can be handled in a natural way, without requiring parameter adjustment. A numerical comparison also shows that the new approach can be more effective than state-of-the-art block preconditioning techniques.

show abstract

“…In [11] a number of segregated preconditioning techniques (PCD, SIMPLE and LSC) are compared using Trilinos implementations. A finite element discretization with full Newton linearization of the driven cavity problem is used, both in 2D and 3D.…”

Section: ) Solve Phasementioning

confidence: 99%

Design of a Parallel Hybrid Direct/Iterative Solver for CFD Problems

Thies

Wubs

2011

2011 IEEE Seventh International Conference on eScience

View full text Add to dashboard Cite

Abstract-We discuss the parallel implementation of a hybrid direct/iterative solver for a special class of saddle point matrices arising from the discretization of the steady Navier-Stokes equations on an Arakawa C-grid, the F -matrices.The two-level method described here has the following properties: (i) it is very robust, even hat comparatively high Reynolds Numbers; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively. The implementation focusses on generality, modularity, code reuse and recursiveness. The solver is implemented using building blocks of the Trilinos libraries. We show its performance on a parallel computer for the NavierStokes equations.

show abstract

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations

Cited by 141 publications

References 30 publications

A New Analysis of Block Preconditioners for Saddle Point Problems

A New Analysis of Block Preconditioners for Saddle Point Problems

A new algebraic multigrid approach for Stokes problems

Design of a Parallel Hybrid Direct/Iterative Solver for CFD Problems

Contact Info

Product

Resources

About