<i>A priori</i> pivoting in solving the Navier–Stokes equations

Wille, S.Ø.; Loula, Abimael F. D.

doi:10.1002/cnm.528

Cited by 5 publications

(13 citation statements)

References 4 publications

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“…To avoid this problem, it is better to use a suitable a priori reordering of unknowns. As pointed out by Wille and others [13][14][15], pivoting is not necessary when the unknowns are ordered in the sequence, so that all velocity unknowns come first and then all the pressure unknowns like in the block preconditioners. The reason being that during (incomplete) factorization the zeros at the main diagonal will vanish, provided fill-in is allowed based on the connectivity of nodal points rather than actual zeros in the matrix.…”

Section: A Saddle Point Ilu-type Preconditionermentioning

confidence: 99%

“…This is true even if we use an optimal node renumbering. The main advantage of this ordering is that no pivoting is necessary since during factorization the zeros on the main diagonal in the zero pressure block disappear, see, for example, [14]. In the remaining part of this paper, we shall refer to this reordering as p-last.…”

Section: Ordering Scheme For Direct Solversmentioning

confidence: 99%

“…In case of a different renumbering scheme like Sloan [28] or the one proposed by Wille et al [14], we define levels in the following way.…”

Section: Ordering Scheme For Direct Solversmentioning

confidence: 99%

“…The same may happen in the case of ILU factorization. Wille and coworkers [13][14][15] use a node-renumbering technique in combination with a reordering of unknowns that is common for block preconditioners. This method can be applied to both direct and ILU-preconditioned iterative methods.…”

Section: Introductionmentioning

confidence: 99%

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A comparison of preconditioners for incompressible Navier–Stokes solvers

Rehman

Vuik

Segal

2007

Numerical Methods in Fluids

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SUMMARYWe 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 so-called 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 medium-sized 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.

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Section: A Saddle Point Ilu-type Preconditionermentioning

confidence: 99%

Section: Ordering Scheme For Direct Solversmentioning

confidence: 99%

“…In case of a different renumbering scheme like Sloan [28] or the one proposed by Wille et al [14], we define levels in the following way.…”

Section: Ordering Scheme For Direct Solversmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A comparison of preconditioners for incompressible Navier–Stokes solvers

Rehman

Vuik

Segal

2007

Numerical Methods in Fluids

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“…In general algebraic preconditioners are based on ILU factorization of the coefficient matrix. In order to avoid problems with zeros on the main diagonal, either dynamic pivoting or a clever apriori reordering technique has to be applied [1][2][3][4][5][6][7][8][9]. In [9] we published an a-priori reordering technique (SILU), that converges fast for small to mid-sized grids.…”

Section: Introductionmentioning

confidence: 99%

SIMPLE‐type preconditioners for the Oseen problem

Rehman

Vuik

Segal

2008

Numerical Methods in Fluids

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In this paper, the steady incompressible Navier–Stokes equations are discretized by the finite element method. The resulting systems of equations are solved by preconditioned Krylov subspace methods. Some new preconditioning strategies, both algebraic and problem dependent are discussed. We emphasize on the approximation of the Schur complement as used in semi implicit method for pressure‐linked equations‐type preconditioners. In the usual formulation, the Schur complement matrix and updates use scaling with the diagonal of the convection–diffusion matrix. We propose a variant of the SIMPLER preconditioner. Instead of using the diagonal of the convection–diffusion matrix, we scale the Schur complement and updates with the diagonal of the velocity mass matrix. This variant is called modified SIMPLER (MSIMPLER). With the new approximation, we observe a drastic improvement in convergence for large problems. MSIMPLER shows better convergence than the well‐known least‐squares commutator preconditioner which is also based on the diagonal of the velocity mass matrix. Copyright © 2008 John Wiley & Sons, Ltd.

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Numerical solution of saddle point problems

Benzi¹,

Golub²,

Liesen³

2005

Acta Numerica

1,999

1,774

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Abstract. 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.

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A priori pivoting in solving the Navier–Stokes equations

Cited by 5 publications

References 4 publications

A comparison of preconditioners for incompressible Navier–Stokes solvers

A comparison of preconditioners for incompressible Navier–Stokes solvers

SIMPLE‐type preconditioners for the Oseen problem

Numerical solution of saddle point problems

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