A parallel solver for PDE systems and application to the incompressible Navier–Stokes equations

Vainikko, Eero; Graham, Ivan G.

doi:10.1016/j.apnum.2003.11.015

Cited by 6 publications

(4 citation statements)

References 18 publications

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“…We note that while there is a rich literature on domain decomposition methods for symmetric saddle point problems and in particular for mixed discretizations of the Stokes equations, there seems to be very few papers concerned with coupled domain decomposition methods for nonsymmetric saddle point systems and in particular for the Oseen equations. One such paper is Vainikko and Graham (2004), where domain decomposition methods are implemented and experimentally compared with (and found to be superior to) block triangular preconditioners for the Oseen problem. Theoretical understanding, however, is still largely lacking.…”

Section: Domain Decomposition Methodsmentioning

confidence: 99%

“…Numerical experiments for simple model problems indicate that the performance of this preconditioner depends only mildly on the mesh size and viscosity coefficient; see Elman (1999). However, it was found to perform poorly (occasionally failing to converge) on some difficult problems; see Vainikko and Graham (2004). As noted in Elman, Howle, Shadid, Shuttleworth and Tuminaro (2005a), for finite element problems the performance of this preconditioner can be greatly improved by the use of appropriate scalings.…”

Section: Approximating a And Smentioning

confidence: 97%

See 1 more Smart Citation

Numerical solution of saddle point problems

Benzi¹,

Golub²,

Liesen³

2005

Acta Numerica

1,974

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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Section: Domain Decomposition Methodsmentioning

confidence: 99%

Section: Approximating a And Smentioning

confidence: 97%

Numerical solution of saddle point problems

Benzi¹,

Golub²,

Liesen³

2005

Acta Numerica

1,974

1,774

View full text Add to dashboard Cite

show abstract

“…In [16] and [17] Elman observed some h-dependence in the convergence rate of the GMRES method using the BFBt preconditioner for the finite difference (MAC) and finite element Q 2 -Q 1 discretizations. Some h-dependence was also observed by Vainikko and Graham in [39] with Q 2 -P −1 elements and by Hemmingsson and Wathen in [26] with a finite difference method. In all these papers the variant of the preconditioner with identity matrices instead ofM u in (2.9) was used.…”

Section: Bfbtmentioning

confidence: 64%

Pressure Schur Complement Preconditioners for the Discrete Oseen Problem

Olshanskii¹,

Vassilevski²

2007

SIAM J. Sci. Comput.

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Abstract. 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 two-dimensional and three-dimensional 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 mass-matrix 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.

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“…We anticipate that this strategy will succeed for eigenvalue problems where the desired eigenvalues are not clustered. In addition to the example discussed here, this strategy has been used successfully to compute eigenvalues for flow in an expanding pipe [24].…”

Section: Computation Of Eigenvalues Parpack and Inexact Inverse Iteramentioning

confidence: 97%

Parallel iterative methods for Navier–Stokes equations and application to eigenvalue computation

Graham

Spence

Vainikko

2003

Concurrency and Computation

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SUMMARYWe describe the construction of parallel iterative solvers for finite-element approximations of the Navier-Stokes equations on unstructured grids using domain decomposition methods. The iterative method used is FGMRES, preconditioned by a parallel adaptation of a block preconditioner recently proposed by Kay et al. The parallelization is achieved by adapting the technology of our domain decomposition solver DOUG (previously used for scalar problems) to block-systems. The iterative solver is applied to shifted linear systems that arise in eigenvalue calculations. To illustrate the performance of the solver, we compare several strategies both theoretically and practically for the calculation of the eigenvalues of large sparse non-symmetric matrices arising in the assessment of the stability of flow past a cylinder.

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A parallel solver for PDE systems and application to the incompressible Navier–Stokes equations

Cited by 6 publications

References 18 publications

Numerical solution of saddle point problems

Numerical solution of saddle point problems

Pressure Schur Complement Preconditioners for the Discrete Oseen Problem

Parallel iterative methods for Navier–Stokes equations and application to eigenvalue computation

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