Block preconditioners for finite element discretization of incompressible flow with thermal convection

Howle, Victoria E.; Kirby, Robert C.

doi:10.1002/nla.1814

Cited by 22 publications

(30 citation statements)

References 24 publications

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“…In order to compare the timing for the preconditioners P a and P b in the Newton scheme, we reproduce and report the results of Howle and Kirby for P a in Tables and with tight and loose tolerances, respectively. A comparison of Table with Table and Table with Table for R a =2×10 4 shows that P b is about four times faster than P a .…”

Section: Computational Investigationsmentioning

confidence: 99%

“…Here, we introduce the structure of these two preconditioners, more details on which can be found in Section 2. See also previous studies for their definition. The original Rayleigh–Bénard systems coming from either Picard (

scriptP

subscript) or Newton (

scriptN

subscript) linearization of Equations have the two forms given by

either A_{P} = ([], \begin{matrix} array N_{scriptP} & array Q \\ array 0 & array K \end{matrix}) or A_{N} = ([], \begin{matrix} array N_{scriptN} & array Q \\ array G & array K \end{matrix}) .

We consider any formulation of the problem (for instance, any choice of boundary conditions) such that

A_{P}

and

A_{N}

are nonsingular.…”

Section: Introductionmentioning

confidence: 99%

“…The computational experiments reported in Howle and Kirby, which concern Newton linearizations of the Rayleigh–Bénard problem, indicate that all the runs with the preconditioner P a require shorter time than those with P b . However a mistake has been found in the numerical implementation of the preconditioner P b , thus we reexamined the performance of the two preconditioners and observed that the timings for P b are actually smaller than the ones obtained with P a , regardless of the linearization algorithm used.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, we observed that the performance of P b is significantly better when applied to Newton's linearization. In order to rectify the previous results in Howle and Kirby, we report in this work the new timings for both preconditioners. We then investigate in detail the performance of P b in terms of number of iterations and computational time with various Rayleigh numbers R a , both for the Picard and for the Newton schemes.…”

Section: Introductionmentioning

confidence: 99%

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Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

Aulisa

Bornia

et al. 2017

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Summary In this paper, we compare two block triangular preconditioners for different linearizations of the Rayleigh–Bénard convection problem discretized with finite element methods. The two preconditioners differ in the nested or nonnested use of a certain approximation of the Schur complement associated to the Navier–Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time. This investigation reveals some inconsistencies in the literature that are hereby discussed. We observe that the nonnested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard–Newton scheme. Numerical results of two‐ and three‐dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.

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Section: Computational Investigationsmentioning

confidence: 99%

scriptP

subscript) or Newton (

scriptN

subscript) linearization of Equations have the two forms given by

either A_{P} = ([], \begin{matrix} array N_{scriptP} & array Q \\ array 0 & array K \end{matrix}) or A_{N} = ([], \begin{matrix} array N_{scriptN} & array Q \\ array G & array K \end{matrix}) .

We consider any formulation of the problem (for instance, any choice of boundary conditions) such that

A_{P}

and

A_{N}

are nonsingular.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

Aulisa

Bornia

et al. 2017

Numerical Linear Algebra App

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“…In , Howle et al derive block preconditioners for a finite element discretization of incompressible flow coupled to heat transport by the Boussinesq approximation. The technique relies on efficiently approximating the Schur complement obtained by eliminating the fluid variables to obtain an equation for temperature alone.…”

mentioning

confidence: 99%

Multigrid methods

Dendy

2012

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In [2], Yaveneh et al. consider the classic Petrov-Galerkin for black box multigrid for nonsymmetric problems with factor-of-three coarsening. With symmetric line Gauss-Seidel as a smoother, the algorithm achieves fast and reliable convergence for both first-order and second-order discretizations of the convection operator for a wide range of diffusion coefficients.In [3], Emams reports on experiences with aggregation AMG relying on Krylov acceleration as preconditioners for linear systems in general purpose fluid flow simulation. In benchmarks that reflect the requirements for industrial situations, the performance of recently published algorithms for semi-definite problems is shown to be very attractive except for proposed variants that occasionally fail for nonsymmetric problems; modifications are suggested to lead to reliable solvers for these situations.In [4], Reps et al. analyze the Krylov convergence rate of a Helmholtz problem preconditioned with multigrid, which is applied to the Helmholtz problem formulated on a complex contour and uses polynomial smoothers at each level. For a one-dimensional model, it is shown that the Krylov

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Efficient solvers for coupled models in respiratory mechanics

Verdugo

Roth

Yoshihara

et al. 2016

Numer Methods Biomed Eng

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We present efficient preconditioners for one of the most physiologically relevant pulmonary models currently available. Our underlying motivation is to enable the efficient simulation of such a lung model on high-performance computing platforms in order to assess mechanical ventilation strategies and contributing to design more protective patient-specific ventilation treatments. The system of linear equations to be solved using the proposed preconditioners is essentially the monolithic system arising in fluid-structure interaction (FSI) extended by additional algebraic constraints. The introduction of these constraints leads to a saddle point problem that cannot be solved with usual FSI preconditioners available in the literature. The key ingredient in this work is to use the idea of the semi-implicit method for pressure-linked equations (SIMPLE) for getting rid of the saddle point structure, resulting in a standard FSI problem that can be treated with available techniques. The numerical examples show that the resulting preconditioners approach the optimal performance of multigrid methods, even though the lung model is a complex multiphysics problem. Moreover, the preconditioners are robust enough to deal with physiologically relevant simulations involving complex real-world patient-specific lung geometries. The same approach is applicable to other challenging biomedical applications where coupling between flow and tissue deformations is modeled with additional algebraic constraints. Copyright © 2016 John Wiley & Sons, Ltd.

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Block preconditioners for finite element discretization of incompressible flow with thermal convection

Cited by 22 publications

References 24 publications

Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

Multigrid methods

Efficient solvers for coupled models in respiratory mechanics

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