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Zh. Vychisl. Mat. Mat. Fiz., 2006, Volume 46, Number 5, Pages 858–886 (Mi zvmmf471)  

This article is cited in 5 scientific papers (total in 5 papers)

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

B. V. Pal'tsev, I. I. Chechel'

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

Abstract: The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For $R/r$ exceeding about 30, where $r$ and $R$ are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large $R/r$) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for $R/r$ of up to $5\times10^3$. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from $R/r\sim30$ and is considerably more efficient for large values of $R/r$. It is also established that the convergence rates of both methods depend little on the stretching coefficient $\eta$ of circularly rectangular mesh cells in a range of $\eta$ that is well sufficient for effective use of the multigrid method for arbitrary values of $R/r$ smaller than $\sim 5\times10^3$.

Key words: stationary Stokes system, spherical gaps, iterative methods with splitting of boundary conditions, second-order accurate finite-element implementations in the axisymmetric case, convergence rates, multigrid method.

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English version:
Computational Mathematics and Mathematical Physics, 2006, 46:5, 820–847

Bibliographic databases:

UDC: 519.634
Received: 02.12.2005

Citation: B. V. Pal'tsev, I. I. Chechel', “On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers”, Zh. Vychisl. Mat. Mat. Fiz., 46:5 (2006), 858–886; Comput. Math. Math. Phys., 46:5 (2006), 820–847

Citation in format AMSBIB
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\by B.~V.~Pal'tsev, I.~I.~Chechel'
\paper On the convergence rate and optimization of a~numerical method with splitting of boundary conditions for the stokes system in a~spherical layer in the axisymmetric case: Modification for thick layers
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2006
\vol 46
\issue 5
\pages 858--886
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2286281}
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\jour Comput. Math. Math. Phys.
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\vol 46
\issue 5
\pages 820--847
\crossref{https://doi.org/10.1134/S0965542506050083}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. B. V. Pal'tsev, A. V. Stavtsev, I. I. Chechel', “Numerical study of the basic stationary spherical couette flows at low Reynolds numbers”, Comput. Math. Math. Phys., 47:4 (2007), 664–686  mathnet  crossref  mathscinet  zmath
    2. M. K. Kerimov, “Boris Vasil'evich Pal'tsev (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 50:7 (2010), 1113–1119  mathnet  crossref  mathscinet  adsnasa  isi  elib
    3. B. V. Pal'tsev, M. B. Soloviev, I. I. Chechel', “On the development of iterative methods with boundary condition splitting for solving boundary and initial-boundary value problems for the linearized and nonlinear Navier–Stokes equations”, Comput. Math. Math. Phys., 51:1 (2011), 68–87  mathnet  crossref  mathscinet  isi  elib
    4. B. V. Pal'tsev, M. B. Solov'ev, I. I. Chechel', “Numerical study of spherical Couette flows for certain zenith-angle-dependent rotations of boundary spheres at low Reynolds numbers”, Comput. Math. Math. Phys., 56:6 (2012), 940–975  mathnet  crossref  mathscinet  isi  elib  elib
    5. B. V. Pal'tsev, M. B. Solov'ev, I. I. Chechel', “On the structure of steady axisymmetric Navier-Stokes flows with a stream function having multiple local extrema in its definite-sign domains”, Comput. Math. Math. Phys., 53:11 (2013), 1696–1719  mathnet  crossref  crossref  mathscinet  isi  elib  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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