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

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

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
\Bibitem{PalChe06} \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 \mathnet{http://mi.mathnet.ru/zvmmf471} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2286281} \elib{https://elibrary.ru/item.asp?id=9199432} \transl \jour Comput. Math. Math. Phys. \yr 2006 \vol 46 \issue 5 \pages 820--847 \crossref{https://doi.org/10.1134/S0965542506050083} \elib{https://elibrary.ru/item.asp?id=13531847} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746032167} 

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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
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
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
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
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
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