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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 5, Pages 846–889 (Mi zvmmf656)

Second-order accurate (up to the axis of symmetry) finite-element implementations of iterative methods with splitting of boundary conditions for Stokes and stokes-type systems in a spherical layer

B. V. Pal'tsev, I. I. Chechel'

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

Abstract: Previously, numerical implementations of methods with splitting of boundary conditions for solving the first boundary value problem for Stokes and Stokes-type systems in a spherical layer with axial symmetry were developed on the basis of bilinear finite elements in a spherical coordinate system. These finite-element implementations are second-order accurate outside the neighborhood of the axis of symmetry, but their accuracy reduces near the axis of symmetry (down to the first order for pressure). Recently, new linear-type second-order accurate (up to the poles) finite-element approximations of the Laplace–Beltrami operators and the angular components of the gradient and divergence operators on a sphere in $\mathbb R^3$ in the axisymmetric case, as well as corresponding finite-element spaces, have been found by the authors. These finite-element approximations and spaces are used here to modify the above finite-element implementations of methods with splitting of boundary conditions for Stokes and Stokes-type systems. The finite-element schemes arising at iterations are written using one-dimensional tridiagonal operators with respect to angular and radial variables, which makes it possible to accelerate the computations nearly twofold. Numerical experiments reveal that the modified finite-element implementations of methods are second-order accurate with respect to the mesh size in the max-norm over the entire spherical layer. The new numerical method for the Stokes system is highly accurate with respect to both velocity and pressure. At the same time, in actual cases arising in implicit time discretizations of an initial-boundary value problem for the nonstationary Stokes system, when the singular parameter is large and the time step $\tau$ is small, the numerical methods constructed for Stokes-type systems become highly inaccurate with respect to pressure, while preserving sufficient accuracy of velocity. It is shown that sufficiently high accuracy of both velocity and pressure can be achieved under the condition $\tau\sim h$, where $h$ is the characteristic mesh size of the spatial grid. A numerical experiment is described that shows how the accuracy of numerical solutions can be considerably improved for such Stokes-type systems occurring in reality.

Key words: Stokes and Stokes-type systems, iterative methods with splitting of boundary conditions, second-order accurate (up to the axis of symmetry) finite-element implementations.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:5, 816–857

Bibliographic databases:
UDC: 519.634

Citation: B. V. Pal'tsev, I. I. Chechel', “Second-order accurate (up to the axis of symmetry) finite-element implementations of iterative methods with splitting of boundary conditions for Stokes and stokes-type systems in a spherical layer”, Zh. Vychisl. Mat. Mat. Fiz., 45:5 (2005), 846–889; Comput. Math. Math. Phys., 45:5 (2005), 816–857

Citation in format AMSBIB
\Bibitem{PalChe05} \by B.~V.~Pal'tsev, I.~I.~Chechel' \paper Second-order accurate (up to the axis of symmetry) finite-element implementations of iterative methods with splitting of boundary conditions for Stokes and stokes-type systems in a~spherical layer \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 5 \pages 846--889 \mathnet{http://mi.mathnet.ru/zvmmf656} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2190079} \zmath{https://zbmath.org/?q=an:1095.35029} \elib{https://elibrary.ru/item.asp?id=13482200} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 5 \pages 816--857 

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Erratum
• Correction
B. V. Pal'tsev, I. I. Chechel'
Zh. Vychisl. Mat. Mat. Fiz., 2005, 45:9, 1728

This publication is cited in the following articles:
1. B. V. Pal'tsev, I. I. Chechel', “Second-order accurate method with splitting of boundary conditions for solving the stationary axially symmetric Navier–Stokes problem in spherical gaps”, Comput. Math. Math. Phys., 45:12 (2005), 2148–2165
2. 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”, Comput. Math. Math. Phys., 46:5 (2006), 820–847
3. 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
4. Algazin S.D., “Numerical study of Navier–Stokes equations”, J. Appl. Mech. Tech. Phys., 48:5 (2007), 656–663
5. M. K. Kerimov, “Boris Vasil'evich Pal'tsev (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 50:7 (2010), 1113–1119
6. M. B. Soloviev, “On numerical implementations of a new iterative method with boundary condition splitting for solving the nonstationary stokes problem in a strip with periodicity condition”, Comput. Math. Math. Phys., 50:10 (2010), 1682–1701
7. M. B. Soloviev, “Numerical implementations of an iterative method with boundary condition splitting as applied to the nonstationary stokes problem in the gap between coaxial cylinders”, Comput. Math. Math. Phys., 50:11 (2010), 1895–1913
8. Pal'tsev B.V., “On an Iterative Method with Boundary Condition Splitting as Applied to the Dirichlet Initial-Boundary Value Problem for the Stokes System”, Doklady Mathematics, 81:3 (2010), 452–457
9. Solov'ev M.B., “On Numerical Implementations of a New Iterative Method with Boundary Condition Splitting for the Nonstationary Stokes Problem”, Doklady Mathematics, 81:3 (2010), 471–475
10. 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
11. 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
12. 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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