
This article is cited in 12 scientific papers (total in 12 papers)
Secondorder accurate (up to the axis of symmetry) finiteelement implementations of iterative methods with splitting of boundary conditions for Stokes and stokestype 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 Stokestype systems in a spherical layer with axial symmetry were developed on the basis of bilinear finite elements in a spherical coordinate system. These finiteelement implementations are secondorder 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 lineartype secondorder accurate (up to the poles) finiteelement 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 finiteelement spaces, have been found by the authors. These finiteelement approximations and spaces are used here to modify the above finiteelement implementations of methods with splitting of boundary conditions for Stokes and Stokestype systems. The finiteelement schemes arising at iterations are written using onedimensional 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 finiteelement implementations of methods are secondorder accurate with respect to the mesh size in the maxnorm 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 initialboundary 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 Stokestype 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 Stokestype systems occurring in reality.
Key words:
Stokes and Stokestype systems, iterative methods with splitting of boundary conditions, secondorder accurate (up to the axis of symmetry) finiteelement implementations.
Full text:
PDF file (5473 kB)
References:
PDF file
HTML file
English version:
Computational Mathematics and Mathematical Physics, 2005, 45:5, 816–857
Bibliographic databases:
UDC:
519.634 Received: 28.10.2004
Citation:
B. V. Pal'tsev, I. I. Chechel', “Secondorder accurate (up to the axis of symmetry) finiteelement implementations of iterative methods with splitting of boundary conditions for Stokes and stokestype 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 Secondorder accurate (up to the axis of symmetry) finiteelement implementations of iterative methods with splitting of boundary conditions for Stokes and stokestype systems in a~spherical layer
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2005
\vol 45
\issue 5
\pages 846889
\mathnet{http://mi.mathnet.ru/zvmmf656}
\mathscinet{http://www.ams.org/mathscinetgetitem?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 816857
Linking options:
http://mi.mathnet.ru/eng/zvmmf656 http://mi.mathnet.ru/eng/zvmmf/v45/i5/p846
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
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:

B. V. Pal'tsev, I. I. Chechel', “Secondorder 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

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

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

Algazin S.D., “Numerical study of Navier–Stokes equations”, J. Appl. Mech. Tech. Phys., 48:5 (2007), 656–663

M. K. Kerimov, “Boris Vasil'evich Pal'tsev (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 50:7 (2010), 1113–1119

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

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

Pal'tsev B.V., “On an Iterative Method with Boundary Condition Splitting as Applied to the Dirichlet InitialBoundary Value Problem for the Stokes System”, Doklady Mathematics, 81:3 (2010), 452–457

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

B. V. Pal'tsev, M. B. Soloviev, I. I. Chechel', “On the development of iterative methods with boundary condition splitting for solving boundary and
initialboundary value problems for the linearized and nonlinear Navier–Stokes equations”, Comput. Math. Math. Phys., 51:1 (2011), 68–87

B. V. Pal'tsev, M. B. Solov'ev, I. I. Chechel', “Numerical study of spherical Couette flows for certain zenithangledependent rotations of boundary spheres at low Reynolds numbers”, Comput. Math. Math. Phys., 56:6 (2012), 940–975

B. V. Pal'tsev, M. B. Solov'ev, I. I. Chechel', “On the structure of steady axisymmetric NavierStokes flows with a stream function having multiple local extrema in its definitesign domains”, Comput. Math. Math. Phys., 53:11 (2013), 1696–1719

Number of views: 
This page:  225  Full text:  89  References:  42  First page:  1 
