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 Mat. Sb., 1994, Volume 185, Number 9, Pages 109–138 (Mi msb927)

On rapidly convergent iterative methods with complete boundary-condition splitting for a multidimensional singularly perturbed system of Stokes type

B. V. Pal'tsev

Dorodnitsyn Computing Centre of the Russian Academy of Sciences

Abstract: This paper is an investigation of a group of iterative methods with complete boundary-condition splitting for solving the first boundary value problem for a system of Stokes type with a small parameter $\varepsilon>0$:
\begin{gather*} -\varepsilon ^2\Delta{\mathbf u}+{\mathbf u}+\operatorname{grad}p={\mathbf f}, \qquad \operatorname{div}{\mathbf u}=0\quad \text {in $\Omega$},
{\mathbf u}|_\Gamma ={\mathbf g}, \qquad \int _\Gamma ({\mathbf g},{\mathbf n}) ds=0, \end{gather*}
where $\mathbf{u}=(u^1(x),…,u^n(x))$ is the velocity vector, $p = p(x)$ is the pressure, $\mathbf{f}=(f^1(x),…,f^n(x))$ is the field of external forces, and $\mathbf{g}=(g^1(x),…,g^n(x))$ is a given value of the velocity vector on the boundary $\Gamma$ of a domain $\Omega$ in the $n$-dimensional Euclidean space $\mathbb{R}^n$.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 83:1, 93–118

Bibliographic databases:

UDC: 517.946+532.516.5
MSC: Primary 35A35, 35Q30, 35B25, 35A40; Secondary 65N12, 76D07, 76M25

Citation: B. V. Pal'tsev, “On rapidly convergent iterative methods with complete boundary-condition splitting for a multidimensional singularly perturbed system of Stokes type”, Mat. Sb., 185:9 (1994), 109–138; Russian Acad. Sci. Sb. Math., 83:1 (1995), 93–118

Citation in format AMSBIB
\Bibitem{Pal94} \by B.~V.~Pal'tsev \paper On rapidly convergent iterative methods with complete boundary-condition splitting for a~multidimensional singularly perturbed system of Stokes type \jour Mat. Sb. \yr 1994 \vol 185 \issue 9 \pages 109--138 \mathnet{http://mi.mathnet.ru/msb927} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1305758} \zmath{https://zbmath.org/?q=an:0849.76011} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 83 \issue 1 \pages 93--118 \crossref{https://doi.org/10.1070/SM1995v083n01ABEH003582} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995TQ10000005} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. V. Pal'tsev, “The conditions for the convergence of iterative methods with complete splitting of the boundary conditions for the Stokes system in a sphere and a spherical layer”, Comput. Math. Math. Phys., 35:6 (1995), 745–767
2. Mamedova I. Serebryakov V., “Parallel Programming of Boundary-Valued Problems for the Poisson and Helmholtz Equations by a Multigrid Algorithm”, Program. Comput. Softw., 21:5 (1995), 225–237
3. B. V. Pal'tsev, I. I. Chechel', “Algorithms based on bilinear finite elements for iterative methods with split boundary conditions for a Stokes-type system in a strip under the periodicity condition”, Comput. Math. Math. Phys., 37:7 (1997), 775–791
4. B. V. Pal'tsev, I. I. Chechel', “On some methods for enhancing the convergence speed for the higher harmonics of bilinear finite element implementations of iterative methods with boundary-condition splitting for a Stokes-type system”, Comput. Math. Math. Phys., 38:6 (1998), 916–929
5. B. V. Pal'tsev, I. I. Chechel', “Real properties of bilinear finite element implementations of methods with the splitting of boundary conditions for a Stokes-type system”, Comput. Math. Math. Phys., 38:2 (1998), 238–251
6. B. V. Pal'tsev, “On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions”, Math. Notes, 65:5 (1999), 571–581
7. B. V. Pal'tsev, I. I. Chechel', “Bilinear finite element implementations of iterative methods with incomplete splitting of boundary conditions for a Stokes-type system on a rectangle”, Comput. Math. Math. Phys., 39:11 (1999), 1755–1780
8. N. A. Meller, B. V. Pal'tsev, E. G. Khlyupina, “On some finite element implementations of iterative methods with splitting of boundary conditions for Stokes and Stokes-type systems in a spherical layer: Axially symmetric case”, Comput. Math. Math. Phys., 39:1 (1999), 92–117
9. A. S. Lozinskii, “On the acceleration of finite-element implementations of iterative processes with splitting of boundary conditions for a Stokes-type system”, Comput. Math. Math. Phys., 40:9 (2000), 1284–1307
10. V. O. Belash, B. V. Pal'tsev, “On the spectral and approximating properties of cubic finite-element approximations of the Laplace and first-derivative operators: The periodic case”, Comput. Math. Math. Phys., 40:5 (2000), 718–738
11. B. V. Pal'tsev, I. I. Chechel', “Exact estimates of the convergence rate of iterative methods with splitting of the boundary conditions for the Stokes-type system in a layer with a periodicity condition”, Comput. Math. Math. Phys., 40:12 (2000), 1751–1764
12. Kobelkov G. Olshanskii M., “Effective Preconditioning of Uzawa Type Schemes for a Generalized Stokes Problem”, Numer. Math., 86:3 (2000), 443–470
13. Chizhonkov E. Lebedev V., “On Acceleration of the Convergence of One Iterative Method”, Russ. J. Numer. Anal. Math. Model, 15:5 (2000), 383–395
14. A. S. Lozinskii, “Finite-element realization of iterative processes with splitting of boundary conditions for a Stokes-type system in nonconcentric annuli”, Comput. Math. Math. Phys., 41:8 (2001), 1145–1157
15. V. O. Belash, B. V. Pal'tsev, “Bicubic finite-element implementations of methods with splitting of boundary conditions for a Stokes-type system in a strip under the periodicity condition”, Comput. Math. Math. Phys., 42:2 (2002), 188–210
16. Belash V. Pal'tsev B. Chechel I., “On Convergence Rate of Some Iterative Methods for Bilinear and Bicubic Finite Element Schemes for the Dissipative Helmholtz Equation with Large Values of a Singular Parameter”, Russ. J. Numer. Anal. Math. Model, 17:6 (2002), 485–520
17. B. V. Pal'tsev, I. I. Chechel', “Increasing the rate of convergence of bilinear finite-element realizations of iterative methods by splitting boundary conditions for Stokes-type systems for large values of a singular parameter”, Comput. Math. Math. Phys., 44:11 (2004), 1949–1967
18. Pal'tsev B. Chechel I., “Finite-Element Linear Second-Order Accurate (Up to the Poles) Approximations of Laplace–Beltrami, Gradient, and Divergence Operators on a Sphere in R-3 in the Axisymmetric Case”, Dokl. Math., 69:2 (2004), 200–207
19. 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”, Comput. Math. Math. Phys., 45:5 (2005), 816–857
20. 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
21. Chizhonkov E. Kargin A., “On Solution of the Stokes Problem by the Iteration of Boundary Conditions”, Russ. J. Numer. Anal. Math. Model, 21:1 (2006), 21–38
22. Pal'tsev B.V. Stavtsev A.V. Chechel I.I., “Improved Bicubic Finite-Element Approximation of the Neumann Problem for Poisson's Equation”, Dokl. Math., 77:2 (2008), 258–264
23. M. K. Kerimov, “Boris Vasil'evich Pal'tsev (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 50:7 (2010), 1113–1119
24. 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
25. Solov'ev M.B., “On Numerical Implementations of a New Iterative Method with Boundary Condition Splitting for the Nonstationary Stokes Problem”, Dokl. Math., 81:3 (2010), 471–475
26. 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
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