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Domain decomposition method and numerical analysis of a fluid dynamics problem

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Abstract

A two-dimensional problem obtained by time discretization and linearization of a viscous flow governed by the incompressible Navier-Stokes equations is considered. The original domain is divided into subdomains such that their interface is a smooth (nonclosed, self-avoiding) curve with the ends belonging to the boundary of the domain. A nonconforming finite element method is constructed for the problem, and the convergence rate of the discrete solution of the problem to the exact one is estimated in the L 2(Ω h ) norm.

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References

  1. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer-Verlag, New York, 1991).

    Book  MATH  Google Scholar 

  2. D. Braess, W. Dahmen, and C. Wieners, “A multigrid algorithm for the mortar finite element method,” SIAM J. Numer. Anal. 37(1), 48–69 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Bernardi, Y. Maday, and A. Patera, “A new nonconforming approach to domain decomposition: The mortar element method,” Nonlinear Partial Differential Equations and Their Applications (Pitman, Paris, 1994), pp. 13–51.

    Google Scholar 

  4. B. Wohlmuth, “Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers,” SIAM J. Numer. Anal. 36(5), 1636–1658 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Flemisch, J. M. Melenk, and B. Wohlmuth, “Mortar methods with curved interfaces,” Appl. Numer. Math. 54, 339–361 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Huagn and J. Zou, “A mortar element method for elliptic problems with discontinuous coefficients,” IMA J. Numer. Anal. 22, 549–576 (2002).

    Article  MathSciNet  Google Scholar 

  7. F. Ben Belgacem, “The Mixed mortar finite element method for the incompressible Stokes problem: Convergence analysis,” SIAM J. Numer. Anal. 37(4), 1085–1100 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. V. Rukavishnikov and V. A. Rukavishnikov, “Nonconformal finite element method for the Stokes problem with a discontinuous coefficient,” Sib. Zh. Ind. Mat. 10(4), 104–117 (2007).

    MathSciNet  MATH  Google Scholar 

  9. A. V. Rukavishnikov, “Construction of a numerical method for the Stokes problem with a discontinuous viscosity coefficient,” Vychisl. Tekhnol. 14(2), 110–123 (2009).

    MATH  Google Scholar 

  10. A. V. Rukavishnikov, “Nonconformal finite element method for a fluid dynamics problem with a curved interface,” Vychisl. Mat. Mat. Fiz. 52(6), 1072–1094 (2012).

    MathSciNet  MATH  Google Scholar 

  11. M. A. Olshanskii and A. Reusken, “Analysis of a Stokes interface problem,” Numer. Math. 103, 129–149 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. E. V. Chizhonkov, “Numerical solution to a Stokes interface problem,” Comput. Math. Math. Phys. 49, 105–116 (2009).

    Article  MathSciNet  Google Scholar 

  13. K. Ohmori and N. Saito, “On the convergence of finite element solutions to the interface problem for the Stokes system,” J. Comput. Appl. Math. 198(1), 116–128 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Ohmori and N. Saito, “Flux-free finite element method with Lagrange multipliers for two-fluid flows,” J. Sci. Comput. 32(2), 147–173 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  15. W. J. Gordon and Ch. A. Hall, “Transfinite element methods: Blending functions interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Amrouche and V. Girault, “On the existence and regularity of the solution of Stokes problem in arbitrary dimension,” Proc. Jpn. Acad. 67, 171–175 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Quateroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations (Clarendon, Oxford, 1999).

    Google Scholar 

  18. V. A. Kondrat’ev and O. A. Oleinik, “Boundary value problems for the system of elasticity theory in unbounded domains: Korn’s inequalities,” Russ. Math. Surv. 43(5), 65–119 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. V. Rukavishnikov, “On the existence and uniqueness of a weak solution to the Stokes problem with a discontinuous multiplier,” in Science to Khabarovsk Krai: Proceedings of the 7th Krai Contest-Conference of Young Scientists (Khabar. Gos. Tekh. Univ., Khabarovsk, 2005), pp. 75–94.

    Google Scholar 

  20. P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1977; Mir, Moscow, 1980).

    Google Scholar 

  21. T. Richter, http://numeric/uni-hd.de/richter/SS10/fsi/fsi.pdf.

  22. P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, Boston, 1985).

    MATH  Google Scholar 

  23. J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, “Analysis of the inexact Uzawa algorithm for saddle point problems,” SIAM J. Number. Anal. 34(3), 1072–1092 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, New Jersey, 1996).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, ul. Dzerzhinskogo 54, Khabarovsk, 680000, Russia

    A. V. Rukavishnikov

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  1. A. V. Rukavishnikov
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Correspondence to A. V. Rukavishnikov.

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Original Russian Text © A.V. Rukavishnikov, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 9, pp. 1515–1536.

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Rukavishnikov, A.V. Domain decomposition method and numerical analysis of a fluid dynamics problem. Comput. Math. and Math. Phys. 54, 1459–1480 (2014). https://doi.org/10.1134/S0965542514070094

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  • DOI: https://doi.org/10.1134/S0965542514070094

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