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Journal of Mathematical Sciences

, Volume 236, Issue 4, pp 430–445 | Cite as

On Projectors to Subspaces of Vector-Valued Functions Subject to Conditions of the Divergence-Free Type

  • S. Repin
Article
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We study operators that project a vector-valued function υW1,2(Ω, ℝd) to subspaces formed by the condition that the divergence is orthogonal to a certain amount (finite or infinite) of test functions. The condition that the divergence is equal to zero almost everywhere presents the first (narrowest) limit case while the integral condition of zero mean divergence generates the other (widest) case. Estimates of the distance between υ and the respective projection Open image in new window on such a subspace are important for analysis of various mathematical models related to incompressible media problems (especially in the context of a posteriori error estimates. We establish different forms of such estimates, which contain only local constants associated with the stability (LBB) inequalities for subdomains. The approach developed in the paper also yields two-sided bounds of the inf-sup (LBB) constant.

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References

  1. 1.
    I. Babuška and A. K. Aziz, Surway Lectures on the Mathematical Foundations of the Finite Element Method, Academic Press, New York (1972).Google Scholar
  2. 2.
    M. E. Bogovskii, “Solution of the first boundary value problem for the equation of continuity of an incompressible medium,” Soviet Math. Dokl., 248, No. 5, 1037–1040 (1979).MathSciNetGoogle Scholar
  3. 3.
    M. Costabel and M. Dauge. “On the inequalities of Babuska–Aziz, Friedrichs and Horgan–Payne,” Arch. Ration. Mech. Anal., 217, No. 3, 873–898 (2015).MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Costabel, M. Crouzeix, M. Dauge, and Y. Lafranche. “The inf-sup constant for the divergence on corner domains,” Num. Methods PDES, 31, No 2, 439–458 (2015).MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Dobrowolski, “On the LBB constant on stretched domains,” Math. Nachr., 254/255, 64–67 (2003).MathSciNetCrossRefGoogle Scholar
  6. 6.
    C. Horgan and L. Payne, “On inequalities of Korn, Friedrichs and Babuska–Aziz,” Arch. Ration. Mech. Anal., 82, 165–179 (1983).MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Kessler, Die Ladyzhenskaya-Konstante in der numerischen Behandlung von Strömungsproblemen, Bayerischen Julius-Maximilians-Universität, Würzburg (2000).Google Scholar
  8. 8.
    O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Flow, 2nd ed., Gordon and Breach, New York (1969).zbMATHGoogle Scholar
  9. 9.
    O. A. Ladyzenskaja and V. A. Solonnikov, “Some problems of vector analysis, and generalized formulations of boundary value problems for the Navier-Stokes equation,” Zap. Nauchn. Semin. LOMI, 59, 81–116 (1976).MathSciNetGoogle Scholar
  10. 10.
    J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris (1967).zbMATHGoogle Scholar
  11. 11.
    M. A. Olshanskii and E. V. Chizhonkov, “On the best constant in the inf sup condition for prolonged rectangular domains,” Mat. Zam., 67, No. 3, 387–396 (2000).CrossRefGoogle Scholar
  12. 12.
    L. E. Payne, “A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov,” IMA J. Appl. Math., 72, 563–569 (2007).MathSciNetCrossRefGoogle Scholar
  13. 13.
    K. I. Piletskas, “On spaces of solenoidal vectors,” Zap. Nauchn. Semin. LOMI, 96, 237–239 (1980).MathSciNetzbMATHGoogle Scholar
  14. 14.
    K. I. Piletskas, “Spaces of solenoidal vectors,” Trudy Mat. Inst. Steklov, 159, 137–149 (1983).MathSciNetzbMATHGoogle Scholar
  15. 15.
    S. Repin, “A posteriori estimates for the Stokes problem,” J. Math. Sci., 109, No. 5, 1950–1964 (2002).MathSciNetCrossRefGoogle Scholar
  16. 16.
    S. Repin, “Estimates of deviations from exact solutions for some boundary–value problems with incompressibility condition,” St. Petersburg Math. J., 16, No. 5, 124–161 (2004).MathSciNetGoogle Scholar
  17. 17.
    S. Repin, A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin (2008).CrossRefGoogle Scholar
  18. 18.
    S. Repin, “Estimates of deviations from exact solution of the generalized Oseen problem,” Zap. Nauchn. Semin. POMI, 410, 110–130 (2013).MathSciNetGoogle Scholar
  19. 19.
    S. Repin, “Estimates of the distance to the set of divergence free fields,” Zap. Nauchn. Semin. POMI, 425, 99–116 (2014).Google Scholar
  20. 20.
    S. Repin, “On variational representations of the constant in the inf sup condition for the Stokes problem,” Zap. Nauchn. Semin. POMI, 444, 110–123 (2016).MathSciNetGoogle Scholar
  21. 21.
    S. Repin, “Estimates of the distance to the set of solenoidal vector fields and applications to a posteriori error control,” Comput. Methods Appl. Math., 15, No. 4, 515–530 (2015).MathSciNetCrossRefGoogle Scholar
  22. 22.
    G. Stoyan, “Towards discrete Velte decompositions and narrow bounds for inf-sup constants,” Comput. Math. Appl., 38, 243–261 (1999).MathSciNetCrossRefGoogle Scholar
  23. 23.
    R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam (1979).zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.St. Petersburg Department of V. A. Steklov Institute of MathematicsPeter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia

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