Advertisement

Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements

  • Manting Xie
  • Hehu Xie
  • Xuefeng Liu
Original Paper Area 2
  • 112 Downloads

Abstract

An algorithm is proposed to give explicit lower bounds of the Stokes eigenvalues by utilizing two nonconforming finite element methods: Crouzeix–Raviart (CR) element and enriched Crouzeix–Raviart (ECR) element. Compared with the existing literatures which give lower eigenvalue bounds under the asymptotic condition that the mesh size is “small enough”, the proposed algorithm in this paper drops the asymptotic condition and provide explicit lower bounds even for a rough mesh. Numerical experiments are also performed to validate the theoretical results.

Keywords

Stokes eigenvalue problem Eigenvalue bound Crouzeix–Raviart element Enriched Crouzeix–Raviart element Explicit lower bound 

Mathematics Subject Classification

65N30 65N25 65L15 

References

  1. 1.
    Babuška, I., Osborn, J.E.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52, 275–297 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babuška, I., Osborn, J.: Eigenvalue problems. In: Lions, P.G., Ciarlet, P.G. (eds.) Handbook of Numerical Analysis, Vol. II, Finite Element Methods (Part 1), pp. 641–787. North-Holland, Amsterdam (1991)Google Scholar
  3. 3.
    Batcho, P., Karniadakis, G.: Generalized Stokes eigenfunctions: a new trial basis for the solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 115, 121–146 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chatelin, F.: Spectral Approximation of Linear Operators. Academic Press Inc, New York (1983)zbMATHGoogle Scholar
  6. 6.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  7. 7.
    Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element for solving the stationary Stokes equations. RAIRO Anal. Numer. 3, 33–75 (1973)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fabes, E., Kenig, C., Verchota, G.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hu, J., Huang, Y., Lin, Q.: The lower bounds for eigenvalues of elliptic operators by nonconforming finite element methods. J. Sci. Comput. 61(1), 196–221 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jia, S., Luo, F., Xie, H.: A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem. Acta Math. Sinica (Engl. Ser.) 30(6), 949–967 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Labrosse, G., Leriche, E., Lallemand, P.: Stokes eigenmodes in cubic domain: their symmetry properties. Theor. Comput. Fluid Dyn. 28(3), 335–356 (2014)CrossRefGoogle Scholar
  12. 12.
    Leriche, E., Labrosse, G.: Stokes eigenmodes in square domain and the stream function–vorticity correlation. J. Comput. Phys. 200, 489–511 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. China Sci. Press, Beijing (2006)Google Scholar
  14. 14.
    Lin, Q., Luo, F., Xie, H.: A multilevel correction method for Stokes eigenvalue problems and its applications. Math. Methods Appl. Sci. 38, 4540–4554 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lin, Q., Luo, F., Xie, H.: A posterior error estimator and lower bound of a nonconforming finite element method. J. Comput. Appl. Math. 265, 243–254 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lin, Q., Tobiska, L., Zhou, A.: On the superconvergence of nonconforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25, 160–181 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lin, Q., Xie, H.: The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods. Math. Pract. Theory (in Chinese) 42(11), 219–226 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lin, Q., Xie, H., Luo, F., Li, Y., Yang, Y.: Stokes eigenvalue approximation from below with nonconforming mixed finite element methods. Math. Pract. Theory (in Chinese) 19, 157–168 (2010)MathSciNetGoogle Scholar
  19. 19.
    Liu, X.: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. Comput. 267, 341–355 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lovadina, C., Lyly, M., Stenberg, R.: A posteriori estimates for the Stokes eigenvalue problem. Numer. Methods Partial Differ. Equ. 25(1), 244–257 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Luo, F., Lin, Q., Xie, H.: Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods. Sci. China Math. 55(5), 1069–1082 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mercier, B., Osborn, J., Rappaz, J., Raviart, P.: Eigenvalue approximation by mixed and hybrid methods. Math. Comput. 36(154), 427–453 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nakao, M.: A numerical verification method for the existence of weak solutions for nonlinear boundary value problems. J. Math. Anal. Appl. 164, 489–507 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nakao, M.T., Yamamoto, N., Watanabe, Y.: A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations. J. Comput. Appl. Math. 91(1), 137–158 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Plum, M.: Explicit \(H^2\) -estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl. 165, 36–61 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods PDEs 8, 97–111 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Takayasu, A., Liu, X., Oishi, S.: Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains. NOLTA, IEICE, E96-N, No. 1, pp. 34–61 (2013)Google Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinChina
  2. 2.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  4. 4.Graduate School of Science and TechnologyNiigata UniversityNiigataJapan

Personalised recommendations