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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 1105–1128 | Cite as

Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions

  • Xiangyun Meng
  • Martin StynesEmail author
Article
  • 29 Downloads

Abstract

We consider the singularly perturbed fourth-order boundary value problem ε2Δ2u −Δu = f on the unit square \({\Omega }\subset \mathbb {R}^{2}\), with boundary conditions u = u/n = 0 on Ω. Here, ε ∈ (0,1) is a small parameter. The problem is solved numerically by means of Adini finite elements—a simple nonconforming finite element method for this problem. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with N2 elements is constructed and convergence of the method is proved in a ‘broken’ version of the Sobolev norm \(v\mapsto \left (\varepsilon ^{2}|v|_{2}^{2} + |v|_{1}^{2} \right )^{1/2}\). For a particular choice of the mesh, the error in the computed solution is at most C [ε1/2(N− 1 lnN)2 + min {ε1/2,ε− 3/2N− 2} + N− 3], where the constant C is independent of ε and N. Numerical results support our theoretical convergence rates, even for an example where not all the hypotheses of our theory are satisfied.

Keywords

Singularly perturbed Fourth-order differential equation Adini element Shishkin mesh 

Mathematics Subject Classification (2010)

Primary 65N30 Secondary 35B25 

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Notes

Acknowledgements

We thank two unknown reviewers for their extremely careful reading of our manuscript and for giving us many helpful suggestions.

Funding information

The research of the second author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF U1530401.

References

  1. 1.
    Andreev, V. B.: On the accuracy of grid approximations of nonsmooth solutions of a singularly perturbed reaction-diffusion equation in the square. Differ. Uravn. 42 (7), 895–906, 1005 (2006).  https://doi.org/10.1134/S0012266106070044 MathSciNetGoogle Scholar
  2. 2.
    Brenner, S. C., Gudi, T., Neilan, M., Sung, L.Y.: c 0 penalty methods for the fully nonlinear Monge-Ampère equation. Math. Comput. 80 (276), 1979–1995 (2011).  https://doi.org/10.1090/S0025-5718-2011-02487-7  https://doi.org/10.1090/S0025-5718-2011-02487-7 zbMATHGoogle Scholar
  3. 3.
    Brenner, S. C., Neilan, M.: A c 0 interior penalty method for a fourth order elliptic singular perturbation problem. SIAM J. Numer. Anal. 49(2), 869–892 (2011).  https://doi.org/10.1137/100786988 MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods Texts in Applied Mathematics, 3rd edn., vol. 15. Springer, New York (2008).  https://doi.org/10.1007/978-0-387-75934-0
  5. 5.
    Chen, H., Chen, S.: Uniformly convergent nonconforming element for 3-D fourth order elliptic singular perturbation problem. J. Comput. Math. 32(6), 687–695 (2014).  https://doi.org/10.4208/jcm.1405-m4303  https://doi.org/10.4208/jcm.1405-m4303 MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, H., Chen, S., Xiao, L.: Uniformly convergent c 0-nonconforming triangular prism element for fourth-order elliptic singular perturbation problem. Numer. Methods Partial Differential Equations 30 (6), 1785–1796 (2014).  https://doi.org/10.1002/num.21878 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, S., Zhao, Y., Shi, D.: Anisotropic interpolations with application to nonconforming elements. Appl. Numer. Math. 49(2), 135–152 (2004).  https://doi.org/10.1016/j.apnum.2003.07.005 MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen, S.C., Zhao, Y.C., Shi, D.Y.: Non c 0 nonconforming elements for elliptic fourth order singular perturbation problem. J. Comput. Math. 23(2), 185–198 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978)zbMATHGoogle Scholar
  10. 10.
    Constantinou, P., Varnava, C., Xenophontos, C.: An hp finite element method for 4th order singularly perturbed problems. Numer. Algorithms 73(2), 567–590 (2016).  https://doi.org/10.1007/s11075-016-0108-9  https://doi.org/10.1007/s11075-016-0108-9 MathSciNetzbMATHGoogle Scholar
  11. 11.
    Constantinou, P., Xenophontos, C.: An hp finite element method for a 4th order singularly perturbed boundary value problem in two dimensions. Computers and Mathematics with Applications.  https://doi.org/10.1016/j.camwa.2017.02.009. http://www.sciencedirect.com/science/article/pii/S0898122117300755 (2017)
  12. 12.
    Du, S., Lin, R., Zhang, Z.: Robust residual-based a posteriori error estimators for mixed finite element methods for fourth order elliptic singularly perturbed problems. arXiv:http://arXiv.org/abs/1609.04506 (2016)
  13. 13.
    Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O’Riordan, E., Shishkin, G. I.: Robust Computational Techniques for Boundary Layers Applied Mathematics (Boca Raton), vol. 16. Chapman & Hall/CRC, Boca Raton (2000)Google Scholar
  14. 14.
    Franz, S., Roos, H. G.: Robust error estimation in energy and balanced norms for singularly perturbed fourth order problems. Comput. Math. Appl. 72 (1), 233–247 (2016).  https://doi.org/10.1016/j.camwa.2016.05.001  https://doi.org/10.1016/j.camwa.2016.05.001 MathSciNetGoogle Scholar
  15. 15.
    Franz, S., Roos, H. G., Wachtel, A.: A c 0 interior penalty method for a singularly-perturbed fourth-order elliptic problem on a layer-adapted mesh. Numer. Methods Partial Differential Equations 30(3), 838–861 (2014).  https://doi.org/10.1002/num.21839 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ganesan, S., Tobiska, L.: Finite Elements: Theory and Algorithms. IISc. Cambridge University Press, Cambridge (2017)zbMATHGoogle Scholar
  17. 17.
    Guzmán, J., Leykekhman, D., Neilan, M.: A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem. Calcolo 49(2), 95–125 (2012).  https://doi.org/10.1007/s10092-011-0047-8 MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hu, J., Huang, Y.: The correction operator for the canonical interpolation operator of the Adini element and the lower bounds of eigenvalues. Sci. China Math. 55(1), 187–196 (2012).  https://doi.org/10.1007/s11425-011-4267-9 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hu, J., Shi, Z.: A lower bound of the l 2 norm error estimate for the Adini element of the biharmonic equation. SIAM J. Numer. Anal. 51(5), 2651–2659 (2013).  https://doi.org/10.1137/130907136 MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hu, J., Yang, X., Zhang, S.: Capacity of the Adini element for biharmonic equations. J. Sci. Comput. 69(3), 1366–1383 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. Rev. Franç,aise Automat. Informat. Recherche Operationnelle Sér. Rouge Anal. Numér. 9(R-1), 9–53 (1975)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Luo, P., Lin, Q.: High accuracy analysis of the Adini’s nonconforming element. Computing 68(1), 65–79 (2002).  https://doi.org/10.1007/s006070200003 MathSciNetzbMATHGoogle Scholar
  23. 23.
    Mao, S., Chen, S.: Accuracy analysis of Adini’s non-conforming plate element on anisotropic meshes. Comm. Numer. Methods Engrg. 22(5), 433–440 (2006).  https://doi.org/10.1002/cnm.825 MathSciNetzbMATHGoogle Scholar
  24. 24.
    Nilssen, T. K., Tai, X. C., Winther, R.: A robust nonconforming h 2-element. Math. Comp. 70 (234), 489–505 (2001).  https://doi.org/10.1090/S0025-5718-00-01230-8 MathSciNetzbMATHGoogle Scholar
  25. 25.
    O’Malley, R.E., Jr.: Introduction to singular perturbations. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London. Applied Mathematics and Mechanics, vol. 14 (1974)Google Scholar
  26. 26.
    Panaseti, P., Zouvani, A., Madden, N., Xenophontos, C.: A c 1-conforming hp finite element method for fourth order singularly perturbed boundary value problems. Appl. Numer. Math. 104, 81–97 (2016).  https://doi.org/10.1016/j.apnum.2016.02.002 MathSciNetzbMATHGoogle Scholar
  27. 27.
    Roos, H. G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed differential Equations, Springer Series in Computational Mathematics, 2nd edn., vol. 24. Springer, Berlin (2008). Convection-diffusion-reaction and flow problemsGoogle Scholar
  28. 28.
    Semper, B.: Conforming finite element approximations for a fourth-order singular perturbation problem. SIAM J. Numer. Anal. 29(4), 1043–1058 (1992).  https://doi.org/10.1137/0729063 MathSciNetzbMATHGoogle Scholar
  29. 29.
    Shi, Z. C., Wang, M.: Finite Element Method. Science Press, Beijing (2013)Google Scholar
  30. 30.
    Wang, L., Wu, Y., Xie, X.: Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems. Numer. Methods Partial Differential Equations 29(3), 721–737 (2013).  https://doi.org/10.1002/num.21723 MathSciNetzbMATHGoogle Scholar
  31. 31.
    Wang, M., Meng, X.: A robust finite element method for a 3-D elliptic singular perturbation problem. J. Comput. Math. 25(6), 631–644 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Wang, M., Shi, Z. C., Xu, J.: Some n-rectangle nonconforming elements for fourth order elliptic equations. J. Comput. Math. 25(4), 408–420 (2007)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Wang, M., Xu, J.C., Hu, Y.C.: Modified M,orley element method for a fourth order elliptic singular perturbation problem. J. Comput. Math. 24(2), 113–120 (2006)MathSciNetGoogle Scholar
  34. 34.
    Wang, W., Huang, X., Tang, K., Zhou, R.: Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem. Adv. Comput. Math.  https://doi.org/10.1007/s10444-017-9572-6  https://doi.org/10.1007/s10444-017-9572-6 (2017)
  35. 35.
    Xie, P., Shi, D., Li, H.: A new robust c 0-type nonconforming triangular element for singular perturbation problems. Appl. Math. Comput. 217(8), 3832–3843 (2010).  https://doi.org/10.1016/j.amc.2010.09.042  https://doi.org/10.1016/j.amc.2010.09.042 MathSciNetzbMATHGoogle Scholar
  36. 36.
    Yang, Y., Zhang, Z., Lin, F.: Eigenvalue approximation from below using non-conforming finite elements. Sci. China Math. 53(1), 137–150 (2010).  https://doi.org/10.1007/s11425-009-0198-0 MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zhang, S., Wang, M.: A posteriori estimator of nonconforming finite element method for fourth order elliptic perturbation problems. J. Comput. Math. 26(4), 554–577 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Division of Applied and Computational MathematicsBeijing Computational Science Research CenterBeijingChina

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