Advertisement

Optimal estimation for the Fujino–Morley interpolation error constants

  • Shih-Kang Liao
  • Yu-Chen Shu
  • Xuefeng LiuEmail author
Original Paper
  • 15 Downloads

Abstract

The quantitative estimation for the interpolation error constants of the Fujino–Morley interpolation operator is considered. To give concrete upper bounds for the constants, which is reduced to the problem of providing lower bounds for eigenvalues of bi-harmonic operators, a new algorithm based on the finite element method along with verified computation is proposed. In addition, the quantitative analysis for the variation of eigenvalues upon the perturbation of the shape of triangles is provided. Particularly, for triangles with longest edge length less than one, the optimal estimation for the constants is provided. An online demo with source codes of the constants calculation is available at http://www.xfliu.org/onlinelab/.

Keywords

Fujino–Morley interpolation operator Finite element method Verified computing Eigenvalue problem 

Mathematics Subject Classification

35P15 97N50 65N30 

Notes

Acknowledgements

The authors would like to thank for the support from the Ministry of Science and Technology, Taiwan, ROC under Grant nos. MOST 106-2115-M-006-011, MOST 107-2911-M-006-506. This research is also supported by Japan Society for the Promotion of Science, Grand-in-Aid for Young Scientist (B) 26800090, Grant-in-Aid for Scientific Research (C) 18K03411 and Grant-in-Aid for Scientific Research (B) 16H03950 for the third author.

References

  1. 1.
    Babuska, I., Osborn, J.: Eigenvalue problems. Handb. Numer. Anal. 3(1), 641–787 (1991)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Behnke, H.: The calculation of guaranteed bounds for eigenvalues using complementary variational principles. Computing 47(1), 11–27 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carstensen, C., Gallistl, D.: Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126(1), 33–51 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carstensen, C., Gedicke, J.: Guaranteed lower bounds for eigenvalues. Math. Comput. 83(290), 2605–2629 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fujino, T.: The triangular equilibrium element in the solution of plate bending problems. In: Gallagher, R.H., Yamada, Y., Tinsley Oden, J. (eds.) Recent Advaced on Matrix Methods Structural Analysis Design, pp. 725–786. The University of Alabama Press, Tuscaloosa (1971)Google Scholar
  6. 6.
    Kikuchi, F., Liu, X.: Estimation of interpolation error constants for the \(P_0\) and \(P_1\) triangular finite element. Comput. Methods Appl. Mech. Eng. 196, 3750–3758 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kobayashi, K.: On the Interpolation Constants Over Triangular Elements. Institue of Mathematics, Czech Academy of Sciences, Prague (2015)zbMATHGoogle Scholar
  8. 8.
    Liu, X.: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. Comput. 267, 341–355 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Liu, X., Kikuchi, F.: Analysis and estimation of error constants for \(P_0\) and \(P_1\) interpolations over triangular finite elements. J. Math. Sci. Univ. Tokyo 17(1), 27–78 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Liu, X., You, C.: Explicit bound for quadratic Lagrange interpolation constant on triangular finite elements. Appl. Math. Comput. 319, 693–701 (2018)MathSciNetGoogle Scholar
  11. 11.
    Morley, L.: The triangular equilibrium element in the solution of plate bending problems. Aeronaut. Q. 19(2), 149–169 (1968)Google Scholar
  12. 12.
    Morley, L.: The constant-moment plate-bending element. J. Strain Anal. 6(1), 20–24 (1971)CrossRefGoogle Scholar
  13. 13.
    Rump, S.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht. http://www.ti3.tuhh.de/rump/ (1999). Accessed 4 Jan 2019

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of Science and TechnologyNiigata UniversityNiigataJapan
  2. 2.Department of Applied MathematicsNational Cheng Kung UniversityTainanTaiwan
  3. 3.Department of MathematicsNational Cheng Kung UniversityTainanTaiwan

Personalised recommendations