Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation

  • Kazuaki Tanaka
  • Akitoshi Takayasu
  • Xuefeng Liu
  • Shin’ichi Oishi
Original Paper Area 2


This paper proposes a verified numerical method of proving the invertibility of linear elliptic operators. This method also provides a verified norm estimation for the inverse operators. This type of estimation is important for verified computations of solutions to elliptic boundary value problems. The proposed method uses a generalized eigenvalue problem to derive the norm estimation. This method has several advantages. Namely, it can be applied to two types of boundary conditions: the Dirichlet type and the Neumann type. It also provides a way of numerically evaluating lower and upper bounds of target eigenvalues. Numerical examples are presented to show that the proposed method provides effective estimations in most cases.


Eigenvalue problem Elliptic operator Finite element method Inverse norm estimation Numerical verification 

Mathematics Subject Classification

65N25 65N30 35J25 



The authors express their sincere thanks to Prof. M. Plum and Prof. K. Nagatou-Plum in Karlsruhe Institute of Technology, Germany for his valuable comment and kind remarks. They also express their appreciation for reviewer’s attentive review and valuable comments. The second author was supported by a Grant-in-Aid for JSPS Fellows. The third author was partially supported by a Grant-in-Aid for Young Scientists (B) (No. 23740092) from Japan Society for the Promotion of Science.


  1. 1.
    Plum, M.: Computer-assisted proofs for semilinear elliptic boundary value problems. Jpn. J. Ind. Appl. Math. 26, 419–442 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Nakao, M.T., Hashimoto, K., Watanabe, Y.: Numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing 75(1), 1–14 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Breuer, B., Horak, J., McKenna, P.J., Plum, M.: A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam. J. Differ. Equ. 224, 60–97 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Takayasu, A., Liu, X., Oishi, S.: Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains. NOLTA IEICE 4(1), 34–61 (2013)CrossRefGoogle Scholar
  5. 5.
    Kobayashi, K.: On the interpolation constants over triangular elements. RIMS Kōkyūroku 1733, 58–77 (2011). (in Japanese)Google Scholar
  6. 6.
    Liu, X., Oishi, S.: Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape. SIAM J. Numer. Anal. 51(3), 1634–1654 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kikuchi, F., Liu, X.: Estimation of interpolation error constants for the P0 and P1 triangular finite elements. Comput. Methods Appl. Mech. Eng. 196, 3750–3758 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dautray, R., Lions, J.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Grisvard, P.: Elliptic problems in nonsmooth domains. Classics in Applied Mathematics, vol. 69. SIAM (2011)Google Scholar
  10. 10.
    Oishi, S.: Numerical verification of existence and inclusion of solutions for nonlinear operator equations. J. Comput. Appl. Math. 60, 171–185 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Behnke, H.: The calculation of guaranteed bounds for eigenvalues using complementary variational principles. Computing 47, 11–27 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Rump, S.M.: Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse. BIT Numer. Math. 51, 367–384 (2011)Google Scholar
  13. 13.
    Liu, X., Oishi, S.: Guaranteed high-precision estimation for P0 interpolation constants on triangular finite elements. Jpn. J. Ind. Appl. Math. 30(3), 635–652 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Rump, S.M.: INTLAB-INTerval LABoratry. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2014

Authors and Affiliations

  • Kazuaki Tanaka
    • 1
  • Akitoshi Takayasu
    • 2
  • Xuefeng Liu
    • 3
    • 4
  • Shin’ichi Oishi
    • 2
    • 5
  1. 1.Graduate School of Fundamental Science and EngineeringWaseda UniversityShinjuku-kuJapan
  2. 2.Department of Applied Mathematics, Faculty of Science and EngineeringWaseda UniversityShinjuku-kuJapan
  3. 3.Research Institute for Science and EngineeringWaseda UniversityShinjuku-kuJapan
  4. 4.Graduate School of Science and Technology, Niigata UniversityNiigata CityJapan
  5. 5.JST, CRESTTokyoJapan

Personalised recommendations