Skip to main content
SpringerLink
Log in

Some considerations of the invertibility verifications for linear elliptic operators

  • Original Paper
  • Area 2
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper presents three computer-assisted procedures for verifying the invertibility of second-order linear elliptic operators and for computing a bound on the norm of its inverse. One of these procedures is an improvement of a theorem by Nakao et al. (Computing 75:1–14, 2005) that uses projection and constructive a priori error estimates and was proposed by two of the authors of this paper. Results verifying these procedures are presented for several numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  2. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  3. Hashimoto, K., Nagatou, K., Nakao, M.T.: A computational approach to constructive a priori error estimate for finite element approximations of bi-harmonic problems in nonconvex polygonal domains. Information 9, 573–580 (2006)

    MathSciNet  Google Scholar 

  4. Kikuchi, F., Liu, X.: Determination of the Babuska-Aziz constant for the linear triangular finite element. Jpn. J. Ind. Appl. Math. 23, 75–82 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kobayashi, K.: A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions. Jpn. J. Ind. Appl. Math. 26, 493–516 (2009)

    Article  MATH  Google Scholar 

  6. Nagatou, K.: Numerical verification method for infinite dimensional eigenvalue problems. Jpn. J. Ind. Appl. Math. 26, 477–491 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Nakao, M.T.: Computable \(L^\infty \) error estimates in the finite element method with application to nonlinear elliptic problems. In: Agarwal, R.P. (ed.) Contributions in Numerical Mathematics, pp. 309–319. World Scientific, Singapore (1993)

    Chapter  Google Scholar 

  8. Nakao, M.T., Yamamoto, N.: Numerical verification of solutions for nonlinear elliptic problems using \(L^\infty \) residual method. J. Math. Anal. Appl. 217, 246–262 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Nakao, M.T., Yamamoto, N., Kimura, S.: On best constant in the error bound for the \(H^1_0\)-projection into piecewise polynomial spaces. J. Approx. Theory 93, 491–500 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Nakao, M.T., Yamamoto, N.: A guaranteed bound of the optimal constant in the error estimates for linear triangular element. In: Alfeld, G., Chen, X. (eds.) Topics in Numerical Analysis with Special Emphasis on Nonlinear Problems, pp. 163–173, Computing, Suppl. 15, Springer, Wien, New York (2001)

  11. Nakao, M.T., Yamamoto, N.: A guaranteed bound of the optimal constant in the error estimates for linear triangular element Part II: Details. In: Kulisch, U., Lohner, R., Facius, A. (eds.) Perspectives on Enclosure Methods, pp. 265–276. Springer, Wien (2001)

    Chapter  Google Scholar 

  12. Nakao, M.T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim. 22, 321–356 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Nakao, M.T., Hashimoto, K., Watanabe, Y.: A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing 75, 1–14 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nakao, M.T., Hashimoto, K., Nagatou, K.: A computational approach to constructive a priori and a posteriori error estimates for finite element approximations of bi-harmonic problems. In: Proceedings of the 4th JSIAM-SIMAI Seminar on Industrial and Applied Mathematics. GAKUTO International Series, Mathematical Sciences and Applications 28, pp. 139–148, Gakkotosho, Tokyo (2008)

  15. Nakao, M.T., Watanabe, Y.: Numerical verification methods for solutions of semilinear elliptic boundary value problems. NOLTA 2, 2–31 (2011)

    Article  Google Scholar 

  16. Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104, Kluwer Academic Publishers, Dordrecht (1999) http://www.ti3.tu-harburg.de/rump/

  17. Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida, T.: A numerical verification of nontrivial solutions for the heat convection problem. J. Math. Fluid Mech. 6, 1–20 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Watanabe, Y., Kinoshita, T., Nakao, M.T.: A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations. Math. Comp. 82, 1543–1557 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Yamamoto, N., Nakao, M.T.: Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains. Numer. Math. 65, 503–521 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Yamamoto, N., Hayakawa, K.: Error estimation with guaranteed accuracy of finite element method in nonconvex polygonal domains. J. Comp. Appl. Math. 159, 173–183 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yamamoto, N., Gemma, K.: On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains. J. Comp. Appl. Math. 199, 286–296 (2007)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors heartily thank the two anonymous referees for their thorough reading and valuable comments. This work was supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 24340018, 23740074, and 24540151).

Author information

Authors and Affiliations

  1. National Institute of Technology, Sasebo College, 1-1 Okishin-cho, Sasebo, Nagasaki, 857-1193, Japan

    Mitsuhiro T. Nakao

  2. Research Institute for Information Technology, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka, 812-8518, Japan

    Yoshitaka Watanabe

  3. Center for the Promotion of Interdisciplinary Education and Research, Kyoto University, Kyoto, 606-8501, Japan

    Takehiko Kinoshita

  4. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Takehiko Kinoshita

  5. JST CREST/Faculty of Science and Engineering, Waseda University, Tokyo, 169-8555, Japan

    Takuma Kimura

  6. Department of Communication Engineering and Informatics, Faculty of Informatics and Engineering, University of Electro-Communications, 1-5-1 Chofu-ga-oka, Chofu, Tokyo, 182-8585, Japan

    Nobito Yamamoto

Authors
  1. Mitsuhiro T. Nakao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshitaka Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Takehiko Kinoshita
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Takuma Kimura
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Nobito Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshitaka Watanabe.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nakao, M.T., Watanabe, Y., Kinoshita, T. et al. Some considerations of the invertibility verifications for linear elliptic operators. Japan J. Indust. Appl. Math. 32, 19–31 (2015). https://doi.org/10.1007/s13160-014-0160-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-014-0160-6

Keywords

Mathematics Subject Classification

Search

Navigation