, Volume 75, Issue 1, pp 1–14 | Cite as

A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems



In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.

AMS Subject Classifications:

35J25 35J60 65N25 


Numerical verification unique solvability of linear elliptic problem finite element method 


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  1. 1.
    Alefeld, G.: On the convergence of some interval-arithmetic modifications of Newton’s method. SIAM J. Numer. Anal. 21, 363–372 (1984).Google Scholar
  2. 2.
    Nagatou, K., Yamamoto, N., Nakao, M. T.: An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optimiz. 20, 543–565 (1999).Google Scholar
  3. 3.
    Nakao, M. T.: A numerical approach to the proof of existence of solutions for elliptic problems. Japan J. Appl. Math. 5, 313–332 (1988).Google Scholar
  4. 4.
    Nakao, M. T.: Solving nonlinear elliptic problems with result verification using an H-1 residual iteration. Computing (Suppl. 9), 161–173 (1993).Google Scholar
  5. 5.
    Nakao, M. T., Yamamoto, N.: Numerical verification of solutions for nonlinear elliptic problems using L residual method. J. Math. Anal. Appl. 217, 246–262 (1998).Google Scholar
  6. 6.
    Nakao, M. T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optimiz. 22(3–4), 321–356 (2001).Google Scholar
  7. 7.
    Nakao, M. T., Watanabe, Y.: An efficient approach to the numerical verification for solutions of elliptic differential equations. Numer. Algorith. 37. Special issue for Proceedings of SCAN 2002, 311–323 (2004).Google Scholar
  8. 8.
    Plum, M.: Explicit H2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl. 165, 36–61 (1992).Google Scholar
  9. 9.
    Plum, M.: Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems. Computing 49, 25–44 (1992).Google Scholar
  10. 10.
    Plum, M.: Computer-assisted enclosure methods for elliptic differential equations. J. Linear Algebra Appl. 327, 147–187 (2001).Google Scholar
  11. 11.
    Rump S. M.: INTLAB-INTerval LABoratory, a Matlab toolbox for verified computations, version 4.1.2. Inst. Infomatik, Technical University of Hamburg – Hamburg.
  12. 12.
    Rump, S. M.: Solving algebraic problems with high accuracy. In: A new approach to scientific computation (Kulisch, U., and Miranker, W. L., eds.). New York: Academic Press 1983.Google Scholar
  13. 13.
    Yamamoto, N., Nakao, M. T.: Numerical verifications for solutions to elliptic equations using residual iterations with higher order finite element. J. Comput. Appl. Math. 60, 271–279 (1995).Google Scholar
  14. 14.
    Yamamoto, N.: A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed point theorem. SIAM J. Numer. Anal. 35, 2004–2013 (1998).Google Scholar

Copyright information

© Springer-Verlag Wien 2005

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Graduate School of MathematicsKyushu UniversityJapan
  3. 3.Computing and Communications CenterKyushu UniversityJapan

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