Numerical verifications for eigenvalues of second-order elliptic operators

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Abstract

In this paper, we consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is based on Nakao’s method [9] using the Newton-like operator and the error estimates for the C∘ finite element solution. We construct, in computer, a set containing solutions which satisfies the hypothesis of Schauder’s fixed point theorem for compact map on a certain Sobolev space. Moreover, we propose a method to verify the eigenvalue which has the smallest absolute value. A numerical example is presented.

Key words

eigenvalue problem elliptic operators error estimates finite element solution 
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References

  1. [1]
    H. Behnke and F. Goerisch, Inclusions for eigenvalues of selfadjoint problems. Topics in Validated Computations-Studies in Computational Mathematics (ed. J. Herzberger), Elsevier, Amsterdam, 1994.Google Scholar
  2. [2]
    P. Grisvard, Elliptic problems in nonsmooth domains. Pitman Monographs and Survays in Pure and Applied Mathematics,24, London, 1985.Google Scholar
  3. [3]
    O. Knüppel, PROFIL/BIAS — A fast interval library. Computing,53 (1994), 277–288.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    M. Krízek and P. Neittaanmäki, Finite Element Approximation of Variational Problems and Applications. Longman Scientific and Technical, Harlow, 1990.MATHGoogle Scholar
  5. [5]
    M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems. Japan J. Appl. Math.,5 (1988), 313–332.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems II. Japan J. Appl. Math.,7 (1990), 477–488.MATHMathSciNetGoogle Scholar
  7. [7]
    M.T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear boundary value problems. J. Math. Anal. Appl.,164 (1992), 489–507.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    M.T. Nakao, Solving nonlinear elliptic problem with result verification using anH −1 type residual iteration. Computing, Suppl.,9 (1993), 161–173.Google Scholar
  9. [9]
    M.T. Nakao and N. Yamamoto, Numerical verification of solutions for nonlinear elliptic problems using anL∞ residual method. J. Math. Anal. Appl.,217 (1998), 246–262.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Plum, Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method. J.Appl.Math.Phys.(ZAMP),41 (1990), 205–226.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Plum, Bounds for eigenvalues of second-order elliptic differential operators. J. Appl. Math. Phys. (ZAMP),42 (1991), 848–863.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    M. Plum, ExplicitH 2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl.,165 (1992), 36–61.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Plum, Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems. J. Comput. Appl. Math.,60 (1995), 187–200.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    S.M. Rump, Solving algebraic problems with high accuracy. A New Approach to Scientific Computation (eds. U. Kulisch and W.L. Miranker), Academic Press, New York. 1983.Google Scholar
  15. [15]
    Y. Watanabe and M.T. Nakao, Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. Appl. Math.,10 (1993), 165–178.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    N. Yamamoto and M.T. Nakao, Numerical verifications for solutions to elliptic equations using residual iterations with a high order finite element. J. Comput. Appl. Math.,60 (1995), 271–279.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    E. Zeidler, Nonlinear Functional Analysis and its Applications I. Springer, New York, 1986.MATHGoogle Scholar

Copyright information

© JJIAM Publishing Committee 1999

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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