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Approximation of sign-indefinite spectral problems

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Abstract

A variational sign-indefinite eigenvalue problem in an infinite-dimensional Hilbert space is approximated by a problem in a finite-dimensional subspace. We analyze the convergence and accuracy of approximate eigenvalues and eigenelements. The general results are illustrated by a sample scheme of the finite-element method with numerical integration for a one-dimensional sign-indefinite second-order differential eigenvalue problem.

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References

  1. Ciarlet, Ph., The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1977. Translated under the title Metod konechnykh elementov dlya ellipticheskikh zadach, Moscow: Mir, 1980.

    Google Scholar 

  2. Solov’ev, S.I., Approximation of Variational Eigenvalue Problems, Differ. Uravn., 2010, vol. 46, no. 7, pp. 1022–1032.

    MathSciNet  Google Scholar 

  3. Solov’ev, S.I., Approximation of Nonnegative-Definite Spectral Problems, Differ. Uravn., 2011, vol. 47, no. 8, pp. 1175–1182.

    MathSciNet  Google Scholar 

  4. Solov’ev, S.I., Nelineinye zadachi na sobstvennye znacheniya. Priblizhennye metody (Nonlinear Spectral Problems. Approximate Methods), Lambert Academic Publishing, 2011.

  5. Solov’ev, S.I., Finite-Element Methods for Nonself-Adjoint Spectral Problems, Uch. Zap. Kazan. Gos. Univ. Fiz.-Mat. Nauki, 2006, vol. 148, no. 4, pp. 51–62.

    Google Scholar 

  6. Solov’ev, S.I., Approximation of Eigenvalue Problem, in Setochnye metody dlya kraevykh zadach i prilozheniya (Grid Methods for Boundary Value Problems and Applications), Kazan, 2010, pp. 394–400.

  7. Solov’ev, S.I., Error of the Bubnov-Galerkin Method with Perturbations for Symmetric Spectral Problems with Nonlinear Occurrence of the Parameter, Zh. Vychisl. Mat. Mat. Fiz., 1992, vol. 32, no. 5, pp. 675–691.

    Google Scholar 

  8. Solov’ev, S.I., Approximation of Symmetric Spectral Problems with Nonlinear Occurrence of the Parameter, Izv. Vyssh. Uchebn. Zaved. Mat., 1993, no. 10, pp. 60–68.

  9. Solov’ev, S.I., Error Estimates for the Finite-Element Method for Symmetric Spectral Problems with Nonlinear Occurrence of the Parameter, Izv. Vyssh. Uchebn. Zaved. Mat., 1994, no. 9, pp. 70–77.

  10. Solov’ev, S.I., The Finite Element Method for Symmetric Eigenvalue Problems with Nonlinear Occurrence of the Spectral Parameter, Zh. Vychisl. Mat. Mat. Fiz., 1997, vol. 37, no. 11, pp. 1311–1318.

    MathSciNet  Google Scholar 

  11. Solov’ev, S.I., Eigenvalues of Shell Loaded by Masses, in Setochnye metody dlya kraevykh zadach i prilozheniya (Grid Methods for Boundary Value Problems and Applications), Kazan, 2010, pp. 400–408.

  12. Solov’ev, S.I., The Bubnov-Galerkin Method with Perturbations for Spectral Problems, in Issledovaniya po prikladnoi matematike i informatike (Investigations on Applied Mathematics and Computer Science), Kazan, 2006, no. 26, pp. 95–100.

  13. Solov’ev, S.I., Approximation of the Spectrum of Compact Operator, in Setochnye metody dlya kraevykh zadach i prilozheniya (Grid Methods for Boundary Value Problems and Applications), Kazan, 2007, pp. 253–256.

  14. Solov’ev, S.I., Superconvergence of Finite-Element Approximations of Eigenfunctions, Differ. Uravn., 1994, vol. 30, no. 7, pp. 1230–1238.

    Google Scholar 

  15. Solov’ev, S.I., Superconvergence of Finite-Element Approximations of Eigensubspaces, Differ. Uravn., 2002, vol. 38, no. 5, pp. 710–711.

    Google Scholar 

  16. Dautov, R.Z., Lyashko, A.D., and Solov’ev, S.I., Convergence of the Bubnov-Galerkin Method with Perturbations for Symmetric Spectral Problems with Nonlinear Occurrence of the Parameter, Differ. Uravn., 1991, vol. 27, no. 7, pp. 1144–1153.

    MathSciNet  MATH  Google Scholar 

  17. Riesz, F. and Sz.-Nagy, B., Leçoons d’analyse fonctionelle, Budapest: Académiai Kiadó, 1977. Translated under the title Lektsii po funktsional’nomu analizu, Moscow: Mir, 1977.

    Google Scholar 

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Authors and Affiliations

  1. Kazan Federal University, Kazan, Russia

    S. I. Solov’ev

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  1. S. I. Solov’ev
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Original Russian Text © S.I. Solov’ev, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 7, pp. 1042–1055.

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Solov’ev, S.I. Approximation of sign-indefinite spectral problems. Diff Equat 48, 1028–1041 (2012). https://doi.org/10.1134/S0012266112070130

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  • DOI: https://doi.org/10.1134/S0012266112070130

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