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Convergence rates for intermediate problems

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Abstract

Convergence rate estimates are derived for a variant of Aronszajn-type intermediate problems that is both computationally feasible and known to be convergent for problems with nontrivial essential spectrum. Implementation of these derived bounds is discussed in general and illustrated on differential eigenvalue problems. Convergence rates are derived for the commonly used method of simple truncation.

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References

  1. ARONSZAJN, N.: Approximation methods for eigenvalues of completely continuous symmetric operators. Proc. Symposium on Spectral Theory and Differential Problems, Oklahoma A&M College, Stillwater, 179–202 (1951)

    Google Scholar 

  2. BAZLEY, N., FOX, D. W.: Truncation in the method of intermediate problems for lower bounds to eigenvalues. J. Res. Nat. Bur. Stds.65B. 105–111 (1961)

    Google Scholar 

  3. BAZLEY, N., FOX, D. W.: Methods for lower bounds to frequencies of continuous elastic systems. J. Appl. Math. and Phys. (ZAMP)17(1). 1–37 (1966)

    Google Scholar 

  4. BAZLEY, N., FOX, D. W.: Comparison operators for lower bounds to eigenvalues. J. Reine Angew. Math.223, 142–149 (1966)

    Google Scholar 

  5. BEATTIE, C.: Some convergence results for intermediate problems with essential spectra. M.S.E. Research Center preprint series#65. Johns Hopkins University Applied Physics Laboratory, Laurel 1982

    Google Scholar 

  6. BEATTIE, C., GREENLEE, W. M.: Convergence theorems for intermediate problems. Proc. Roy. Soc. Edinburgh Sect. A100, 107–122 (1985). Also correction: Proc. Roy. Soc. Edinburgh Sect. A104 349–350, (1986)

    Google Scholar 

  7. BELLMAN, R.E.: Stability Theory of Differential Equations. New York: McGraw-Hill 1953

    Google Scholar 

  8. BIRKHOFF, G., FPIX, G.: Accurate eigenvalue computations for elliptic problems. Proc. Numerical Solution of Field Problems in Continuum Physics. Symp. on Appl. Math., Vol 2. American Mathematical Society, 111–151 (1970)

  9. BROWN, R.D.: Variational approximation methods for eigenvalues: convergence theorems. Banach Center Publications, Vol. 13, Warsaw, 543–558 (1984)

    Google Scholar 

  10. COURANT, R., HILBERT D.: Methods of Mathematical Physics, Vol. 1. New York: Interscience 1953

    Google Scholar 

  11. FIX, G.: Orders of convergence of the Rayleigh-Ritz and Weinstein-Bazley methods. Proc. Nat. Acad. Sci. U.S.A.61, 1219–1223 (1968)

    Google Scholar 

  12. GREENLEE, W. M.: Rate of convergence in singular perturbations. Ann. Inst. Fourier (Grenoble)18, 135–191 (1968)

    Google Scholar 

  13. GREENLEE, W. M.: On fractional powers of operators in Hilbert space. Acta Sci. Math. (Szeged)36, 55–61 (1974)

    Google Scholar 

  14. GREENLEE, W. M.: A convergent variational method of eigenvalue approximation. Arch. Rational Mech.81(3), 279–287 (1983)

    Google Scholar 

  15. KADLEC, J.: The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain. Czech. Math. J.14(89), 386–393 (1964) (In Russian)

    Google Scholar 

  16. KATO, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer 1966

    Google Scholar 

  17. LIONS, J. L., MAGENES, E.: Problèmes aux limites non homogènes et applications, Vol. 1. Paris: Dunod (1968)

    Google Scholar 

  18. POZNYAK, L. T.: Estimation of the rate of convergence of a variant of the method of intermediate problems. Ž. Vyčisl. Mat. i Mat. Fiz.8, 1117–1126 (1968); USSR Computational Math. and Math. Phys.8, 246–260 (1968)

    Google Scholar 

  19. POZNYAK, L. T.: The convergence of the Bazley-Fox process, and an estimate of the rate of this convergence. Ž. Vyčisl. Mat. i Mat. Fiz.9, 860–872 (1968); USSR Computational Math. and Math. Phys.9, 167–184 (1969)

    Google Scholar 

  20. REED, M., SIMON, B.: Methods of Modern Mathematical Physics, Vol. 2. New York: Academic Press 1975

    Google Scholar 

  21. WEINBERGER, H.: Error estimation in the Weinstein method for eigenvalues. Proc. Amer. Math. Soc.3, 643–646 (1952)

    Google Scholar 

  22. WEINBERGER, H.: A theory of lower bounds for eigenvalues. Tech. Note BN-103, Inst. for Fluid Dyn. and Appl. Math., Univ. of Maryland, College Park 1959

    Google Scholar 

  23. WEINBERGER, H.: Variational Methods for Eigenvalue Approximation. Philadelphia: SIAM 1974

    Google Scholar 

  24. WEINSTEIN, A., STENGER, W.: Methods of Intermediate Problems for Eigenvalues. New York: Academic Press 1972

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute and State University, 24061, Blacksburg, Virginia, USA

    Christopher Beattie

  2. Department of Mathematics, University of Arizona, 85721, Tucson, Arizona, USA

    W. M. Greenlee

Authors
  1. Christopher Beattie
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  2. W. M. Greenlee
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Additional information

The work of the first author was partially supported by AFOSR Grant 84-0326. The work of the second author was partially supported by NSF Grant MCS-8301402

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Beattie, C., Greenlee, W.M. Convergence rates for intermediate problems. Manuscripta Math 59, 209–227 (1987). https://doi.org/10.1007/BF01158047

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  • DOI: https://doi.org/10.1007/BF01158047

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