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Guaranteed Lower Bounds for Eigenvalues of Elliptic Operators

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Abstract

In this paper, we provide a method to produce guaranteed lower bounds for eigenvalues of 2m-th order elliptic operators in n dimensions for \( m\le n\), especially for elliptic operators with variable coefficients. This method is based on the corresponding Morley–Wang–Xu elements in literature and a unified way to estimate the explicit constants related to the \(L^2\) error estimates for the interpolation of Morley–Wang–Xu elements.

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References

  1. Armentano, M.G., Duran, R.G.: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. ETNA 17, 93–101 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Babǔska, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 641–787. North Holland, Amsterdam (1991)

    Google Scholar 

  3. Brenner, S.C.: Poincare–Friedrichs inequalities for piecewise \(H^1\) functions. SIAM J. Numer. Anal. 41, 306–324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carstensen, C., Gallistl, D.: Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126, 33–51 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carstensen, C., Gedicke, J.: An oscillation-free adaptive FEM for symmetric eigenvalue problems. Numer. Math. 118, 401–427 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carstensen, C., Gedicke, J.: Guaranteed lower bounds for eigenvalues. Math. Comput. 83, 2605–2629 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chavel, I., Feldman, E.A.: An optimal Poincaré inequality for convex domains of non-negative curvature. Arch. Ration. Mech. Anal. 65, 263–273 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Gallistl, D.: Adaptive Finite Element Computation of Eigenvalues. PhD thesis, der Humboldt-Universität zu, Berlin (2014)

  10. Hu, J., Huang, Y.Q., Lin, Q.: Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods. J. Sci. Comput. 61, 196–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, J., Huang, Y.Q., Shen, Q.: Constructing both lower and upper bounds for the eigenvalues of the elliptic operators by the nonconforming finite element methods. Numer. Math. 131, 273–302 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu, J., Ma, R.: Guaranteed Lower and Upper Bounds for Eigenvalues of Second Order Elliptic Operators in Any Dimension. arxiv:1406.6520 (2014)

  13. Hu, J., Shi, Z.C.: The best \(L^2\) norm error estimate of lower order finite element methods for the fourth order problem. J. Comput. Math. 30, 449–460 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Larrson, S., Thomée, V.: Partial Differential Equations with Numerical Methods. Springer, New York (2008)

    Google Scholar 

  15. Laugesen, R.S., Siudeja, B.A.: Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality. J. Differ. Equ. 249, 118–135 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, Y.A.: Lower approximation of eigenvalue by the nonconforming finite element method. Math. Numer. Sin. 30, 195–200 (2008). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  17. Lin, Q., Xie, H.: Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Probl. Imaging 7, 795–811 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, X.: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. Comput. 267, 341–355 (2015)

    MathSciNet  Google Scholar 

  19. Luo, F., Lin, Q., Xie, H.: Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods. Sci. China Math. 55, 1069–1082 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rannacher, R.: Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33, 23–42 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Strang, G., Fix, G.: An Analysis of the Finite Element Method, 2nd edn. Wellesley-Cambridge Press, Wellesley (2008)

    MATH  Google Scholar 

  23. Stummel, F.: Basic compactness properties of nonconforming and hybrid finite element spaces. RAIRO Anal. Numer. 4, 81–115 (1980)

    MathSciNet  MATH  Google Scholar 

  24. Wang, M., Xu, J.C.: Minimal finite-element spaces for 2m-th order partial differential equations in \(R^n\). Math. Comput. 82, 25–43 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, Y., Zhang, Z., Lin, F.: Eigenvalue approximation from below using non-conforming finite elements. Sci. China Math. 53, 137–150 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, Z., Yang, Y., Chen, Z.: Eigenvalue approximation from below by Wilson’s elements. Math. Numer. Sin. 29, 319–321 (2007). (in Chinese)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Jun Hu & Rui Ma

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Yunqing Huang

Authors
  1. Jun Hu
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  2. Yunqing Huang
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  3. Rui Ma
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Corresponding author

Correspondence to Rui Ma.

Additional information

This work was supported by National Natural Science Foundation of China (Grant Nos. 11271035, 91430213 and 11421101).

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Hu, J., Huang, Y. & Ma, R. Guaranteed Lower Bounds for Eigenvalues of Elliptic Operators. J Sci Comput 67, 1181–1197 (2016). https://doi.org/10.1007/s10915-015-0126-0

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  • DOI: https://doi.org/10.1007/s10915-015-0126-0

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