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A framework of verified eigenvalue bounds for self-adjoint differential operators

Author

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  • Liu, Xuefeng

Abstract

For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues. In the case of the Laplacian operator, by applying Crouzeix–Raviart finite elements, an efficient algorithm is developed to bound the eigenvalues for the Laplacian defined in 1D, 2D and 3D spaces. Moreover, for nonconvex domains, for which case there may exist singularities of eigenfunctions around re-entrant corners, the proposed algorithm can easily provide eigenvalue bounds. By further adopting the interval arithmetic, the explicit eigenvalue bounds from numerical computations can be mathematically correct.

Suggested Citation

  • Liu, Xuefeng, 2015. "A framework of verified eigenvalue bounds for self-adjoint differential operators," Applied Mathematics and Computation, Elsevier, vol. 267(C), pages 341-355.
    • Handle: RePEc:eee:apmaco:v:267:y:2015:i:c:p:341-355
    DOI: 10.1016/j.amc.2015.03.048
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    Cited by:

    1. Xuqing Zhang & Yu Zhang & Yidu Yang, 2020. "Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element," Mathematics, MDPI, vol. 8(8), pages 1-23, July.
    2. Liu, Xuefeng & You, Chun’guang, 2018. "Explicit bound for quadratic Lagrange interpolation constant on triangular finite elements," Applied Mathematics and Computation, Elsevier, vol. 319(C), pages 693-701.
    3. Kazuaki Tanaka & Taisei Asai, 2022. "A posteriori verification of the positivity of solutions to elliptic boundary value problems," Partial Differential Equations and Applications, Springer, vol. 3(1), pages 1-25, February.

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