Constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods

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Abstract

This paper introduces a method of constructing nonconforming finite elements which can produce lower bounds for the eigenvalues of elliptic operators. Based on such nonconforming discrete eigenfunctions, we propose a simple method to produce upper bounds of eigenvalues. More precisely, we construct conforming approximations of exact eigenfunctions by a projection average interpolation operator of nonconforming discrete eigenfunctions. After showing the approximation property of the projection average interpolation operator, we prove that the Rayleigh quotients of the aforementioned conforming approximations are convergent to the exact eigenvalues from above. Finally, we combine lower and upper bounds of eigenvalues to obtain high accuracy approximations of eigenvalues. Numerical examples verify our theoretical results.

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Correspondence to Jun Hu.

The first author was supported by the NSFC Projects 11271035 and 11421101; the second author was supported by the NSFC Key Project 91430213 and International Science and Technology Cooperation Program of China Project 2010DFR00700; the third author was supported by the NSFC Project 11401416.

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Hu, J., Huang, Y. & Shen, Q. Constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods. Numer. Math. 131, 273–302 (2015). https://doi.org/10.1007/s00211-014-0688-z