Eigenvalue approximation from below using non-conforming finite elements.(English)Zbl 1187.65125

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators, with emphasis on obtaining lower bounds. In addition, this article also contains some new materials for eigenvalue approximations of the Laplace operator, which include: 6mm
1)
the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below;
2)
the proof of the fact that the non-conforming $$EQ^{\text{rot}}_1$$ element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;
3)
the explanation of the phenomena that numerical eigenvalues $$\lambda _{1,h }$$ and $$\lambda _{3,h }$$ of the non-conforming $$Q^{\text{rot}}_1$$ element approximate the true eigenvalues from below for the L-shaped domain.
Finally, several unsolved problems are listed.

MSC:

 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs 65N15 Error bounds for boundary value problems involving PDEs
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References:

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