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Nonconforming finite element approximations of the Steklov eigenvalue problem. (English) Zbl 1190.65168
This paper deals with nonconforming finite element approximations of the Steklov eigenvalue problem, $$-\Delta u+u=0$$ in $$\Omega$$ with $$\partial u/\partial \nu = \lambda u$$ on $$\partial\Omega$$, where $$\Omega$$ is a bounded convex polygonal domain in $$\mathbb{R}^2$$. The authors consider three typical elements: Crouzeix-Raviart element for triangular elements, and $$Q_1^{\text{rot}}$$ and $$EQ_1^{\text{rot}}$$ elements for rectangular elements. First, the authors establish error estimates for both approximate eigenvalues and eigenfunctions. Then, they prove that the $$j$$th eigenvalue computed by $$EQ_1^{\text{rot}}$$ gives a lower bound of the $$j$$th exact eigenvalue, whereas the Crouzeix-Raviart element and $$Q_1^{\text{rot}}$$ provide lower bounds for larger eigenvalues. Finally, numerical examples to confirm theoretical results are presented.

##### 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 65N15 Error bounds for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs
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