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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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