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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.

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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