On an external finite element method for a second-order eigenvalue problem on a concave 2D-domain with Dirichlet boundary conditions
type: Item
instance of: scholarly article
title: On an external finite element method for a second-order eigenvalue problem on a concave 2D-domain with Dirichlet boundary conditions
zbMATH Open document ID: 0836.65112
DOI: 10.1016/0096-3003(94)00155-W
author: Michèle Vanmaele
published in: Applied Mathematics and Computation
publication date: 1996-04-29T00:00:00Z
review text: The author deals with second-order elliptic eigenvalue problems in two variables on non-convex domains, given in terms of a variational equation in the space \(H^1_0 (\Omega)\). Using linear, triangular finite elements, the approximate problem is formulated on a polygonal domain \(\Omega_h\) where, in general, \(\Omega_h \not \subset \Omega\) so that the finite element space \(V_h\) is not contained in \(H^1_0(\Omega)\). In an earlier paper of \textit{M. Vanmaele} and \textit{A. Ženíšek} [RAIRO, Modélisation Math. Anal. Numér. 27, No. 5, 565-589 (1993; Zbl 0792.65086)], estimates for both the approximate eigenvalues and eigenfunctions are derived. In that paper, the proofs of the estimate heavily rest upon the \(O(h^2)\)-estimate for the error of some projection (Theorem 4.8). In this paper, a new proof of that estimate is given proceeding similarly as in the proof of the well-known Aubin-Nitsche lemma. Moreover, an optimal \(O(h^2)\)-estimate for the approximate eigenvalues is obtained under weaker conditions on both the boundary and the coefficients in the variational equation.
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 803666
zbMATH Keywords: error estimate
zbMATH Keywords: second-order elliptic eigenvalue problems
zbMATH Keywords: linear, triangular finite elements
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Aubin-Nitsche lemma
reviewed by: Waldemar Velte
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 803666