On an external finite element method for a second-order eigenvalue problem on a concave 2D-domain with Dirichlet boundary conditions
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.
second-order elliptic eigenvalue problems
scientific article; zbMATH DE number 803666
scientific article