Eigenvalue problems

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

“…) is a direct consequence of the estimate (2.11) and Theorem 7.1 in [3], with a constant C depending on the constant in (2.11) (which is independent of t) and on the inverse of the distance from µ t to the rest of the spectrum of T t . Now, according to Lemma 2.1, (2.6) implies that for t small enough this distance can be bounded below in terms of the distance from µ 0 to the rest of the spectrum of T 0 , which obviously does not depend on t.…”

“…For t > 0 fixed, the spectral theory for compact operators in [3] can be directly applied to prove convergence of the eigenpairs of T th to those of T t . However, further considerations are needed to show that the error estimates do not deteriorate as t becomes small.…”

“…For conforming methods (which is not our case) the convergence of the eigenfunctions in H 1 directly yields a double order of convergence for the eigenvalues (Lemma 9.1 in [3]). Alternatively, when such double order holds for the L 2 convergence in the corresponding load problem without further assumptions on the regularity of the solution, abstract spectral theory can be used to prove a similar result even for non-conforming methods (see for instance Section 7 of [3]).…”

“…The aim of this paper is to analyze one of these methods, that of lowest order, when used to approximate the free vibration bending modes of a plate. Being non-conforming, the spectral theory based on minimum-maximum principles (see Section 8 of [3]) cannot be directly applied to this method. Instead, our analysis will be based on the abstract spectral theory for compact operators as stated in Section 7 of [3].…”

“…Alternatively, when such double order holds for the L 2 convergence in the corresponding load problem without further assumptions on the regularity of the solution, abstract spectral theory can be used to prove a similar result even for non-conforming methods (see for instance Section 7 of [3]). …”