Eigenvalue approximation by mixed and hybrid methods

Abstract

Rate of convergence estimates are derived for the approximation of eigenvalues and eigenvectors by mixed and hybrid methods. Several closely related abstract results on spectral approximation are proved. These results are then applied to a variety of finite element methods of mixed and hybrid type: a mixed method for 2nd order problems, mixed methods for 4th order problems, a hybrid method for 2nd order problems, and two mixed methods for the Stokes eigenvalue problem.

A Superconvergence result for mixed finite element approximations of the eigenvalue problem

“…Based on the abstract theory of [2,20], the recent important results [4,5], and [6], Lemma 6.1, and (2.6), (2.7) and (3.12), we know a priori error estimates for the eigenpair approximation…”

“…There are several works for the second order elliptic eigenvalue problems in the mixed formulation and their numerical methods such as Babuška and Osborn [1,2,21], Mercier et al [20], Boffi et al [5,6], Boffi [4], etc. Osborn [21], Mercier et al [20], Boffi et al [5], Boffi [4] give the analysis for the eigenpair approximations by mixed/hybrid finite element methods based on the theory of compact operator (see, e.g., [9]).…”

“…Osborn [21], Mercier et al [20], Boffi et al [5], Boffi [4] give the analysis for the eigenpair approximations by mixed/hybrid finite element methods based on the theory of compact operator (see, e.g., [9]). …”

Isoparametric mixed finite element approximation of eigenvalues and eigenvectors of 4th order eigenvalue problems with variable coefficients

“…In all papers [14,22,23,29] on mixed finite element analysis of 4th order eigenvalue problems, it has been assumed that neither any numerical integration is essential nor any approximation of the boundary is necessary (since the boundary of the convex domain is a polygonal one in all the cases, the convexity of the domain being a requirement for the regularity of the solution [18,21,24]). But in many situations,we are to consider convex domains with curved boundary Γ.…”

A hybrid finite element method to compute the free vibration frequencies of a clamped plate

“…Convergence and error estimâtes are then derived by a Rayleigh-Ritz principle ; since we cannot prove the uniform convergence of the approximate operators T h to T, we check an approximability property over suitable finite dimensional spaces, similar to Property P 2 by Rappaz [17] (actually, our approach is analogous to the one he uses for a second order problem ; see also Rappaz [16], and Anselone [1] whose notion of " collective compactness ", however, does not apply in the present case). Spectral approximations by methods of hybrid type are investigated by Kolata [11], Mercier-Rappaz [13], Mercier-Osborn-Rappaz-Raviart [14], who ail assume T h -> T in norm.…”