Some Remarks on the a Posteriori Error Analysis of the Mixed Laplace Eigenvalue Problem

Abstract

In this note we consider the a posteriori error analysis of mixed finite element approximations to the Laplace eigenvalue problem based on local postprocessing. The estimator makes use of an improved L 2 approximation for the Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) finite element methods. For the BDM method we also obtain improved eigenvalue convergence for postprocessed eigenvalues. We verify the theoretical results in several numerical examples.

A note on asymptotically exact a posteriori error estimates for mixed Laplace eigenvalue problems

“…As the author is not aware of a detailed proof that can be found in the literature, it will be given in the appendix in section 6. Note however, that these results are already used for example in [4] (without proof). The resulting super convergence reads as…”

“…Unfortunately, the method of [6] has the drawback of a reduced accuracy of the eigenvalue and the eigenfunction since the Raviart-Thomas space does not allow an optimal approximation. In [4] (using ideas from [20]) the same authors (and collaborators) were able to achieve an optimal approximation by using the Brezzi-Douglas-Marini (BDM) Finite element instead. However, this was only possible by paying the price of unknown constants in the a posteriori estimates since the additional term in (1) is not of higher order any more as was also observed in [5].…”

“…The goal of this work is to combine the advantages from both works. More precisely, we use the BDM Finite Element and the post-processing techniques for the eigenvalue and the eigenfunction as in [4], and consider modifications of the approaches from [6] to derive asymptotically exact upper bounds. For the latter, we introduce an additional (local) post-processing for the flux variable σ h , where we correct its divergence to fit the additional term in (1), which consequently converges again with higher order.…”