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

TL;DR

This paper develops optimal, asymptotically exact a posteriori error estimates for mixed Laplace eigenvalue problems by combining hypercircle techniques and novel flux post-processing, validated through numerical examples.

Contribution

It introduces a new local post-processing method for fluxes that enhances error estimation accuracy in mixed finite element approximations of Laplace eigenvalues.

Findings

Error estimates are asymptotically exact and optimal.

Numerical results confirm the effectiveness of the proposed error bounds.

Adaptive mesh refinement improves approximation quality.

Abstract

We derive optimal and asymptotically exact a posteriori error estimates for the approximation of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for mixed eigenvalue approximations with Raviart-Thomas Finite elements. In addition, we use the post-processings introduced for the eigenvalue and eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini Finite element. To combine these approaches, we define a novel additional local post-processing for the fluxes that appropriately modifies the divergence. Consequently, the new flux can be used to derive upper bounds and still shows good approximation properties. Numerical examples validate the theory and motivate the use of an adaptive mesh refinement.