# A posteriori error analysis for the mixed Laplace eigenvalue problem

## TL;DR

This paper develops a posteriori error estimates for mixed finite element methods solving the Laplace eigenvalue problem, providing reliable and efficient error bounds with local reconstructions to improve numerical accuracy.

## Contribution

It introduces a new local reconstruction technique for the primal variable, enabling effective a posteriori error estimation in mixed Laplace eigenvalue approximations.

## Key findings

- Error estimator provides an upper bound for the approximation error.
- The error estimator is shown to be locally efficient.
- Reconstruction is performed on vertex patches for improved accuracy.

## Abstract

This paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem with homogeneous Dirichlet boundary conditions. In particular, the resulting error estimator constitutes an upper bound for the error and is shown to be local efficient. Therefore, we present a reconstruction in the standard $H^1_0$-conforming space for the primal variable of the mixed Laplace eigenvalue problem. This reconstruction is performed locally on a set of vertex patches.