# Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems

## TL;DR

This paper introduces a simple, robust a posteriori error estimator for finite element solutions to singularly perturbed reaction-diffusion problems, applicable to arbitrary-order methods without special submeshes.

## Contribution

It presents a new equilibrated flux estimator that is robust and simple to implement for any finite element order, avoiding complex submesh constructions or estimator combinations.

## Key findings

- Provides guaranteed global error bounds without unknown constants.
- Achieves local robustness with respect to mesh size and perturbation parameters.
- Includes explicit bounds for inverse inequality constants on simplices.

## Abstract

We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction-diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction-diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.