# A Simple Approach to Reliable and Robust A Posteriori Error Estimation for Singularly Perturbed Problems

## TL;DR

This paper introduces a simple flux reconstruction method for finite element solutions that provides reliable, fully computable a posteriori error bounds for singularly perturbed reaction-diffusion problems, ensuring robustness and efficiency.

## Contribution

The paper proposes a straightforward flux reconstruction technique that yields guaranteed error bounds and efficient local error indicators, even in the presence of singular perturbations.

## Key findings

- Provides fully computable upper bounds on error in energy norm.
- Ensures local efficiency and robustness of error indicators.
- Works even when Galerkin orthogonality is violated.

## Abstract

A simple flux reconstruction for finite element solutions of reaction-diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction-diffusion problem. The flux reconstruction is based on simple, independent post-processing operations over patches of elements in conjunction with standard Raviart--Thomas vector fields and gives upper bounds even in cases where Galerkin orthogonality might be violated. If Galerkin orthogonality holds, we prove that the corresponding local error indicators are locally efficient and robust with respect to any mesh size and any size of the reaction coefficient, including the singularly perturbed limit.