A Simple Approach to Reliable and Robust A Posteriori Error Estimation for Singularly Perturbed Problems
Provides a reliable and robust a posteriori error estimation technique for singularly perturbed problems, which are challenging due to boundary layers.
The paper presents a flux reconstruction method for finite element solutions of singularly perturbed reaction-diffusion problems that yields fully computable upper bounds on the energy norm of error, with local efficiency and robustness proven for any mesh size and reaction coefficient.
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.