Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems
For researchers in numerical analysis and computational PDEs, this provides a robust error estimator that overcomes limitations of previous approaches, particularly for singularly perturbed problems.
This work presents a simple and robust equilibrated flux a posteriori error estimator for singularly perturbed reaction-diffusion problems, achieving guaranteed global upper bounds and local efficiency without unknown constants, submeshes, or estimator combinations. The method applies to arbitrary-order finite elements and any space dimension.
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.