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

Smears, I; Vohralík, M; (2020) Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems. ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN) 10.1051/m2an/2020034. (In press). Green open access

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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.

Type: Article
Title: Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems
Open access status: An open access version is available from UCL Discovery
DOI: 10.1051/m2an/2020034
Publisher version: https://doi.org/10.1051/m2an/2020034
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10097104
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