A simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems

Abstract

A simple flux reconstruction for finite element solutions of reactiondiffusion 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.

A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem

“…Remark 1. Theorem 2 finally justifies the use of (18) in the proof of Lemma 1 since u − u h ∈ H 1 0 (Ω) d and since…”

“…The idea for using the additional term η P for the consideration of the lower order term of the third equation in ( 1) is taken from [18]. Moreover, the additional terms…”