On a Renewed Approach to A Posteriori Error Bounds for Approximate Solutions of Reaction-Diffusion Equations

Abstract

We discuss a new approach to obtaining the guaranteed, robust and consistent a posteriori error bounds for approximate solutions of the reaction-diffusion problems, modelled by the equation − Δu + σu = f in Ω, u|_∂Ω = 0, with an arbitrary constant or piece wise constant σ ≥ 0. The consistency of a posteriori error bounds for solutions by the finite element methods assumes in this paper that their orders of accuracy in respect to the mesh size h coincide with those in the corresponding sharp a priori bounds. Additionally, it assumes that for such a coincidence it is sufficient that the testing fluxes possess only the standard approximation properties without resorting to the equilibration. Under mild assumptions, with the use of a new technique, it is proved that the coefficient before the L ²-norm of the residual type term in the a posteriori error bound is ${\mathcal O}(h)$ uniformly for all testing fluxes from admissible set, which is the space H(Ω, div). As a consequence of these facts, there is a wide range of computationally cheap and efficient procedures for evaluating the test fluxes, making the obtained a posteriori error bounds sharp. The technique of obtaining the consistent a posteriori bounds was exposed in [arXiv:1711.02054v1 [math.NA] 6 Nov 2017] and very briefly in [Doklady Mathematics, 96 (1), 2017, 380–383].