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

  • Vadim G. Korneev
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 128)

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 L2-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].

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vadim G. Korneev
    • 1
  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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