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On a renewed approach to a posteriori error bounds for approximate solutions of reaction-diffusion equations. (English) Zbl 1433.65261

Apel, Thomas (ed.) et al., Advanced finite element methods with applications. Selected papers from the 30th Chemnitz finite element symposium, St. Wolfgang/Strobl, Austria, September 25–27, 2017. Cham: Springer. Lect. Notes Comput. Sci. Eng. 128, 221-245 (2019).
Summary: 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 \(- \Delta u + \sigma u = f\) in \(\Omega\), \(u|_{\partial \Omega} = 0\), with an arbitrary constant or piece wise constant \(\sigma \geq 0\). 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 \(\mathbf{H}( \Omega, \operatorname{div})\). 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 [V. G. Korneev, “On the error control at numerical solution of reaction-diffusion equations”, arXiv:1711.02054] and very briefly in [V. G. Korneev, Dokl. Math. 96, No. 1, 380–383 (2017; Zbl 1376.65139); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 475, No. 6, 605–608 (2016)].
For the entire collection see [Zbl 1422.65009].

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J15 Second-order elliptic equations

Citations:

Zbl 1376.65139
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