Результаты исследований: Публикации в книгах, отчётах, сборниках, трудах конференций › статья в сборнике материалов конференции › Рецензирование
On a Renewed Approach to A Posteriori Error Bounds for Approximate Solutions of Reaction-Diffusion Equations. / Korneev, Vadim G.
Advanced Finite Element Methods with Applications - Selected Papers from the 30th Chemnitz Finite Element Symposium 2017. ред. / Thomas Apel; Ulrich Langer; Arnd Meyer; Olaf Steinbach. Springer Nature, 2019. стр. 221-245 (Lecture Notes in Computational Science and Engineering; Том 128).Результаты исследований: Публикации в книгах, отчётах, сборниках, трудах конференций › статья в сборнике материалов конференции › Рецензирование
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TY - GEN
T1 - On a Renewed Approach to A Posteriori Error Bounds for Approximate Solutions of Reaction-Diffusion Equations
AU - Korneev, Vadim G.
N1 - Publisher Copyright: © Springer Nature Switzerland AG 2019. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2019
Y1 - 2019
N2 - 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 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].
AB - 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 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].
UR - http://www.scopus.com/inward/record.url?scp=85069163824&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-14244-5_12
DO - 10.1007/978-3-030-14244-5_12
M3 - Conference contribution
AN - SCOPUS:85069163824
SN - 9783030142438
T3 - Lecture Notes in Computational Science and Engineering
SP - 221
EP - 245
BT - Advanced Finite Element Methods with Applications - Selected Papers from the 30th Chemnitz Finite Element Symposium 2017
A2 - Apel, Thomas
A2 - Langer, Ulrich
A2 - Meyer, Arnd
A2 - Steinbach, Olaf
PB - Springer Nature
T2 - 30th Chemnitz Finite Element Symposium, 2017
Y2 - 25 September 2017 through 27 September 2017
ER -
ID: 71957691