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

Результат исследований: Публикации в книгах, отчётах, сборниках, трудах конференцийстатья в сборнике материалов конференциирецензирование

Аннотация

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].

Язык оригиналаанглийский
Название основной публикацииAdvanced Finite Element Methods with Applications - Selected Papers from the 30th Chemnitz Finite Element Symposium 2017
РедакторыThomas Apel, Ulrich Langer, Arnd Meyer, Olaf Steinbach
ИздательSpringer Nature
Страницы221-245
Число страниц25
ISBN (печатное издание)9783030142438
DOI
СостояниеОпубликовано - 2019
Событие30th Chemnitz Finite Element Symposium, 2017 - St. Wolfgang, Австрия
Продолжительность: 25 сен 201727 сен 2017

Серия публикаций

НазваниеLecture Notes in Computational Science and Engineering
Том128
ISSN (печатное издание)1439-7358
ISSN (электронное издание)2197-7100

конференция

конференция30th Chemnitz Finite Element Symposium, 2017
Страна/TерриторияАвстрия
ГородSt. Wolfgang
Период25/09/1727/09/17

Предметные области Scopus

  • Моделирование и симуляция
  • Технология (все)
  • Дискретная математика и комбинаторика
  • Теория оптимизации
  • Вычислительная математика

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