Gradient recovery in adaptive finite element methods for parabolic problems

Omar Lakkis; Tristan Pryer

doi:10.1093/imanum/drq019

Gradient recovery in adaptive finite element methods for parabolic problems

Omar Lakkis, Tristan Pryer

Department of Mathematical Sciences

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Abstract

We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.

Original language	English
Pages (from-to)	246-278
Number of pages	32
Journal	IMA Journal of Numerical Analysis
Volume	32
Issue number	1
Early online date	10 Jun 2011
DOIs	https://doi.org/10.1093/imanum/drq019
Publication status	Published - 31 Jan 2012

Bibliographical note

6 figures, 1 sketch, appendix with pseudocode

Keywords

math.NA
math.AP
65M60, 65Y20, 65G20

Access to Document

10.1093/imanum/drq019

0905.2764v2.PDF
This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record Lakkis, O & Pryer, T 2012, 'Gradient recovery in adaptive finite element methods for parabolic problems', IMA Journal of Numerical Analysis, vol. 32, no. 1, pp. 246-278 is available online at: https://academic.oup.com/imajna/article/32/1/246/719988
Accepted author manuscript, 4.69 MB

Cite this

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AU - Pryer, Tristan

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Y1 - 2012/1/31

N2 - We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.

AB - We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.

KW - math.NA

KW - math.AP

KW - 65M60, 65Y20, 65G20

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JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 1

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Gradient recovery in adaptive finite element methods for parabolic problems

Abstract

Bibliographical note

Keywords

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