Gradient recovery in adaptive finite-element methods for parabolic problems

Omar Lakkis, Tristan Pryer

Type: Article

Publication Date: 2011-06-10

Citations: 20

DOI: https://doi.org/10.1093/imanum/drq019

Abstract

We derive energy-norm a posteriori 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 in the appendix.

Locations

IMA Journal of Numerical Analysis - View
arXiv (Cornell University) - View - PDF
Pure (University of Bath) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+ PDF Chat	A Posteriori Error Control for Discontinuous Galerkin Methods for Parabolic Problems	2011	Emmanuil H. Georgoulis Omar Lakkis Juha M. Virtanen
+	Energy-norm error estimates for finite element discretization of parabolic problems	2015	Herbert Egger
+	Energy-norm error estimates for finite element discretization of parabolic problems	2015	Herbert Egger
+	Long time $L^\infty(L^2)$ a posteriori error estimates for fully discrete parabolic problems	2018	Oliver J. Sutton
+	Long time $L^\infty(L^2)$ a posteriori error estimates for fully discrete parabolic problems	2018	Oliver J. Sutton
+ PDF Chat	Recovery Techniques for Finite Element Methods	2024	Hailong Guo Zhimin Zhang
+	Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems	2006	Omar Lakkis Charalambos Makridakis
+ PDF Chat	Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems	2018	Alexandre Ern Iain Smears Martin Vohralı́k
+	Scalable Recovery-based Adaptation on Quadtree Meshes for Advection-Diffusion-Reaction Problems	2022	Pasquale Claudio Africa Simona Perotto Carlo de Falco
+	A posteriori error estimates for parabolic problems via elliptic reconstruction and duality	2007	Omar Lakkis Charalambos Makridakis
+	Massively parallel-in-space-time, adaptive finite element framework for non-linear parabolic equations	2016	Robert Dyja Baskar Ganapathysubramanian Kristoffer G. van der Zee
+ PDF Chat	Reliable anisotropic-adaptive discontinuous Galerkin method for simplified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml63" display="inline" overflow="scroll" altimg="si63.gif"><mml:msub><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold">N</mml:mi></mml:mrow></mml:msub></mml:math> approximations of radiative transfer	2018	Stefano Giani
+	A least-squares Galerkin gradient recovery method for fully nonlinear elliptic equations	2020	Omar Lakkis Amireh Mousavi
+ PDF Chat	Maximum-Norm Stability, Smoothing and Resolvent Estimates for Parabolic Finite Element Equations	2007	Vidar Thomée
+ PDF Chat	A Posteriori Error Estimates in the Maximum Norm for Parabolic Problems	2009	Alan Demlow Omar Lakkis Charalambos Makridakis
+ PDF Chat	Adaptive Finite Element Methods	2024	Andrea Bonito Claudio Canuto Ricardo H. Nochetto Andreas Veeser
+	Higher-Order GFDM for Linear Elliptic Operators	2023	Heinrich Kraus Jörg Kuhnert Pratik Suchde
+	Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes	2008	G. I. Shishkin
+ PDF Chat	A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems	2021	Emmanuil H. Georgoulis Omar Lakkis Thomas P. Wihler
+ PDF Chat	Parallel-In-Space-Time, Adaptive Finite Element Framework for Nonlinear Parabolic Equations	2018	Robert Dyja Baskar Ganapathysubramanian Kristoffer G. van der Zee

Works That Cite This (13)

Action	Title	Year	Authors
+ PDF Chat	Adaptive discontinuous Galerkin approximations to fourth order parabolic problems	2015	Emmanuil H. Georgoulis Juha M. Virtanen
+ PDF Chat	<i>A posteriori</i>error estimates for fully discrete fractional-step<i>ϑ</i>-approximations for parabolic equations	2015	Fotini Karakatsani
+ PDF Chat	Adaptive discontinuous Galerkin approximations to fourth order parabolic problems	2015	Emmanuil H. Georgoulis Juha M. Virtanen
+ PDF Chat	A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg-de Vries equation	2014	Ohannes A. Karakashian Charalambos Makridakis
+ PDF Chat	Conservative Galerkin methods for dispersive Hamiltonian problems	2021	James Jackaman Tristan Pryer
+	Conservative Galerkin methods for dispersive Hamiltonian problems	2018	James Jackaman Tristan Pryer
+ PDF Chat	A Finite Element Method for Nonlinear Elliptic Problems	2013	Omar Lakkis Tristan Pryer
+	A posteriori error estimates for the fractional-step -scheme for linear parabolic equations	2011	Fotini Karakatsani
+	Adaptive discontinuous Galerkin methods for fourth order problems	2010	Juha M. Virtanen
+ PDF Chat	Residual Estimates for Post-processors in Elliptic Problems	2021	Andreas Dedner Jan Giesselmann Tristan Pryer Jennifer K. Ryan

Works Cited by This (8)

Action	Title	Year	Authors
+	Adaptive procedure with superconvergent patch recovery for linear parabolic problems	2017	Saulius Ziukas N.‐E. Wiberg
+ PDF Chat	A Posteriori Error Estimates in the Maximum Norm for Parabolic Problems	2009	Alan Demlow Omar Lakkis Charalambos Makridakis
+	Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems	2006	Omar Lakkis Charalambos Makridakis
+ PDF Chat	A posteriori error estimates by recovered gradients in parabolic finite element equations	2008	Dmitriy Leykekhman Lars B. Wahlbin
+ PDF Chat	On superconvergence techniques	1987	Michal Křı́žek Pekka Neittaanmäki
+ PDF Chat	Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for $H^1$-stability of the $L^2$-projection onto finite element spaces	2001	Carsten Carstensen
+	Partial Differential Equations	1999	Steve Abbott Lawrence C. Evans
+ PDF Chat	A Posteriori Error Control for Discontinuous Galerkin Methods for Parabolic Problems	2011	Emmanuil H. Georgoulis Omar Lakkis Juha M. Virtanen