Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations

?im?ek, G.; Wu, X.; van der Zee, K.G.; van Brummelen, E.H.

Abstract

We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator.

Citation

?im?ek, G., Wu, X., van der Zee, K., & van Brummelen, E. (2015). Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations. Computer Methods in Applied Mechanics and Engineering, 288, https://doi.org/10.1016/j.cma.2014.11.019

• Journal Article Type: Article
• Acceptance Date: Nov 13, 2014
• Online Publication Date: Dec 3, 2014
• Publication Date: May 1, 2015
• Deposit Date: Apr 7, 2016
• Publicly Available Date: Apr 7, 2016
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 288
• DOI: https://doi.org/10.1016/j.cma.2014.11.019
• Public URL: https://nottingham-repository.worktribe.com/output/748752
• Publisher URL: http://www.sciencedirect.com/science/article/pii/S0045782514004459