Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations
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 subse...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Published: |
Elsevier
2015
|
| Online Access: | https://eprints.nottingham.ac.uk/32682/ |
| _version_ | 1848794466548187136 |
|---|---|
| author | Şimşek, G. Wu, X. van der Zee, K.G. van Brummelen, E.H. |
| author_facet | Şimşek, G. Wu, X. van der Zee, K.G. van Brummelen, E.H. |
| author_sort | Şimşek, G. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:16:38Z |
| format | Article |
| id | nottingham-32682 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:16:38Z |
| publishDate | 2015 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-326822020-05-04T17:05:37Z https://eprints.nottingham.ac.uk/32682/ Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations Şimşek, G. Wu, X. van der Zee, K.G. van Brummelen, E.H. 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. Elsevier 2015-05-01 Article PeerReviewed Şimşek, G., Wu, X., van der Zee, K.G. and van Brummelen, E.H. (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 . pp. 83-109. ISSN 1879-2138 http://www.sciencedirect.com/science/article/pii/S0045782514004459 doi:10.1016/j.cma.2014.11.019 doi:10.1016/j.cma.2014.11.019 |
| spellingShingle | Şimşek, G. Wu, X. van der Zee, K.G. van Brummelen, E.H. Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations |
| title | Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations |
| title_full | Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations |
| title_fullStr | Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations |
| title_full_unstemmed | Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations |
| title_short | Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations |
| title_sort | duality-based two-level error estimation for time-dependent pdes: application to linear and nonlinear parabolic equations |
| url | https://eprints.nottingham.ac.uk/32682/ https://eprints.nottingham.ac.uk/32682/ https://eprints.nottingham.ac.uk/32682/ |