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A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations

Wu, X.; van der Zee, K.G.; Simsek, G.; van Brummelen, E.H.

A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations Thumbnail


Authors

X. Wu

KRISTOFFER VAN DER ZEE KG.VANDERZEE@NOTTINGHAM.AC.UK
Professor of Numerical Analysis &computational Applied Mathematics

G. Simsek

E.H. van Brummelen



Abstract

While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology for duality-based a posteriori error estimation for nonlinear parabolic PDEs, where the full discretization of the PDE relies on the use of an implicit-explicit (IMEX) time-stepping scheme and the finite element method in space. The main result in our work is a decomposition of the error estimate that allows to separate the effects of spatial and temporal discretization error, and which can be used to drive adaptive mesh refinement and adaptive time-step selection. The decomposition hinges on a specially-tailored IMEX discretization of the dual problem. The performance of the error estimates and the proposed adaptive algorithm is demonstrated on two canonical applications: the elementary heat equation and the nonlinear Allen-Cahn phase-field model.

Citation

Wu, X., van der Zee, K., Simsek, G., & van Brummelen, E. (2018). A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations. SIAM Journal on Scientific Computing, 40(5), A3371–A3399. https://doi.org/10.1137/17M1133968

Journal Article Type Article
Acceptance Date Jul 17, 2018
Online Publication Date Oct 9, 2018
Publication Date Oct 9, 2018
Deposit Date Aug 22, 2018
Publicly Available Date Aug 22, 2018
Journal SIAM Journal on Scientific Computing
Print ISSN 1064-8275
Electronic ISSN 1095-7197
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 40
Issue 5
Pages A3371–A3399
DOI https://doi.org/10.1137/17M1133968
Keywords A posteriori error estimate, Duality-based error estimate, IMEX scheme, Implicit-explicit schemes, Space-time error, Adaptivity, Parabolic PDE
Public URL https://nottingham-repository.worktribe.com/output/1042993
Publisher URL https://epubs.siam.org/doi/abs/10.1137/17M1133968
Additional Information Copyright 2018 Society for Industrial and Applied Mathematics
Contract Date Aug 22, 2018

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