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.

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