A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations
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 nonlinea...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2018
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| Online Access: | https://eprints.nottingham.ac.uk/55020/ |
| _version_ | 1848799102528126976 |
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| author | Wu, X. van der Zee, Kristoffer George Şimşek, G. van Brummelen, E.H. |
| author_facet | Wu, X. van der Zee, Kristoffer George Şimşek, G. van Brummelen, E.H. |
| author_sort | Wu, X. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:30:20Z |
| format | Article |
| id | nottingham-55020 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:30:20Z |
| publishDate | 2018 |
| publisher | Society for Industrial and Applied Mathematics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-550202018-09-18T08:13:58Z https://eprints.nottingham.ac.uk/55020/ A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations Wu, X. van der Zee, Kristoffer George Şimşek, G. van Brummelen, E.H. 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. Society for Industrial and Applied Mathematics 2018-07-17 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/55020/1/wu_vanderZee_simsek_vanBrummelen_NottmEprint2018.pdf Wu, X., van der Zee, Kristoffer George, Şimşek, G. and van Brummelen, E.H. (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 . ISSN 1064-8275 (In Press) A posteriori error estimate Duality-based error estimate IMEX scheme Implicit-explicit schemes Space-time error Adaptivity Parabolic PDE |
| spellingShingle | A posteriori error estimate Duality-based error estimate IMEX scheme Implicit-explicit schemes Space-time error Adaptivity Parabolic PDE Wu, X. van der Zee, Kristoffer George Şimşek, G. van Brummelen, E.H. A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations |
| title | A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations |
| title_full | A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations |
| title_fullStr | A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations |
| title_full_unstemmed | A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations |
| title_short | A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations |
| title_sort | a-posteriori error estimation and adaptivity for nonlinear parabolic equations using imex-galerkin discretization of primal and dual equations |
| topic | A posteriori error estimate Duality-based error estimate IMEX scheme Implicit-explicit schemes Space-time error Adaptivity Parabolic PDE |
| url | https://eprints.nottingham.ac.uk/55020/ |