2017/06/13 by X. Wu, Wu, Xunxun, Kristoffer van der Zee +5 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1706.04281
openalex publication_date 2017/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.