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 remain 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 us 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.