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

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