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

## TL;DR

This paper develops a duality-based a posteriori error estimation method for nonlinear parabolic PDEs discretized with IMEX-Galerkin schemes, enabling adaptive mesh refinement and time-stepping.

## Contribution

It introduces a novel error decomposition technique that separates spatial and temporal errors for nonlinear parabolic PDEs using IMEX discretization.

## Key findings

- Effective error estimates for heat and Allen-Cahn equations
- Adaptive algorithms improve accuracy and efficiency
- Error decomposition guides adaptive refinement

## 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.