The Crank–Nicolson–Galerkin FEM for Darcy–Forchheimer problem coupled with time-dependent convection–diffusion–reaction equation

Abstract

This work addresses a time-dependent convection–diffusion–reaction equation coupled with the Darcy–Forchheimer system. By combining the Crank–Nicolson scheme in time with a finite element discretization in space, we propose a linearized, fully discrete scheme that is unconditionally stable. We first establish the well-posedness of the continuous system, followed by the analysis of the semi-discrete solution. Optimal error estimates for the velocity field, pressure, and concentration are derived for both the semi-discrete and fully discrete schemes. Finally, numerical simulations are presented to validate the theoretical results and demonstrate the efficiency of the proposed method.