Error analysis of Crank–Nicolson-FEM for Fitzhugh–Nagumo system with Robin boundary conditions

We investigate the convergence properties Crank–Nicolson scheme coupled with the finite element approximation of the Fitzhugh–Nagumo system. This model describes the dynamics of excitable media, such as nerve cells, and has applications in various fields, including neuroscience and cardiac modeling. The study focuses on the time splitting algorithm, which combines implicit time-stepping using Crank–Nicolson with piecewise finite element spatial discretization. The well-posedness and error estimates for both the temporal and fully discretization errors are established. This type of boundary conditions are incorporated into the formulation, allowing for non-homogeneous fluxes at the domain boundaries. Numerical experiments validate the theoretical findings.