A Numerical Study of the Generalized FitzHugh–Nagumo Equation Using a Higher Order Galerkin Finite Element Method

In this article, we aim to present a higher-order finite element approximation and analysis of the generalized FitzHugh–Nagumo (gFHN) equation, which simplifies the complexities of the Hodgkin–Huxley model while maintaining a diverse range of excitation-propagation characteristics. Higher-order elements in finite element methods (FEMs) are known to produce higher-order approximations of solutions. However, very limited studies have been conducted on nonlinear problems. As a result, for the finite element analysis of the gFHN equation, we utilize the Galerkin method with quadratic Lagrange shape functions. The existence and uniqueness of the solution are established through the application of the Banach fixed-point theorem. Furthermore, to obtain the optimal order of convergence in $${{L}^{2}}(\Omega )$$ norm, a priori error estimates for the semi-discrete solution are derived. For temporal discretization, the Crank–Nicolson (CN) scheme is employed, while the predictor-corrector scheme effectively handles the non-linearity, resulting in $$\mathcal{O}(\Delta {{t}^{2}})$$ in the temporal direction. Also, the stability of the applied CN scheme is analyzed by using the energy technique. We validate the present scheme by implementing it on various numerical problems. The obtained results are compared with existing literature both numerically and graphically. The outcomes of these test problems confirm the effectiveness of quadratic shape functions and give rise to optimal convergence of third order.