Two new finite element schemes and their analysis for modeling of wave propagation in graphene

In this paper, we investigate a system of governing equations for modeling wave propagation in graphene. Compared to our previous work (Yang et al., 2020) , here we re-investigate the governing equations by eliminating two auxiliary unknowns from the original model. A totally new stability for the model is established for the first time. Since the finite element scheme proposed in Yang et al. (2020) is only first order in time, here we propose two new schemes with second order convergence in time for the simplified modeling equations. Discrete stabilities inheriting exactly the same form as the continuous stability are proved for both schemes. Convergence error estimates are also established for both schemes. Numerical results are presented to justify our theoretical analysis.