A Third Order Linearized BDF Scheme for Maxwell's Equations with Nonlinear Conductivity Using Finite Element Method

Abstract

In this paper, we study a third order accurate linearized backward differential formula (BDF) type scheme for the nonlinear Maxwell's equations, using the N\u00e9delec finite element approximation in space. A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. An optimal $L^2$ error estimate is presented, via a linearized stability analysis for the numerical error function, under a condition for the time step, $\tau \leq C^*_0h^2$ for a fixed constant $C^*_0$. Numerical results are provided to confirm our theoretical analysis and demonstrate the high order accuracy and stability (convergence) of the linearized BDF finite element method.