Developing and Analyzing New Unconditionally Stable Finite Element Schemes for Maxwell’s Equations in Complex Media

In this paper we propose and analyze an unconditionally stable leapfrog method for Maxwell’s equations that removes the time step constraint for stability, which makes the proposed scheme more efficient in computation and easier in algorithm implementation compared to the same order Crank–Nicolson scheme. We also prove the unconditional stability and the optimal error estimate of the proposed scheme. To show the generality of our technique, we further develop similar unconditionally stable leapfrog schemes for other complicated Maxwell’s equations. Numerical results are presented to justify our theoretical analysis and demonstrate the practical applications in simulating wave propagation in metamaterials.