The Crank–Nicolson mixed finite element method for the improved system of time-domain Maxwell’s equations

Abstract

In this paper, we mainly establish the Crank–Nicolson (CN) mixed finite element (MFE) method for the system of time-domain Maxwell’s equations with a lossy medium. To this end, we first deduce the two-dimensional (2D) improved system of time-domain Maxwell’s equations with the lossy medium, construct the time semi-discretized CN format for the 2D improved system of Maxwell’s equations, and discuss the existence, stability, and error estimates of time semi-discrete solutions. And then, we construct a fully discretized CN MFE (FDCNMFE) format with both symmetry and positive definiteness, and analyze the existence, stability, and errors of FDCNMFE solutions. Furthermore, we make use of some numerical tests to verify the correctness of theoretical results in the 2D case. Finally, as a generalization, we also give the corresponding methods and results for three-dimensional (3D) improved system of time-domain Maxwell’s equations in the appendix A. This paper has advantages in at least the following four aspects. First, the study process herein is to discrete time and then discrete space variables so as to save the semi-discretized MFE discussion for spatial variables, which is a new attempt. Second, the magnetic field strength in the improve system of Maxwell’s equations is independent of the electric intensity such that their MFE subspaces can avoid the constraint of the Babuška-Brezzi (B-B) condition and can be freely chosen, so that the FDCNMFE solutions reach optimal order error estimates. Third, the 2D and 3D FDCNMFE formats reach the second-order accuracy in time and have both symmetry and positive definiteness so as to be solved easily. Fourth, both time semi-discretized CN solutions and the FDCNMFE solutions are unconditionally stable and unconditionally convergent.