On Discrete Energy Dissipation of Maxwell's Equations in a Cole-Cole Dispersive Medium

Abstract

A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell’s equations in a Cole-Cole dispersive medium. Several numerical formulas that approximate the time fractional derivatives are investigated based on this criterion, including the L1 formula, the fractional BDF-2, and the shifted fractional trapezoidal rule (SFTR). Detailed error analysis is provided within the framework of time domain mixed finite element methods for smooth solutions. The convergence results and discrete energy dissipation law are confirmed by numerical tests. For nonsmooth solutions, the method SFTR can still maintain the optimal convergence order at a positive time on uniform meshes. Authors believe this is the first appearance that a second-order time-stepping method can restore the optimal convergence rate for Maxwell's equations in a Cole-Cole dispersive medium regardless of the initial singularity of the solution.