On Discrete Energy Dissipation of Maxwell's Equations in a Cole-Cole Dispersive Medium (Journal of Computational Mathematics, 2023)
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 dispersivemedium. Several numerical formulas that approximate the time fractional derivatives areinvestigated based on this criterion, including the L1 formula, the fractional BDF-2, andthe shifted fractional trapezoidal rule (SFTR). Detailed error analysis is provided withinthe framework of time domain mixed finite element methods for smooth solutions. Theconvergence results and discrete energy dissipation law are confirmed by numerical tests.For nonsmooth solutions, the method SFTR can still maintain the optimal convergenceorder at a positive time on uniform meshes. Authors believe this is the first appearancethat a second-order time-stepping method can restore the optimal convergence rate forMaxwell's equations in a Cole-Cole dispersive medium regardless of the initial singularityof the solution.
Keywords: Discrete energy dissipation; Crank-Nicolson scheme; Maxwell's equations; Shifted fractional trapezoidal rule; Mixed finite element methods.