TY - JOUR
T1 - On Discrete Energy Dissipation of Maxwell's Equations in a Cole-Cole Dispersive Medium
AU - Yin , Baoli
AU - Liu , Yang
AU - Li , Hong
AU - Zhang , Zhimin
JO - Journal of Computational Mathematics
VL - 5
SP - 980
EP - 1002
PY - 2023
DA - 2023/09
SN - 41
DO - http://doi.org/10.4208/jcm.2210-m2021-0257
UR - https://global-sci.org/intro/article_detail/jcm/22017.html
KW - Discrete energy dissipation, Crank-Nicolson scheme, Maxwell's equations, Shifted fractional trapezoidal rule, Mixed finite element methods.
AB - <p style="text-align: justify;">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&#39;s equations in a Cole-Cole dispersive medium regardless of the initial singularity
of the solution.</p>