Discretization with variable time steps of an evolution equation with a positive-type memory term
Journal of Computational and Applied Mathematics
Global superconvergence for Maxwell's equations
Mathematics of Computation
Petrov--Galerkin Methods for Linear Volterra Integro-Differential Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
Journal of Computational Physics
Interior penalty method for the indefinite time-harmonic Maxwell equations
Numerische Mathematik
Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
Finite difference/spectral approximations for the time-fractional diffusion equation
Journal of Computational Physics
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Interior Penalty Discontinuous Galerkin Method for Maxwell's Equations in Cold Plasma
Journal of Scientific Computing
Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media
Journal of Computational Physics
Multiscale Computations for 3D Time-Dependent Maxwell's Equations in Composite Materials
SIAM Journal on Scientific Computing
Incorporating the Havriliak-Negami dielectric model in the FD-TD method
Journal of Computational Physics
Journal of Scientific Computing
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In this paper, we consider the time-dependent Maxwell's equations when Cole-Cole dispersive medium is involved. The Cole-Cole model contains a fractional time derivative term, which couples with the standard Maxwell's equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1-26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572-591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533-1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12-20], we propose two fully discrete mixed finite element schemes for the Cole-Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.