Superconvergence analysis for Maxwell's equations in dispersive media

Abstract

Abstract. In this paper, we consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell's equations when dispersive media are involved.

Superconvergent estimates of conforming finite element method for nonlinear time‐dependent Joule heating equations

“…As we know, the superconvergence analysis is always a hot topic for the numerical methods of PDEs, such as, the second order elliptic equation [13][14][15][16], linear elasticity problem [17], nonlinear Schrödinger equation [18], parabolic equation [19], Maxwell's equation [20][21][22], Stokes equation [23,24], nonlinear Sobolev equation [25] and so forth.…”

Stability and superconvergence analysis of the FDTD scheme for the 2D Maxwell equations in a lossy medium

“…Furthermore, in [22] it was demonstrated by numerical experiments that the FDTD scheme is of superconvergence. It should be noted that global superconvergence in the L 2 norm of the finite element methods has been established in [15,16] for the 3D Maxwell equations. Since the FDTD scheme can be obtained as a special case using the lowest-order rectangular edge element, the results in [15,16] may explain the superconvergence of the FDTD scheme easily.…”

“…It should be noted that global superconvergence in the L 2 norm of the finite element methods has been established in [15,16] for the 3D Maxwell equations. Since the FDTD scheme can be obtained as a special case using the lowest-order rectangular edge element, the results in [15,16] may explain the superconvergence of the FDTD scheme easily. Remis [28] gave practical stability conditions for the FDTD scheme on nonuniform grids in terms of the minimum primary and dual stepsizes and the maximum electromagnetic wave speed by studying the spectral norm of the coefficient matrix.…”

Unified Analysis of Time Domain Mixed Finite Element Methods for Maxwell's Equations in Dispersive Media

“…However, almost all studies are restricted to the simple medium case. Very recently, we initiated the error analysis of TDFEM for dispersive media [20][21][22]. In [22], we discussed the superconvergence results for some semi-discrete schemes developed for dispersive medium models.…”

“…Very recently, we initiated the error analysis of TDFEM for dispersive media [20][21][22]. In [22], we discussed the superconvergence results for some semi-discrete schemes developed for dispersive medium models. While in [20], we analyzed the backward Euler mixed finite element methods (FEMs) for three most popular dispersive medium models.…”