Superconvergence Analysis for the Maxwell's Equations in Debye Medium with a Thermal Effect

Abstract

In this paper, a mixed finite element method is investigated for the Maxwell's equations in Debye medium with a thermal effect. In particular, in two dimensional case, the zero order N\u00e9d\u00e9lec element $(Q_{01}\times Q_{10})$, the piecewise constant space $Q_0$ element, and the bilinear element $Q_{11}$ are used to approximate the electric field E and the polarization electric field P, the magnetic field H, and the temperature field $u$, respectively. With the help of the high accuracy results, mean-value technique and interpolation postprocessing approach, the convergent rate $\mathcal{O}(\tau+h^2)$ for global superconvergence results are obtained under the time step constraint $\tau=\mathcal{O}(h^{1+\gamma}),$ $ \gamma>0$ by using the linearized backward $Euler$ finite element discrete scheme. At last, a numerical experiment is given to verify the theoretical analysis and the validity of our method.