Two-Grid Methods for Maxwell’s Equations in a Cole-Cole Dispersive Medium

Abstract

A two-grid method (TGM) for the time-dependent Maxwell's equations in Cole-Cole dispersive media with a fractional time derivative term is proposed. We employ the lowest Raviart-Thomas-Nédélec mixed finite elements to discrete the space. It is known that for these type of Nédélec edge finite elements, the standard TGM cannot be applied directly. Therefore, we modified the traditional TGM, and the discrete process can be divided into two steps. Firstly, we get the rough discrete solutions on the coarse mesh. At the same time, superconvergence results can be obtained by using a post-processing technique. Secondly, the superconvergent solutions on the coarse grid are added on the fine mesh as a correction, and the optimal error estimates could be obtained accordingly. Finally, the numerical experiments can verify that the theoretical results are correct and reasonable.