Superconvergence Analysis of High-Order Rectangular Edge Elements for Time-Harmonic Maxwell’s Equations

In this paper, high-order rectangular edge elements are used to solve the two dimensional time-harmonic Maxwell’s equations. Superconvergence for the Nédélec interpolation at the Gauss points is proved for both the second and third order edge elements. Using this fact, we obtain the superconvergence results for the electric field E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {E}$$\end{document}, magnetic field H and curlE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$curl\mathbf {E}$$\end{document} in the discrete l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^2$$\end{document} norm when the Maxwell’s equations are solved by both elements. Extensive numerical results are presented to justify our theoretical analysis.