Superconvergence Of Edge Elements For Maxwell Equations

Maxwell equations not only predict the exsistance of electromagnetics, but also have widespread and useful application, e.g. cloaking, which is one of the central issues at present. E?ective posterior error estimators play an essential role in dealing with those problems. Superconvergence plays an esssential role in the construction of posterior error estimators. Therefore, the meaning for the study of superconvergence involving maxwell equations is obviouse.In this paper, we mainly focus on the analysis for superconvergence of lin- ear triangular and tetrohedral edge elements. In the theoretical proof, we adopt barrycenter coordinates whose de?nition and properties are presented in the chapter2.In the chapter 3, the object of our study is maxwell equations with four unknows involving metamaterials. In the theoretical analysis, we ?rstly obtain the superclose relation between the interpolation of the ?rst linear type of edge elemts and the ?nite element solution, which is used to prove the superconvergence result at the midpoint of interior edges after the averaging technique, one of the techniques of obtaining superconvergence. In the numerical computing, Fully discrete C-N scheme is used to deal with the weak form. The following numerical results test our proof.In the chapter 4, our subject is maxwell equation with only one unkown.Back in 1992, Monk pointed out the superconvergence phenomenon of the ?rst type linear tetrahedral elements in the discrete maximum norm. However, Monk did not prove any theretical investigation of this superconvergence phenomenon.Since then, superconvergence analysis has been carried out for edge elements on hexa- hedral grid and triangular grids. Untill now, the theoretical justi?cation of this superconvergence phenomenon for linear edge elemts on tetrahedral grids is still open. The paper is motivated by this open issue. In the chapter 4, we ?ll the gap by providing a theoretical analysis of this superconvergence phenomenon. We further provide some numerical results.