The Superconvergence Analysis For Edge Elements To Maxwell’s Equations Involving Metamaterial In2-D

In this paper, mixed finite element method is used to solve the governing modelof time dependent Maxwell’s equations involving metamaterial in2-D. The mixedfinite element spaces here are the first family of linear Nédélecelements. Thoughthere exits no natural superconvergence points at the midpoints of the interior edgesfor the triangular mesh, superconvergence happens here after a simple average tech-nique.In our computational experiment, we find that after a simple average of thefinite element solution over two over the two neighboring elements, an (h~2) errorcan be obtained between the analytic solution and the average value of the finiteelement solutions at the midpoints of interior edges over the two neighboring ele-ments and this error is one order higher than the standard L2error O(h) betweenthe analytic solution and the finite element solution. Thus, We obtain the super-convergence results at the midpoints of the interior edges and we also provide thecorresponding comprehensive analysis.In this paper, we also present a result that the centers of the equilateral trian-gular meshes are points without the phenomenon of superconvergence. The corre-sponding analysis and computational experiments are also carried out.Furthermore, we extend our result to the lowest order Nédélecrectangularmesh. We’ve already known that superconvergence happens at the centers of therectangular mesh. In this paper, after a simple average technique, we find thatsuperconvergence also happens at the midpoints of the interior edges of rectangularmeshes and the corresponding theoretical proof is presented.