Averaging for superconvergence: Verification and application of 2D edge elements to Maxwell's equations in metamaterials

In this paper, a simple averaging technique for edge elements is justified to improve the standard convergence rate (hence achieving superconvergence) when they are used to solve the Maxwell's equations. For simplicity, here we focus on the lowest-order triangular edge element, which is widely used in practice. Though there exists no natural superconvergence points for the numerical solution obtained on such an edge element, superconvergence can be obtained after a simple average of solutions over the neighboring elements. A comprehensive analysis for the lowest-order triangular edge element is carried out, and one-order higher convergence rate than the standard interpolation error estimate is proved for the averaged solution at midpoints of those interior edges of parallelograms (formed by two triangles), i.e., superconvergence happens at the parallelogram centers. We also provide detailed analysis to explain why several cases do not have superconvergence. Extensive numerical results consistent with our analysis are presented.