Optimal Error Estimates for Nédélec Edge Elements for Time-Harmonic Maxwell's Equations

Abstract

In this paper, we obtain optimal error estimates in both $L^2$-norm and $\boldsymbol{H}$(curl)-norm for the Nédélec edge finite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the $L^2$ error estimates into the $L^2$ estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.