An a priori error estimate for interior penalty discretizations of the Curl-Curl operator on non-conforming meshes

Abstract

We prove an a-priori error estimate for regularized Curl-Curl Problems which are discretized by the Interior Penalty/Nitsche’s Method on meshes non-conforming across interfaces. It is shown that the total error can be bounded by the best approximation error which in turn depends on the concrete choice of the approximation space $V_{h}$ . In this work we show that if $V_{h}$ is the space of edge functions of the first kind of order k we can expect (suboptimal) convergence $O(h^{k-1})$ as the mesh is refined. The numerical experiments in (Casagrande et al., SAM Report 2014-32, ETH Zurich, 2014) indicate that this bound is sharp for $k=1$ . Moreover it is shown that the regularization term can be made arbitrarily small without affecting the error in the $\lvert \cdot \rvert_{\mathbf {curl}}$ semi-norm. A numerical experiment shows that the regularization parameter can be chosen in a wide range of values such that, at the same time, the discrete problem remains solvable and the error due to regularization is negligible compared to the discretization error.