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An A Priori Error Estimate for Interior Penalty Discretizations of the Curl-Curl Operator on Non-Conforming Meshes

by R. Casagrande and R. Hiptmair

(Report number 2014-40)

Abstract
We prove an a-priori error estimate for conductivity-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, Winkelmann, Hiptmair and Ostrowski, SAM Report 2014-32, ETH Z�rich, 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 \(|\cdot|_{curl}\) 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.

Keywords: Discontinuous Galerkin, Sliding Interface, Non-Conforming Mesh, FEM, Magnetostatics, Curl-Curl Operator, Interior Penalty

@Techreport{CH14_590,
  author = {R. Casagrande and R. Hiptmair},
  title = {An A Priori Error Estimate for Interior Penalty Discretizations of the Curl-Curl Operator on Non-Conforming Meshes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-40.pdf },
  year = {2014}
}

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