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.
Highlights
1 Introduction In this work we study the D, magnetostatic boundary value problem
7 Conclusion and outlook We have proved a-priori error estimators for the interior penalty formulation of the regularized curl-curl source problem ( )-( ); If the solution is approximated by k-th order edge functions we can expect at least convergence of order O(hk– )
For k = no convergence was observed in a numerical experiment [ ], which implies that our result is sharp
Summary
Mesh assumptions: We assume that the elements are shape regular in the sense of Ciarlet: There is a constant σmax, independent of h, such that for all h ∈ H and for all T ∈ Th we have hT ρT. We set Vh∗ := V ∗ + Vh. Note that, because A and ∇ × A are in H (P ) the traces of A and ∇ × A are well defined on the faces of the mesh elements In order to bound the term T we first note that the global Thomas-Raviart interpolation operator wh(∇ × A| i )i= , is well defined by [ ], Lemma. This can be implemented by using a hierarchical basis for the edge functions [ ]
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