Non-Conforming Nitsche Interfaces for Edge Elements in Curl–Curl-Type Problems

Abstract

In this article, a methodology to incorporate non-conforming interfaces between several conforming mesh regions is presented for Maxwell's curl-curl problem. The derivation starts from a general interior penalty discontinuous Galerkin formulation of the curl-curl problem and eliminates all interior jumps in the conforming parts but retains them across non-conforming interfaces. Therefore, it is possible to think of this Nitsche approach for interfaces as a specialization of discontinuous Galerkin on meshes, which are conforming nearly everywhere. The applicability of this approach is demonstrated in two numerical examples, including parameter jumps at the interface. A convergence study is performed for h-refinement, including the investigation of the penalization-(Nitsche-) parameter.

Sliding Non-Conforming Interfaces for Edge Elements in Eddy Current Problems

“…An important task is the correct handling of the surface terms in (11), where the integration has to be carried out on the actual interface facets F I ∈ F I h . There are several ways to accomplish this integration, e.g., using smooth B-splines as in [15] or remeshing the interface surface.…”

“…8. However, the oscillations can be drastically decreased by adapting the penalization parameter β in (11), without significantly deteriorating quantities, containing a time derivative of the primal solution quantity, as shown in Fig. 7.…”

“…From a theoretical point of view, this parameter has to be chosen "sufficiently large" (citation from [13] and [19]) to ensure convergence. For coplanar interfaces, it was heuristically shown in [11] that this Nitsche factor could be chosen in a large range without deteriorating the solution significantly. For curved interfaces, this range is narrowed, as shown in the two numerical examples in this work and also dependent on the mesh size and type (structured or unstructured).…”

“…Furthermore, all the applications in this work are simulated with Nédélec edge elements of the first kind and linear geometry elements. However, special care has to be taken, where the interface is placed, as described in [11].…”

A Virtual Finite Element Method for Two Dimensional Maxwell Interface Problems with a Background Unfitted Mesh

“…Indeed, Hiptmair et al in [10,11] show that using Nitsche's penalties on interface edges can only yield suboptimal convergence rates in both computation and analysis. Recently, this issue was further explored numerically in [41]. An alternative approach is to use immersed finite element methods in a Petrov-Galerkin formulation [24], where the standard conforming Nédélec space is used as the test function space to remove the non-conformity errors.…”