Submitted

A Family of Immersed Finite Element Spaces and Applications to Three Dimensional H(Curl) Interface Problems

Long Chen, Ruchi Guo, and Jun Zou

Submitted

Coming soon   Bibtex coming soon

ABSTRACT:

 Maxwell interface problems are of great importance in many
electromagnetic applications. Unfitted mesh methods are especially
attractive in 3D computation as they can circumvent generating complex
3D interface-fitted meshes. However, many unfitted mesh methods rely
on non-conforming approximation spaces, which may cause a loss of
accuracy for solving Maxwell equations, and the widely-used penalty
techniques in the literature may not help in recovering the optimal
convergence. In this article, we provide a remedy by developing
N\'ed\'elec-type immersed finite element spaces with a Petrov-Galerkin
scheme that is able to produce optimal-convergent solutions. To
establish a systematic framework, we analyze all the $H^1$,
$\bfH(\curl)$ and $\bfH(\div)$ IFE spaces and form a discrete de Rham
complex. Based on these fundamental results, we further develop a fast
solver using a modified Hiptmair-Xu preconditioner which works for
both the GMRES and CG methods.