论文标题
一个沉浸式有限元空间和应用到三维$ \ mathbf {h}（\ text {curl}）$接口问题的家族
A family of immersed finite element spaces and applications to three dimensional $\mathbf{H}(\text{curl})$ interface problems
论文作者
论文摘要
麦克斯韦界面问题在许多电磁应用中非常重要。未固定的网格方法在3D计算中特别有吸引力，因为它们可以规避生成复杂的3D接口拟合网格。但是，许多未固定的网格方法依赖于不合格的近似空间，这可能会导致求解麦克斯韦方程的准确性丧失，并且文献中广泛使用的惩罚技术可能无法帮助恢复最佳收敛。在本文中，我们通过开发Nédélec-type浸入有限的元素空间，并使用Petrov-Galerkin方案来提供一种补救措施，该方案能够生成最佳的溶液。要建立系统的框架，我们分析了所有$ h^1 $，$ \ mathbf {h}（\ text {curl}）$和$ \ mathbf {h}（\ text {div}）$ if spaces并形成一个离散的de rham复合体。基于这些基本结果，我们使用修改的HiptMair-XU预处理程序进一步开发了快速求解器，该预处理适用于GMRE和CG方法。
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 Nédélec-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 $H^1$, $\mathbf{H}(\text{curl})$ and $\mathbf{H}(\text{div})$ 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.
