论文标题
求解具有最佳收敛的二维H（卷曲） - 纤维化接口系统
Solving Two Dimensional H(curl)-elliptic Interface Systems with Optimal Convergence On Unfitted Meshes
论文作者
论文摘要
在本文中，我们开发和分析了最低学位的第一个家庭nédélec元素的有限元方法，用于解决麦克斯韦界面问题，该问题是由$ \ mathbf {h}建模的（\ text {curl}）$ - 椭圆形方程。为了最佳地捕获跳跃条件，我们构建和使用$ \ Mathbf {h}（\ text {curl}）$沉浸有限元（ife）在接口元素上功能，同时继续使用标准的nédélec函数对所有非接口元素。我们为IFE函数建立了一些重要属性，包括根据边缘自由度的Unisorlacence，与$ H^1 $ IFE函数有关的确切序列和最佳近似功能。为了达到最佳收敛速率，我们采用了一种彼得罗夫 - 盖尔金方法，其中仅将IFE函数用作试验函数，并且将标准的nédélec函数用作可以消除非符合误差的测试函数。我们分析了在特定条件下的INF-SUP条件，并显示了最佳收敛速率，这些收敛速率也通过数值实验验证。
In this article, we develop and analyze a finite element method with the first family Nédélec elements of the lowest degree for solving a Maxwell interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use $\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface elements while keep using the standard Nédélec functions on all the non-interface elements. We establish a few important properties for the IFE functions including the unisolvence according to the edge degrees of freedom, the exact sequence relating to the $H^1$ IFE functions and the optimal approximation capabilities. In order to achieve the optimal convergence rates, we employ a Petrov-Galerkin method in which the IFE functions are only used as the trial functions and the standard Nédélec functions are used as the test functions which can eliminate the non-conformity errors. We analyze the inf-sup conditions under certain conditions and show the optimal convergence rates which are also validated by numerical experiments.
