Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions
Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions
复制标题
二维麦克斯韦特征值问题的拉格朗日有限元收敛
DOI：
10.1093/imanum/drab104
复制
发表时间：
2022
影响因子：
2.1
通讯作者：
Neilan, Michael
中科院分区：
文献类型：
--
作者：
Boffi, Daniele;Guzmán, Johnny;Neilan, Michael
我们考虑二维麦克斯韦本征值问题的有限元近似。我们证明，在一定的设置，收敛的离散特征值使用拉格朗日有限元。特别是，我们证明了收敛在三种情况下：分段线性元素的Powell-Sabin三角剖分，分段二次元素的Clough-Tocher三角剖分和分段四次（和更高）的一般形状规则三角剖分的元素。我们提供了支持理论结果的数值实验。计算还表明，在一般的三角剖分，特征值近似是非常敏感的近奇异顶点，即，恰好落在两条“几乎”直线上的顶点。
We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two ‘almost’ straight lines.