论文标题
2d中麦克斯韦特征值问题的拉格朗日有限元的收敛
Convergence of Lagrange finite elements for the Maxwell Eigenvalue Problem in 2D
论文作者
论文摘要
我们考虑二维中麦克斯韦特征值问题的有限元近似值。在某些情况下，我们证明了使用Lagrange有限元素的离散特征值的收敛。特别是，我们在三种情况下证明了融合：鲍威尔的分段线性元素 - 萨宾三角剖分，克拉夫三角剖分的分段二次元素，以及一般形状规范三角形的分段四分之一（以及更高的四分之一元素）。我们提供支持理论结果的数值实验。这些计算还表明，在一般三角形上，特征值近似对几乎奇异的顶点非常敏感，即恰好落在两个“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.
