Abstract 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.

中文翻译：

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二维麦克斯韦特征值问题的拉格朗日有限元收敛

摘要 我们考虑二维麦克斯韦特征值问题的有限元逼近。我们证明，在某些情况下，离散特征值的收敛使用拉格朗日有限元。特别是，我们证明了在三种情况下的收敛性：Powell-Sabin 三角剖分上的分段线性元素、Clough-Tocher 三角剖分上的分段二次元和一般形状规则三角剖分上的分段四次（和更高）元素。我们提供支持理论结果的数值实验。计算还表明，在一般三角剖分中，特征值近似对几乎奇异的顶点非常敏感，即恰好落在两条“几乎”直线上的顶点。