We propose a new finite element approach, which is different than the classic Babuška–Osborn theory, to approximate Dirichlet eigenvalues. The problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for conforming finite elements is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed to compute the eigenvalues. A numerical example is presented to validate the theory.

中文翻译：

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Dirichlet特征值问题的一种新的有限元方法

我们提出了一种新的有限元方法，该方法不同于经典的Babuška-Osborn理论，以近似Dirichlet特征值。该问题被公式化为索引为零的全纯Fredholm算子函数的特征值问题。使用全纯算子函数的抽象逼近理论证明了有限元的收敛性。频谱指示符方法用于计算特征值。数值例子验证了该理论。