We consider Galerkin approximations of eigenvalue problems for holomorphic Fredholm operator functions for which the operators do not have the structure “coercive+compact”. In this case the regularity (in the vocabulary of discrete approximation schemes) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. The technique is based on the knowledge of a suitable Test function operator. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework can be successfully applied to analyze e.g. complex scaling/perfectly matched layer methods, problems involving sign-changing coefficients due to meta-materials and also (boundary element) approximations of Maxwell-type equations. We demonstrate the application of our framework to the Maxwell eigenvalue problem for a conductive material.

我们考虑全纯 Fredholm 算子函数的特征值问题的伽辽金近似，其中算子不具有“强制+紧凑”结构。在这种情况下，伽辽金近似的规则性（在离散近似方案的词汇中）不是无条件满足的，收敛问题是微妙的。我们报告了一种证明近似规律性的技术，该技术适用于广泛的特征值问题。该技术基于对合适T的知识est 函数运算符。特别地，我们引入了弱T-coercivity和T-compatibility的概念，并证明了对于弱T-coercive算子，Galerkin近似的T-兼容性意味着它们的规律性。我们的框架可以成功地应用于分析例如复杂的缩放/完美匹配层方法、涉及由于超材料引起的符号变化系数的问题以及麦克斯韦方程的（边界元）近似。我们展示了我们的框架在导电材料的麦克斯韦特征值问题中的应用。