We discuss the approximation of the eigensolutions associated with the Maxwell eigenvalues problem in the framework of least-squares finite elements. We write the Maxwell curl curl equation as a system of two first order equation and design a novel least-squares formulation whose minimum is attained at the solution of the system. The eigensolution are then approximated by considering the eigenmodes of the underlying solution operator. We study the convergence of the finite element approximation and we show several numerical tests confirming the good behavior of the method. It turns out that nodal elements can be successfully employed for the approximation of our problem also in presence of singular solutions.

翻译：本文讨论在最小二乘有限元框架下，与Maxwell特征值问题相关的特征解的逼近问题。我们将Maxwell旋度-旋度方程写为两个一阶方程组成的系统，并设计一种新颖的最小二乘公式，其最小值在该系统的解处达到。随后，通过考虑底层解算子的特征模态来逼近特征解。我们研究了有限元逼近的收敛性，并展示了若干数值实验，证实了该方法的良好表现。结果表明，即使在存在奇异解的情况下，节点型单元也能成功用于逼近我们的问题。