We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell-Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method we consider consists of an augmented formulation with the introduction of a mesh dependent term that can be regarded as the Laplacian of the multiplier of the divergence constraint. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems. We investigate the numerical performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

翻译：本文考虑基于节点的拉格朗日插值用于Maxwell特征值问题的有限元逼近。首先介绍的方法是标准Galerkin方法在Powell-Sabin网格上的应用，近期已被证明可在二维情形下获得收敛逼近；另外两种是可采用变分多尺度方法推导的稳定化格式。对于后者，采用与原始问题等价的混合格式，其中算子具有鞍点结构。为强制散度约束而引入的Lagrange乘子可在适当函数空间中消失。第一种稳定化方法考虑增强型格式，引入可视为散度约束乘子拉普拉斯算子的网格依赖项。第二种方法基于正交投影，可重新表述为残差型稳定化技术。我们借助经典谱理论分析特征值问题的逼近方法，其稳定性与收敛性特性继承自相关源问题。通过包含光滑解和奇异解在内的多个基准测试，验证所提格式的数值性能，并提供与理论结果相吻合的收敛性结果。