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We consider nodal-based Lagrangian interpolations for the finite elementapproximation of the Maxwell eigenvalue problem. The first approach introducedis a standard Galerkin method on Powell-Sabin meshes, which has recently beenshown to yield convergent approximations in two dimensions, whereas the othertwo are stabilized formulations that can be motivated by a variationalmultiscale approach. For the latter, a mixed formulation equivalent to theoriginal problem is used, in which the operator has a saddle point structure.The Lagrange multiplier introduced to enforce the divergence constraintvanishes in an appropriate functional setting. The first stabilized method weconsider consists of an augmented formulation with the introduction of a meshdependent term that can be regarded as the Laplacian of the multiplier of thedivergence constraint. The second formulation is based on orthogonalprojections, which can be recast as a residual based stabilization technique.We rely on the classical spectral theory to analyze the approximating methodsfor the eigenproblem. The stability and convergence aspects are inherited fromthe associated source problems. We investigate the numerical performance of theproposed formulations and provide some convergence results validating thetheoretical ones for several benchmark tests, including ones with smooth andsingular solutions.

Finite element formulations for Maxwell's eigenvalue problem using continuous Lagrangian interpolations