A Bochev-Dohrmann-Gunzburger stabilized method for Maxwell eigenproblem

A stabilized mixed finite element method is proposed for solving the Maxwell eigenproblem in terms of the electric field and the multiplier. Using the Bochev-Dohrmann-Gunzburger stabilization, we introduce some ad hoc stabilizing parameters for stabilizing the kernel-coercivity of the electric field and for stabilizing the inf-sup condition of the multiplier. We show that the stabilized mixed method is stable and convergent, with applications to some lowest-order edge elements on affine rectangular and cuboid mesh and on nonaffine quadrilateral mesh which fail in the classical methods. In particular, we prove the uniform convergence for guaranteeing spectral-correct and spurious-free discrete eigenmodes from the Babuska-Osborn spectral theory for compact operators. Numerical results have illustrated the performance of the stabilized method and confirmed the theoretical results obtained.