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 Babus̆ka‐Osborn spectral theory for compact operators. Numerical results have illustrated the performance of the stabilized method and confirmed the theoretical results obtained.