MIXED FINITE ELEMENT METHOD WITH GAUSS'S LAW ENFORCED FOR THE MAXWELL EIGENPROBLEM

A mixed finite element method is proposed for the Maxwell eigenproblem under the general setting. The method is based on a modification of the Kikuchi mixed formulation in terms of the electric field and the multiplier, with a mesh-dependent Gauss law of the electric field enforced in the formulation. The electric field is discretized by discontinuous elements, and the multiplier is always discretized by the lowest-order continuous nodal element (e.g., the linear element). The method renders four key features: the discrete de Rham complex exact sequence is not required and is replaced by a gradient inclusion condition of a low-order scalar element; i.e., the finite element space of the electric field includes the gradient of an auxiliary scalar H-1-conforming finite element space of low order; the discrete compactness property holds; the strong convergence of the Gauss law is ensured globally for the finite element solution; the method converges nearly optimally for both singular and smooth solutions. With these features, we develop a general analysis to prove that whether or not the discrete eigenmodes are spurious-free and spectral-correct attributes essentially to the first-order approximation property in the H(curl ; Omega) norm. As a direct application, except three lowest-order elements that do not have the first-order approximation property on nonaffine meshes, the first-kind Nedelec elements on nonaffine quadrilateral and hexahedral meshes and the second-kind Nedelec elements on affine and nonaffine quadrilateral and hexahedral meshes, including their discontinuous versions, are spurious-free and spectral-correct in the new mixed method, while these Nedelec elements generate spurious and incorrect discrete eigenmodes in the classical methods.