Exponential convergence of hp-FEM for Maxwell\'s equations with weighted regularization in polygonal domains

Abstract

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h Version of Finite Elements, [20]. We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.