Regularization of Maxwell equations, corner singularities and approximation

The aim of this paper is to describe different formulations for the time-harmonic scattering of electromagnetic waves, with special emphasis on the consequences of the choice of the formulation on the finite elements discretization. Despite the fact that any solution of the problem is divergence-free, one must take account explicitly of this condition, for the associated operator is not strongly elliptic. Roughly speaking, one may handle this constraint either by duality or by penalty. Actually, the special form of Maxwell´s equations makes the Lagrange multiplier a priori known, and the solution independent of the penalization coefficient. Another particular feature of the problem comes from the low regularity of the electromagnetic field in the vicinity of edges and conical points, from which it follows that special care must be taken of the choice of the function space for the penalty method