Approximation of grad-div operator in non-convex domains

In this paper we are dealing with the approximation of the grad-div operator in non-convex polygonal domains. A penalization strategy is considered in order to obtain a formulation of the original eigenproblem which is associated with an elliptic operator. However the presence of singular eigensolutions, in the case of non-convex domains, is the origin of major troubles in the numerical approximation of the problem. A mixed-type approximation, based on an under-integrated scheme, is introduced and analyzed from the theoretical and numerical point of view. Several numerical experiments confirm that in presence of singularities the under-integrated scheme is needed in order to reproduce the feautures of the continuous problem.