Approximation of the grad div operator in nonconvex domains

In this paper we are dealing with the approximation of the grad-div operator in nonconvex 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 nonconvex domains, is the origin of major troubles in the numerical approximation of the problem. A mixed-type approximation, based on a projection procedure, is introduced and analyzed from the theoretical and numerical point of view. Several numerical experiments confirm that in presence of singularities the projection is needed in order to reproduce the features of the continuous problem.