Approximation of the grad div operator in nonconvex domains

In this paper we are dealing with the approximation of the grad-div operato r in nonconvex polygonal domains. A penalization strategy is considered in order to obtain a formulation of the original eigenproblem which is associa ted with an elliptic operator. However the presence of singular eigensoluti ons, in the case of nonconvex domains, is the origin of major troubles in t he numerical approximation of the problem. A mixed-type approximation, base d on a projection procedure, is introduced and analyzed from the theoretica l and numerical point of view. Several numerical experiments confirm that i n presence of singularities the projection is needed in order to reproduce the features of the continuous problem.