Abstract. In this paper we consider second order scalar elliptic boundary value problems posed over threedimensional domains and their discretization by means of mixed RaviartThomas finite elements [18]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal RaviartThomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as s of the conforming finite element functions introduced by Ndlec [43]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof.