Multilevel methods for mixed finite elements in three dimensions

In this paper we consider second order scalar elliptic boundary value problems posed over three-dimensional domains and their discretization by means of mixed Raviart-Thomas 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 Raviart-Thomas vector fields, A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as curls of the H(curl)conforming finite element functions introduced by Nedelec [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.