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 s of the –conforming finite element functions introduced by Nédélec 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. Received November 4, 1996 / Revised version received February 2, 1998