### Abstract

In this paper we describe an iterative method for indefinite
saddle-point systems arising from mixed finite element discretisations
of 2nd-order elliptic boundary value problems subject to mixed
boundary conditions and posed over polyhedral three-dimensional
domains. The method is based on a decoupling of the vector of
velocities in the saddle-point system from the vector of pressures,
resulting in a symmetric positive definite (spd) velocity system and a
triangular pressure system.
The crucial step in this approach is the construction of the
divergence-free Raviart-Thomas-Nedelec elements
from the curls of Nedelec's edge elements. Because of
the large kernel of the curl-operator, this representation is not unique.
To find a basis we consider the graph made
up of the nodes and edges of the mesh and eliminate the
edge elements associated with a spanning tree in this graph. To
prove that this technique works in the general case considered here,
we employ fundamental results from Algebraic Topology and Graph Theory.

We also include some numerical experiments, where we solve the
(decoupled) velocity system by ILU-preconditioned conjugate gradients
and the pressure system by simple back substitutions. We compare our method
with a standard ILU-based block preconditioner for the original
saddle-point system, and we find that our method is faster by a factor
of at least 5 in all cases, with the greatest improvement occurring in
the nonuniform mesh case.