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.