Decoupling three-dimensional mixed problems using divergence-free finite elements (Q2780601)

The author considers the iterative solution of saddle point problems resulting from the lowest order mixed finite element discretization of second order elliptic boundary value problems on polyhedral domains. The basic idea is to decouple the discrete velocities from the discrete pressures using the construction of a basis for the divergence free Raviart-Thomas-Nédélec elements by means of the curls of Nédélec's edge elements. The construction of the basis uses the concept of spanning trees from graph theory and fundamental results from homology theory. Numerical results indicate a better efficiency of the proposed iterative solver compared to standard block preconditioning techniques.