The preconditioning which we will present for these systems consists in three stages. First of all the equations for pressure and saturation are locally decoupled on each element. This decoupling aims not only at reducing the coupling, but also at concentrating in the "pressure block" the elliptic part of the system which is then in the second stage preconditioned by efficient methods like AMG. The third step finally consists in "recoupling" the equations (e.g. Block Gauss-Seidel, combinative techniques,...).
In almost all our numerical tests on real test problems from case studies we observed a
considerable reduction of the CPU-time for the linear solver, up to a factor 4.3 with respect to
ILU(0) preconditioning (which is used at the moment in TEMIS3D). The performance
of the preconditioner shows no degradation with respect to the number of elements, the size of
the time step, high migration ratios, or strong heterogeneities and anisotropies in the porous media.