A Two-Level Variant of Additive Schwarz Preconditioning for Use in   Reservoir Simulation
  The computation time for reservoir simulation is dominated by the linear solver. The sets of linear equations which arise in reservoir simulation have two distinctive features: the problems are usually highly anisotropic, with a dominant vertical flow direction, and the commonly used fully implicit method requires a simultaneous solution for pressure and saturation or molar concentration variables. These variables behave quite differently, with the pressure feeling long-range effects while the saturations vary locally. In this paper we review preconditioned iterative methods used for solving the linear system equations in reservoir simulation and their parallelisation. We then propose a variant of the classical additive Schwarz preconditioner designed to achieve better results on a large number of processors and discuss some directions for future research. 
Haran Jackson, Michele Taroni, David Ponting
math.NA cs.NA