Solver preconditioning using the combinatorial multilevel method Conference Paper uri icon


  • The purpose of this paper is to report the first preliminary study of the recently introduced Combinatorial Multilevel (CML) method for solver preconditioning in large-scale reservoir simulation with coupled geomechanics. The CML method is a variant of the popular Algebraic Multigrid (AMG) method yet with essential differences. The basic idea of this new approach is to construct a hierarchy of matrices by viewing the underlying matrix as a graph and by using the discrete geometry of the graph such as graph separators and expansion. In this way, the CML method combines the merits of both geometric and algebraic multigrid methods. The resulting hybrid approach not only provides a simpler and faster set-up phase compared to AMG, but the method can be proved to exhibit strong convergence guarantees for arbitrary symmetric diagonally-dominant matrices. In addition, the underlying theoretical soundness of the CML method contrasts to the heuristic AMG approach, which often can show slow convergence for difficult problems. This new approach is implemented in a reservoir simulator for both pressure and displacement preconditioners in the multi-stage preconditioning technique. We present results based on several known benchmark problems and provide a comparison of performance and complexity with the widespread preconditioning schemes used in large-scale reservoir simulation. An adaptation of CML for unsymmetric matrices is shown to exhibit excellent convergence properties for realistic cases. Copyright 2013, Society of Petroleum Engineers.

author list (cited authors)

  • Wang, Y., & Killough, J. E.

publication date

  • August 2013