Solver preconditioning using the combinatorial multilevel method on reservoir simulation

Yuhe Wang, John E. Killough

Research output: Contribution to journalArticle


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 using the discrete geometry of the graph, based on support theory for preconditioners. 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 setup phase compared to AMG, but the method can be proven 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 poroelastic displacement preconditioners in the multistage 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 non-symmetric matrices is shown to exhibit excellent convergence properties for realistic cases.

Original languageEnglish
Pages (from-to)695-708
Number of pages14
JournalComputational Geosciences
Issue number4
Publication statusPublished - 22 Aug 2015



  • Combinatorial multilevel method
  • Linear solver
  • Preconditioner
  • Reservoir simulation

ASJC Scopus subject areas

  • Computer Science Applications
  • Computers in Earth Sciences
  • Computational Theory and Mathematics
  • Computational Mathematics

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