In recent years, deflation methods have received increasingly particular attention as a means to improving the convergence of linear iterative solvers. This is due to the fact that deflation operators provide a way to remove the negative effect that extreme (usually small) eigenvalues have on the convergence of Krylov iterative methods for solving general symmetric and non-symmetric systems.

In this work, we use deflation methods to extend the capabilities of algebraic multigrid (AMG) for handling highly non-symmetric and indefinite problems, such as those arising in fully implicit formulations of multiphase flow in porous media. The idea is to ensure that components of the solution that remain unresolved by AMG (due to the coupling of roughness and indefiniteness introduced by different block coefficients) are removed from the problem. This translates to a constraint to the AMG iteration matrix spectrum within the unit circle to achieve convergence. This approach interweaves AMG (V, W or V-W) cycles with deflation steps that are computable either from the underlying Krylov basis produced by the GMRES accelerator (Krylov-based deflation) or from the reservoir decomposition given by high property contrasts (domain-based deflation). This work represents an efficient extension to the Generalized Global Basis (GGB) method that was recently proposed for the solution of the elastic wave equation with geometric multigrid and an out-of-core computation of eigenvalues.

Hence, the present approach offers the possibility of applying AMG to more general large-scale reservoir settings without further modifications to the AMG implementation or algebraic manipulation of the linear system (as suggested by two-stage preconditioning methods). Promising results are supported by a suite of numerical experiments with extreme permeability contrasts.

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