Inexact linear solvers for control volume discretizations in porous media

Abstract

We discuss the construction of multi-level inexact linear solvers for control volume discretizations for porous media. The methodology forms a contrast to standard iterative solvers by utilizing an algebraic hierarchy of approximations which preserve the conservative structure of the underlying control volume. Our main result is the generalization of multiscale control volume methods as multi-level inexact linear solvers for conservative discretizations through the design of a particular class of preconditioners. This construction thereby bridges the gap between multiscale approximation and linear solvers. The resulting approximation sequence is referred to as inexact solvers. We seek a conservative solution, in the sense of control-volume discretizations, within a prescribed accuracy. To this end, we give an abstract guaranteed a posteriori error bound relating the accuracy of the linear solver to the underlying discretization. These error bounds are explicitly computable for the grids considered herein. The afore-mentioned hierarchy of conservative approximations can also be considered in the context of multi-level upscaling, and this perspective is highlighted in the text as appropriate. The new construction is supported by numerical examples highlighting the performance of the inexact linear solver realized in both a multi- and two-level context for two- and three-dimensional heterogeneous problems defined on structured and unstructured grids. The numerical examples assess the performance of the approach both as an inexact solver, as well in comparison to standard algebraic multigrid methods.