Auxiliary variables for 3D multiscale simulations in heterogeneous porous media

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called the Lie-Butcher series. This paper presents the theory of backward error analysis for methods based on Lie-Butcher series. Abstract: The multiscale control-volume methods for solving problems involving flow in porous media have gained much interest during the last decade. Recasting these methods in an algebraic framework allows one to consider them as preconditioners for iterative solvers. Despite intense research on the 2D formulation, few results have been shown for 3D, where indeed the performance of multiscale methods deteriorates. The interpretation of multiscale methods as vertex based domain decomposition methods, which are non-scalable for 3D domain decomposition problems, allows us to understand this loss of performance. We propose a generalized framework based on auxiliary variables on the coarse scale. These are enrichments of the coarse scale, which can be selected to improve the interpolation onto the fine scale. Where the existing coarse scale basis functions are designed to capture local sub-scale heterogeneities, the auxiliary variables are aimed at better capturing non-local effects resulting from non-linear behavior of the pressure field. The auxiliary coarse nodes fits into the framework of mass-conservative domain-decomposition (MCDD) preconditioners, allowing us to construct, as special cases, both the traditional (vertex based) multiscale methods as well as their wire basket generalization.