A Multilevel Multiscale Finite Volume Method

The Multiscale Finite Volume (MsFV) method has been developed to efficiently solve reservoir-scale problems while conserving fine-scale details. The method employs two grid levels: a fine grid and a coarse grid. The latter is used to calculate a coarse solution to the original problem, which is interpolated to the fine mesh. The coarse system is constructed from the fine-scale problem using restriction and prolongation operators that are obtained by introducing appropriate localization assumptions. Through a successive reconstruction step, the MsFV method is able to provide an approximate, but fully conservative fine-scale velocity field. For very large problems (e.g. one billion cell model), a two-level algorithm can remain computational expensive. Depending on the upscaling factor, the computational expense comes either from the costs associated with the solution of the coarse problem or from the construction of the local interpolators (basis functions). To ensure numerical efficiency in the former case, the MsFV concept can be reapplied to the coarse problem, leading to a new, coarser level of discretization. One challenge in the use of a multilevel MsFV technique is to find an efficient reconstruction step to obtain a conservative fine-scale velocity field. In this work, we introduce a three-level Multiscale Finite Volume method (MlMsFV) and give a detailed description of the reconstruction step. Complexity analyses of the original MsFV method and the new MlMsFV method are discussed, and their performances in terms of accuracy and efficiency are compared.