Iterative Galerkin-enriched multiscale finite-volume method

The multiscale finite volume (MSFV) method was designed to efficiently compute approximate numerical solutions of elliptic and parabolic problems with highly heterogeneous coefficients. For a wide range of test cases, it has been demonstrated that MSFV solutions are in excellent agreement with reference data obtained with a standard fine-scale finite-volume method. However, for some problems involving large coherent structures with strong contrasts (e.g., channels and shale layers), the MSFV method produces unsatisfactory results due to the dwindling validity of the localization assumption. Recently, the iterative MSFV (i-MSFV) method was introduced to iteratively improve the localization assumption, and it was shown that the method converges to the fine-scale reference solution. In this work, it is explained how the convergence rate of the i-MSFV method can be enhanced by consistent enrichment of the initial coarse space spanned by nodal basis functions. A new set of enrichment basis functions associated with additional coarse-scale degrees of freedom (DoF) is introduced. By construction, the sum of all nodal and enrichment basis functions is one in the entire computational domain. Further, a hybrid finite-volume/Galerkin formulation for the coarse-scale problem and a generic placement strategy for the additional DoF are devised. The resulting iterative Galerkin-enriched MSFV (i-Ge-MSFV) method has all features of the i-MSFV method, but is more robust and has improved convergence properties.