A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Convergence of a Nonconforming Multiscale Finite Element Method
SIAM Journal on Numerical Analysis
Multigrid
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
Mathematics of Computation
Multi-scale finite-volume method for elliptic problems in subsurface flow simulation
Journal of Computational Physics
Numerical methods for multiscale elliptic problems
Journal of Computational Physics
Multiscale finite-volume method for compressible multiphase flow in porous media
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Accurate multiscale finite element methods for two-phase flow simulations
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Adaptive iterative multiscale finite volume method
Journal of Computational Physics
An iterative multiscale finite volume algorithm converging to the exact solution
Journal of Computational Physics
A hierarchical fracture model for the iterative multiscale finite volume method
Journal of Computational Physics
Recent developments in the multi-scale-finite-volume procedure
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
An adaptive multiscale method for density-driven instabilities
Journal of Computational Physics
On a numerical subgrid upscaling algorithm for Stokes-Brinkman equations
Computers & Mathematics with Applications
Finite analytic numerical method for two-dimensional fluid flow in heterogeneous porous media
Journal of Computational Physics
Block-spectral mapping for multi-scale solution
Journal of Computational Physics
Algebraic multiscale solver for flow in heterogeneous porous media
Journal of Computational Physics
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The multiscale finite-volume (MSFV) method for the solution of elliptic problems is extended to an efficient iterative algorithm that converges to the fine-scale numerical solution. The localization errors in the MSFV method are systematically reduced by updating the local boundary conditions with global information. This iterative multiscale finite-volume (i-MSFV) method allows the conservative reconstruction of the velocity field after any iteration, and the MSFV method is recovered, if the velocity field is reconstructed after the first iteration. Both the i-MSFV and the MSFV methods lead to substantial computational savings, where an approximate but locally conservative solution of an elliptic problem is required. In contrast to the MSFV method, the i-MSFV method allows a systematic reduction of the error in the multiscale approximation. Line relaxation in each direction is used as an efficient smoother at each iteration. This smoother is essential to obtain convergence in complex, highly anisotropic, heterogeneous domains. Numerical convergence of the method is verified for different test cases ranging from a standard Poisson equation to highly heterogeneous, anisotropic elliptic problems. Finally, to demonstrate the efficiency of the method for multiphase transport in porous media, it is shown that it is sufficient to apply the iterative smoothing procedure for the improvement of the localization assumptions only infrequently, i.e. not every time step. This result is crucial, since it shows that the overall efficiency of the i-MSFV algorithm is comparable with the original MSFV method. At the same time, the solutions are significantly improved, especially for very challenging cases.