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SIAM Journal on Scientific and Statistical Computing
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SIAM Journal on Scientific and Statistical Computing
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
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A mixed multiscale finite element method for elliptic problems with oscillating coefficients
Mathematics of Computation
Accurate multiscale finite element methods for two-phase flow simulations
Journal of Computational Physics
A Multilevel Multiscale Mimetic (M3) Method for an Anisotropic Infiltration Problem
ICCS '09 Proceedings of the 9th International Conference on Computational Science: Part I
Journal of Computational Physics
Error Analysis for a Mimetic Discretization of the Steady Stokes Problem on Polyhedral Meshes
SIAM Journal on Numerical Analysis
Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes
SIAM Journal on Numerical Analysis
Mimetic Discretizations of Elliptic Control Problems
Journal of Scientific Computing
Ensemble level multiscale finite element and preconditioner for channelized systems and applications
Journal of Computational and Applied Mathematics
Mimetic finite difference method
Journal of Computational Physics
Mimetic scalar products of discrete differential forms
Journal of Computational Physics
Hi-index | 31.46 |
We describe a multilevel multiscale mimetic (M^3) method for solving two-phase flow (water and oil) in a heterogeneous reservoir. The governing equations are the elliptic equation for the reservoir pressure and the hyperbolic equation for the water saturation. On each time step, we first solve the pressure equation and then use the computed flux in an explicit upwind finite volume method to update the saturation. To reduce the computational cost, the pressure equation is solved on a much coarser grid than the saturation equation. The coarse-grid pressure discretization captures the influence of multiple scales via the subgrid modeling technique for single-phase flow recently proposed in [Yu. A. Kuznetsov. Mixed finite element method for diffusion equations on polygonal meshes with mixed cells. J. Numer. Math., 14 (4) (2006) 305-315; V. Gvozdev. discretization of the diffusion and Maxwell equations on polyhedral meshes. Technical Report Ph.D. Thesis, University of Houston, 2007; Yu. Kuznetsov. Mixed finite element methods on polyhedral meshes for diffusion equations, in: Computational Modeling with PDEs in Science and Engineering, Springer-Verlag, Berlin, in press]. We extend significantly the applicability of this technique by developing a new robust and efficient method for estimating the flux coarsening parameters. Specifically, with this advance the M^3 method can handle full permeability tensors and general coarsening strategies, which may generate polygonal meshes on the coarse grid. These problem dependent coarsening parameters also play a critical role in the interpolation of the flux, and hence, in the advection of saturation for two-phase flow. Numerical experiments for two-phase flow in highly heterogeneous permeability fields, including layer 68 of the SPE Tenth Comparative Solution Project, demonstrate that the M^3 method retains good accuracy for high coarsening factors in both directions, up to 64 for the considered models. Moreover, we demonstrate that with a simple and efficient temporal updating strategy for the coarsening parameters, we achieve accuracy comparable to the fine-scale solution, but at a fraction of the cost.