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Iterative multiscale finite-volume method. (English) Zbl 1151.65091
Summary: 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.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
76T30 Three or more component flows
76S05 Flows in porous media; filtration; seepage
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