Algebraic Domain Decomposition Methods for Darcy flow in heterogeneous media

Abstract : To meet the needs of the oil industry to a more accurate description of the geometry and petrophysical properties of basins and reservoirs, the numerical simulation of flow in porous media must evolve towards more efficient and robust algorithms with respect to size of simulations, the complexity of mesh and heterogeneities of the porous medium. The domain decomposition methods are an alternative to multigrid methods and could help remove the difficulties in terms of robustness and efficiency on parallel architectures. They are inherently more suited to parallel computing and are more robust especially when the subdomains are solved by direct methods. They also deal in a single framework with the couplings of models such as wells or conducting faults. The thesis deals specifically with methods defined at the algebraic level. We do not assume prior knowledge of the continuous problem from which the matrix is derived and there was no access to the matrices before assembly. This lack of prior information makes it more difficult to build effective methods. We propose two new methods for construction domain decomposition methods at the algebraic level i.e., algebraic construction of optimized interface conditions and a coarse grid. This last method is particularly important to have robust methods with respect to the number of subdomains. The methods are adaptive and they are based on the analysis of the Krylov space generated in the first resolution of the classical Schwarz method. From the Ritz vectors corresponding to the lowest eigenvalues, we construct interface conditions and coarse grid, which nullifies the error on these components. The methods were tested on parallel computers up to 1000 subdomains on up to 512 cores for matrices from the real simulation of porous media flow.
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Contributor : Mikolaj Szydlarski <>
Submitted on : Wednesday, December 29, 2010 - 9:11:19 PM
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  • HAL Id : tel-00550728, version 1

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Mikolaj Szydlarski. Algebraic Domain Decomposition Methods for Darcy flow in heterogeneous media. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2010. English. ⟨tel-00550728⟩

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