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DIRECT AND ITERATIVE SPARSE LINEAR SOLVERS APPLIED TO GROUNDWATER FLOW SIMULATIONS

Jocelyne Erhel 1 Anthony Beaudoin 2 Jean-Raynald de Dreuzy 3, 4
1 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique , INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
Abstract : Subsurface hydraulic properties are mainly governed by the heterogeneity of geological formations. Moreover, it is not possible to measure the permeability in the whole site. In order to take into account the e¤ects of heterogeneity on groundwater ‡ows and the uncertainty in the input data, we use a probabilistic model in our numerical simulations. In the ‡ow equations, governed by the Darcy law and mass conservation, the permeability is thus a stochastic given function. The Darcy velocity and the hydraulic head become then unknown stochastic functions. When dealing with fractured rocks, the domain itself and the geometry are random variables. With non intrusive methods such as Monte Carlo sampling or stochastic collocation, statistics of outputs are obtained through many simulations of deterministic equations. High performance computing is required to solve the ‡ow equations. We consider two di¤erent cases. In porous media, the domain has a simple geometry. We use then a regular mesh and a …nite volume method, leading to a structured sparse matrix. In fractured media, as the domain is a complex network of fractures, we use an unstructured mesh and a mixed hybrid …nite volume method, leading to a general sparse matrix. In both cases, we compare direct methods, Krylov iterative methods and multigrid iterative methods, used to solve the linear systems obtained by discretizing the ‡ow equations. We study the impact of the random distribution function on the condition number of the matrix and on the accuracy of the result. We also make a complexity analysis and a scalability analysis on parallel computers.
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Jocelyne Erhel, Anthony Beaudoin, Jean-Raynald de Dreuzy. DIRECT AND ITERATIVE SPARSE LINEAR SOLVERS APPLIED TO GROUNDWATER FLOW SIMULATIONS. Conference on Matrix Analysis and Applications (M2A), 2007, Marseille, France. ⟨insu-03320769⟩

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