Fractures in a porous medium are considered individually and are supposed to be a porous medium of higher permeability than in the surrounding rock. Since their thickness is supposed to be small with respect to the dimension of the domain of calculation they are modelled as interfaces. We formulate the flow and the transport in the medium, taking into account interaction between the fracture and the surrounding rock. We proved existence and uniqueness of the flow problem and give numerical experiments illustrating the model. The case of intersecting fractures is also considered.
This paper is concerned with global-in-time, nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem. Two approaches are considered: one uses the time-dependent Steklov-Poincaré operator and the other uses Optimized Schwarz Waveform Relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interfaces between subdomains is derived, and different time grids are employed to adapt to different time scales in the subdomains. Demonstrations of the well-posedness of the subdomain problems involved in each method and a convergence proof of the OSWR algorithm are given for the mixed formulation. Numerical results for 2D problems with strong heterogeneities are presented to illustrate the performance of the two methods. Résumé : Ce papier traite de méthodes de décomposition de domaine sans recouvrement, globales en temps, appliquées à un problème de diffusion en formulation mixte. Deux approches sont considérées: l'une basée sur l'opérateur de SteklovPoincaré, et l'autre basée sur une méthode de relaxation d'onde optimisée (OSWR), utilisant des conditions de transmission de type Robin. Pour chaque méthode, un problème d'interface est écrit sous forme mixte, les inconnues étant sur les interfaces espace-temps entre les sous-domaines, et différentes grilles en temps sont utilisées, adaptées aux différentes échelles temporelles dans les sous-domaines. Des démonstra-tions à la fois du caractère bien posé des problèmes locaux dans les sous-domaines (intervenant dans chaque méthode) et de la convergence de l'algorithm OSWR sont données en formulation mixte. Des résultats numériques pour des problèmes 2D avec de fortes hétérogénéités sont présentés pour illustrer les performances des deux méth-odes.
Key
Mots-clés :formulation mixte, décomposition de domaine espace-temps, problème de diffusion, opérateur de Steklov-Poincaré dépendant du temps, méthode de relaxation d'onde optimisée, maillages non-conformes en temps
Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations31. Introduction. In many simulations of time-dependent physical phenomena, the domain of calculation is actually a union of several subdomains with different physical properties and in which the time scales may be very different. In particular, this is the case for the simulation of contaminant transport around a nuclear waste repository, where the time scales vary over several orders of magnitude due to changes in the hydrogeological properties of the various geological layers involved in the simulation. Consequently, it is inefficient to use a single time step throughout the entire domain. The aim of this article is to investigate, in the context of mixed finite elements [5,30], two global-in-time domain decomposition methods well-suited to nonmatching time grids. Advantages of mixed methods include their mass conservation property and a natural way to handle heterogeneous and anisotropic diffusion tensors.T...
We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q≥0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.
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