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...
Abstract. We design and analyze Schwarz waveform relaxation algorithms for domain decomposition of advection-diffusion-reaction problems with strong heterogeneities. These algorithms rely on optimized Robin or Ventcell transmission conditions, and can be used with curved interfaces. We analyze the semi-discretization in time with discontinuous Galerkin as well. We also show twodimensional numerical results using generalized mortar finite elements in space.
In this paper we are interested in the "fast path" fracture and we aim to use global-in-time, nonoverlapping domain decomposition methods to model flow and transport problems in a porous medium containing such a fracture. We consider a reduced model in which the fracture is treated as an interface between the two subdomains. Two domain decomposition methods are considered: one uses the time-dependent SteklovPoincaré operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Ventcell transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains and in the fracture. Demonstrations of the well-posedness of the Ventcell subdomain problems is given for the mixed formulation. An analysis for the convergence factor of the OSWR algorithm is given in the case with fractures to compute the optimized parameters. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods.
Robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in the convergence. Non conforming grids make the grid generation much easier and faster since it is then a parallel task. The goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of Robin interface conditions with non-matching grids. We consider both a symmetric definite positive operator and the convection-diffusion equation discretized by finite volume schemes. Numerical results are shown.
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