Space-Time Domain Decomposition for Reduced Fracture Models in Mixed Formulation

Abstract

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.

Domain decomposition methods for the diffusion equation with low-regularity solution

“…Note also that one can use Hilbert spaces including L 2 -jumps to model fractures (see the recent paper [28]). Obviously, the numerical results can be improved in different manners: one can use graded meshes [29], an hp-finite element method [30], or an enriched finite element method, such as XFEM [31] or the singular complement method [32].…”

A reduced fracture model for two-phase flow with different rock types

“…We also introduced the tangential component of the gravity vector, u G = u G,τ + u G,n . Writing K γ , k γ and ρ γ for dK f τ , k f and ρ f respectively and integrating equation (29) across the cross-sections of the fracture we obtain…”

“…The unknowns from each subdomain are eliminated and the problem is formulated as a problem set on the interface γ. This formulation is very useful in order to use different time steps in the subdomains and in the fracture [27,29] even though for simplicity of notation, we will use in this article equal time steps all over the domain. Also, for simplicity, throughout this section, all problems are written in strong form, we suppose that homogeneous Dirichlet boundary conditions are imposed everywhere on ∂Ω and we consider the same situation as in the previous section.…”

“…In this paper we have not pursued this idea, but have used the same time steps in the fracture as in the matrix. However, we refer the interested reader to the series of articles [27,29,28], where a method for implementing this idea is described. h .…”

A posteriori stopping criteria for space-time domain decomposition for the heat equation in mixed formulations

“…The optimization of the Robin (or Ventcell) parameters was analyzed in [10,46]. Generalizations to heterogeneous problems with nonmatching time grids were introduced in [11,12,29,33,34,35,36,38,39,40]. More precisely, in [12,35,36], a discontinuous Galerkin (DG) method for the time discretization of the OSWR algorithm was introduced and analyzed for the case of nonconforming time grids.…”