Space-Time Domain Decomposition for Reduced Fracture Models in Mixed Formulation
Thi-Thao-Phuong Hoang, Caroline Japhet, Michel Kern, Jean E. Roberts
SIAM Journal on Numerical Analysis
In this paper we are interested in the "fast path" fracture and we aim to use globalin-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 Steklov-Poincaré operator and the other uses optimized Schwarz waveform relaxation (OSWR) based
... n 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. 289 medium; second, the fracture width is much smaller than any reasonable parameter of spatial discretization. Thus, to tackle the problem, one might need to refine the mesh locally around the fractures. However, this is well-known to be very computationally costly and is not useful at the macroscopic scale (i.e., when the fractures can be modeled individually). One possible approach is to treat the fractures as domains of co-dimension one, i.e., interfaces between subdomains (see [1, 3, 5, 16, 17, 40, 43, 42, 48] and the references therein) so that one can avoid refining locally around the fractures. We point out that in these reduced fracture models, unlike in some discrete fracture models, interaction between the fractures and the surrounding porous medium is taken into account. We are concerned with algorithms for modeling flow and transport in porous media containing such fractures. In particular, in this paper we investigate two spacetime domain decomposition methods, well-suited to nonmatching time grids. We use mixed finite elements [11, 45] as they are mass conservative and they handle well heterogeneous and anisotropic diffusion tensors. The first method is a global-in-time preconditioned Schur method (GTP-Schur) which uses a Steklov-Poincaré-type operator. For stationary problems, this kind of method (see [41, 44, 47] ) is known to be efficient for problems with strong heterogeneity. It uses the so-called balancing domain decomposition (BDD) preconditioner introduced and analyzed in [37, 38] , and in  for mixed finite elements. It involves at each iteration the solution of local problems with Dirichlet and Neumann data and a coarse grid problem to propagate information globally and to ensure the consistency of the Neumann subdomain problems. An extension to the case of unsteady problems with the construction of the time-dependent Steklov-Poincaré operator was introduced in [28, 29], where an interface problem on the space-time interfaces between subdomains is derived. However, for the time-dependent Neumann-Neumann problems there are no difficulties concerning consistency, and we are dealing with only a small number of subdomains, so we consider only a Neumann-Neumann type preconditioner, an extension to the nonsteady case of the method of  . A Richardson iteration for the primal formulation was independently introduced in [18, 34], and its convergence was analyzed. In the case of elliptic problems with fractures, a local preconditioner  significantly improves the convergence of the method. The second method is a global-in-time optimized Schwarz method (GTO-Schwarz) and uses the optimized Schwarz waveform relaxation (OSWR) approach. The OSWR and GTP-Schur methods are iterative methods that compute in the subdomains over the whole time interval, exchanging space-time boundary data through transmission conditions on the space-time interfaces. The OSWR algorithm uses more general (Robin or Ventcell) transmission operators in which coefficients can be optimized to improve convergence rates; see [21, 32, 39] . The optimization of the Robin (or Ventcell) parameters was analyzed in  and the optimization method was extended to the case of discontinuous coefficients in [7, 8, 9, 10, 20, 28, 29] . Generalizations to heterogeneous problems with nonmatching time grids were introduced in [7, 8, 10, 20, 24, 25, 26, 27, 28, 29] . More precisely, in [10, 26, 27] , a discontinuous Galerkin (DG) method for the time discretization of the OSWR algorithm was introduced and analyzed for the case of nonconforming time grids. A suitable time projection between subdomains is defined using an optimal projection algorithm as in [22, 23] with no additional grid. The classical Schwarz algorithm for stationary problems with mixed finite elements was analyzed in  . An OSWR method with Robin transmission conditions for a mixed formulation was proposed and analyzed in [28, 29] , where a mixed form of an interface problem on the space-time interfaces between subdomains was derived. In  , an optimized Schwarz method with Ventcell conditions in the context of mixed formulations was proposed. This method is not obtained in such a straightforward manner as in the case of primal formulations as Lagrange multipliers have to be introduced on the interfaces to handle tangential derivatives involved in the Ventcell conditions. In this work, we define both a GTP-Schur and a GTO-Schwarz algorithm for a problem modeling flow of a single phase, compressible fluid in a porous medium with a fracture. A straightforward application of  would be to consider the fracture as a third subdomain and to take smaller time steps there. We consider instead a reduced model in which the fracture is treated as an interface between two subdomains. The definition of the GTP-Schur method is a straightforward extension of that in  . However, to define the GTO-Schwarz method, something more is needed: a linear combination between the pressure continuity equation and the fracture problem is used as a transmission condition (which leads naturally to Ventcell conditions), and a free parameter is used to accelerate the convergence rate. The well-posedness of the subdomain problems involved in the first approach was addressed in [12, 29, 36] , using Galerkin's method and suitable a priori estimates. In this paper, the proof of wellposedness of both the coupled model and the Ventcell subdomain problems involved in the GTO-Schwarz approach is shown to follow from a more general theorem that covers the two cases. Note that more general reduced models that can handle both large and small permeability fractures  introduce more complicated transmission conditions on the fracture-interface (in the form of Robin type conditions, where the Robin coefficient has a physical origin), and it is not yet clear how to formulate an associated domain decomposition problem with a parameter that can be optimized. This paper is organized as follows: in the remainder of the introduction (subsection 1.1), we state an abstract existence and uniqueness theorem for evolution problems in mixed form, the proof being deferred to Appendix A. In section 2 we consider a reduced model with a highly permeable fracture and prove its well-posedness. Then in section 3 we consider the GTP-Schur approach, based on physical transmission conditions, for solving the resulting problem. Different preconditioners for this method are proposed. In section 4 we consider the GTO-Schwarz method, based on more general (e.g., Ventcell) transmission conditions, for solving the resulting problem. We prove the well-posedness of the subdomain problems with Ventcell boundary conditions. In section 5 we consider the semidiscrete problems in time using different time grids in the subdomains. Finally, in section 6, results of two-dimensional (2D) numerical experiments comparing the different methods are discussed. Abstract evolution problems in mixed form. The goal of this section is to give an existence and uniqueness result for evolution problems posed in mixed form, in the spirit of the well-known theorem for weak parabolic problems (see, for example, [14, vol. 5]). We consider two Hilbert spaces, Σ and M (M will be identified with its dual), and assume we have continuous bilinear forms a : Σ × Σ −→ R, b: Σ × M −→ R, c: M × M −→ R and a continuous linear form L : M −→ R. Downloaded 03/14/18 to 220.127.116.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 4. Global-in-time optimized Schwarz (GTO-Schwarz): Using optimized Schwarz waveform relaxation. While the extension of the GTP-Schur method to handle the fracture model is straightforward, the extension of the GTO-Schwarz method to the fracture problem needs something more. Indeed, instead of imposing Dirichlet boundary conditions on γ × (0, T ) when solving the fracture problem as was done for the GTP-Schur method, for the GTO-Schwarz approach one uses optimized Robin transmission conditions. Thus, we introduce new transmission conditions that combine the equation for continuity of the pressure across the fracture with the flow equations (2.4) in the fracture. These new transmission conditions contain a free parameter, which is used to accelerate the convergence. This is an extension of the OSWR method with optimized Robin parameters studied in [28, 29] in which Robin-to-Robin transmission conditions are considered in mixed form. Here, however, because of the fracture problem, we obtain what we will call Ventcell-to-Robin transmission conditions as described below.