Interface solvers and preconditioners of domain decompositio

Abstract. A multiblock mortar approach to modeling multiphase flow in porous media decomposes the simulation domain into a series of blocks with possibly different physical and numerical models employed in each block. Matching conditions along the interfac

INTERFACE SOLVERS AND PRECONDITIONERS OF DOMAIN DECOMPOSITION TYPE FOR MULTIPHASE FLOW IN MULTIBLOCK POROUS MEDIAIVAN YOTOV

Abstract. A multiblock mortar approach to modeling multiphase ow in porous media decomposes the simulation domain into a series of blocks with possibly di erent physical and numerical models employed in each block. Matching conditions along the interfaces are imposed through the use of mortar nite elements. A parallel domain decomposition algorithm reduces the algebraic nonlinear system to an interface problem which is solved via a nonlinear multigrid with Newton-GMRES smoothing. Key words. Multiblock, mortar nite elements, non-matching grids, domain decomposition, Newton-Krylov methods, multiphase ow AMS subject classi cations. 65M55, 65M06, 65H10, 35K65, 76S05, 76T99

1. Introduction. A multiblock mortar methodology has been recently developed for modeling subsurface ow. The simulation domain is decomposed based upon the physics of the model into a series of blocks. The blocks are independently meshed and possibly di erent physical and numerical models are employed in each block. Interface matching conditions are imposed in a stable and accurate way through the use of mortar nite elements. Mortar nite elements have been successfully applied for standard nite element and spectral nite element discretizations on non-matching grids (see, e.g. 6, 5]). We consider locally conservative mixed nite element ( nite volume) methods for subdomain discretizations. Theoretical and numerical results for single phase ow indicate mortar mixed nite element methods are highly accurate (superconvergent) for both pressure and velocity 21, 1, 4, 23]. An extension of the method to a degenerate parabolic equation arising in two phase ow is presented in 22], where optimal convergence is shown. Multiphysics applications can be found in 16]. Critical for the success of this approach is the ability to e ciently solve the resulting discrete nonlinear system. A parallel non-overlapping domain decomposition implementation, based on a method originally proposed by Glowinski and Wheeler 13, 10, 9], provides an e cient scalable solution technique 21, 19]. Some e cient preconditioners have also been developed 15, 20]. In this paper we present an e cient parallel algorithm that reduces the global system to a nonlinear interface problem. One advantage of this approach compared to solving the global system directly is that the subdomains are loosely coupled and it is relatively easy to couple di erent physical and numerical models. The interface problem is solved via a nonlinear multigrid V-cycle with Newton-GMRES smoothing. A physics based Neumann-Neumann preconditioner is constructed for accelerating the GMRES convergence. The rest of the paper is organized as follows. In the next section we present a multiblock formulation and discretization for a two-phase ow model. The domainyotov@math.pitt.edu

Department of Mathematics, University of Pittsburgh, Pitt

sburgh, PA 15260; . This work was supported in part by the DOE grant DE-FG03-99ER25371, the NSF grant DMS 9873326, the University of Pittsburgh CRDF grant, and the ORAU Ralph E. Powe award. 1

Abstract. A multiblock mortar approach to modeling multiphase flow in porous media decomposes the simulation domain into a series of blocks with possibly different physical and numerical models employed in each block. Matching conditions along the interfac

2

IVAN YOTOV

decomposition solvers and preconditioners are described in Section 3. Computational results are given in Section 4. 2. Multiblock formulation and discretization. To illustrate the numerical technique we consider a two-phase ow model. In a multiblock formulation, the domain R3, is decomposed into a series of subdomains k, k= 1;:::; nb. Let?kl=@ k\@ l be the interface between k and l . The governing mass conservation equations 8] are imposed on each subdomain@( S )+ r U= q; (2.1) where= w (wetting), n (non-wetting) denotes the phase, S is the phase saturation,= (P ) is the phase density, is the porosity, q is the source term, and U=? k (S )K (rP? grD) (2.2) is the Darcy velocity. Here P is the phase pressure, k (S ) is the phase relative permeability, is the phase viscosity, K is the rock permeability tensor, g is the gravitational constant, and D is the depth. On each interface?kl the following physically meaningful continuity conditions are imposed: (2.3) P j k= P j l; (2.4) U]kl U j k k+ U j l l= 0; where k denotes the outward unit normal vector on@ k . The above equations are coupled via the volume balance equation and the capillary pressure relation (2.5) Sw+ Sn= 1; pc(Sw )= Pn? Pw; which are imposed on each k and?kl . We assume for simplicity that no ow U= 0 is imposed on@, although more general types of boundary conditions can also be treated. 2.1. Discretization spaces. The subdomains are discretized using a variant of the mixed nite element method, the expanded mixed method. It has been developed for accurate and e cient treatment of irregular domains (see 3, 2] for single block and 21, 23] for multiblock domains). The original problem is transformed into a problem on a union of regular computational (reference) grids. The permeability after the mapping is usually a full tensor (except in some trivial cases). The mixed method could then be accurately approximated by cell-centered nite di erences for the pressure 3]. To simplify the presentation we will only describe here the rectangular reference case. For a de nition of the spaces on logically rectangular and triangular grids, we refer to 2] (also see 18, 7]). Let us denote the rectangular partition of k by Thk, where …… 此处隐藏：24911字，全部文档内容请下载后查看。喜欢就下载吧 ……