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FETI DOMAIN DECOMPOSITION METHODSFOR SCALAR ADVECTION–DIFFUSION PROBLEMSANDREA TOSELLI∗Abstract. In this paper, we show that iterative substructuring methods of Finite Element Tear-ing and Interconnecting type can be successfully employed for the solution of linear systems arisingfrom the finite element approximation of scalar advection–diffusion problems. Using similar ideas asthose of a recently developed Neumann–Neumann method, we propose a one–level algorithm and aclass of two–level algorithms, obtained by suitably modifying the local problems on the subdomains.We present some numerical results for some significant test cases. Our methods appear to be opti-mal for flows without closed streamlines and possibly very small values of the viscosity. They alsoshow very good performances for rotating flows and moderate Reynolds numbers. Therefore, thealgorithms proposed appear to be well–suited for many convection–dominated problems of practicalinterest.Key words. Advection–diffusion, stabilization, domain decomposition, FETI, preconditioners.AMS subject classifications. 65F10, 65N22, 65N30, 65N551. Introduction. In this paper, we consider the boundary value problemLu := −ν?u + a · ∇u + cu = f,u = 0,on ∂?D,in ?,∂u∂n= 0,on ∂?N.(1)Here, ? ⊂ Rn, n = 2, 3, is a bounded, connected polyhedral domain with a Lipschitzcontinuous boundary ∂? and outward normal denoted by n. We consider a partition∂? = ∂?N∪∂?D, where ∂?Ncan possibly be empty. For simplicity, we only deal withhomogeneous Dirichlet and Neumann conditions, but more general non homogeneousboundary data can also be used; see section 4 and, e.g., [31, Ch. 6].The viscosity ν is positive, but can be arbitrarily small for advection–dominatedproblems. For simplicity, we assume that ν is constant. The velocity field a is givenand we suppose that a ∈ L∞(?)nand ∇ · a ∈ L∞(?). The scalar function c ∈ L∞(?)is a reaction coefficient that may arise from a finite difference discretization of a timederivative, and f ∈ L2(?) is a source term.The aim of this paper is to build a family of iterative methods of Finite ElementTearing and Interconnecting (FETI) type for a conforming finite element (FE) ap-proximation of problem (1). We show that by borrowing some ideas from a Neumann–Neumann method for the same problem, see [2], and by using some recent develop-ments in the analysis ofFETI methods, see [25, 39], FETI algorithms can be employedsuccessfully for advection–diffusion problems as well. We are primarily interested inconvection–dominated problems.FETI methods were first introduced for the solution ofconforming approximationsof elasticity problems in [15]. In this approach, the original domain ? is decomposedinto non–overlapping subdomains ?i, i = 1, . . . , N. On each subdomain ?i a localstiffness matrix is obtained from the finite element discretization of local Neumann∗Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012, USA. E–mail: toselli@cims. nyu.edu. URL: http://www.math.nyu.edu/˜toselli. This work was supportedin part by the Applied Mathematical Sciences Program of the U.S. Department of Energy underContract DEFGO288ER25053.1