3. A Generalized FETI - DP Method for a Mortar Discretization of Elliptic Problems

1. Introduction. In this paper, an iterative substructuring method with La-grange multipliers is proposed for discrete problems arising from approximations of elliptic problem in two dimensions on non-matching meshes. The problem is formulated using a mortar technique. The algorithm belongs to the family of dual-primal FETI (Finite Element Tearing and Interconnecting) methods which has been analyzed recently for discretization on matching meshes. In this method the unknowns at the vertices of substructures are eliminated together with those of the interior nodal points of these substructures. It is proved that the preconditioner proposed is almost optimal; it is also well suited for parallel computations. We will consider a dual-primal FETI (FETI-DP) method, see [5], [9], and [6], for solving discrete problems arising from the approximation of the Dirichlet problem defined on a union of substructures Ω i. Each substructure is the union of a number of elements of a coarse, shape-regular triangulation and the number of these triangles, which form such a substructure, is assumed to be uniformly bounded. The discretiza-tion is obtained by a mortar method on nonmatching meshes across the interface Γ; see [1], [2]. As in all other iterative substructuring methods, the unknowns corresponding to the interior nodal points are eliminated; in this dual-primal FETI method those at the vertices of Ω i are eliminated as well. The remaining Schur complement system is solved by a FETI method; see Section 3 for details. A full analysis of the convergence of several FETI-DP methods has been worked out for finite element approximations on matching meshes; see [9] for the two-dimensional case and [6] for three dimensions. This method, on nonmatching meshes and for the mortar discretizations in the 2-D case, was analyzed in [4]. The preconditioner used there is a standard one and the estimates are not optimal in the general case. In this paper, our analysis is extended to the preconditioner suggested in [7] for matching meshes. The results obtained for this method is better than those of [4]. The superiority of this method is consistent with the numerical results reported on in [11]. The remainder of this paper is organized as follows. In Section 2 differential and discrete problems are formulated while in Section 3 the dual-primal formulation is introduced. Sections 4 is are devoted to the analysis of the proposed preconditioner.