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... 2 with a diameter of size O(1). The extension of the results to 3 can be carried out easily by using the theory developed here in this paper and the well-known three-dimensional AS techniques; see =-=[9, 10, 12]-=-. Let T h(Ω) be a shape-regular quasi-uniform triangulation of size O(h) of Ω, and V ⊂ H10 (Ω) the finite element space consisting of continuous piecewise linear functions associated with the triangul...

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...t are required then becomes just as elementary as for the Neumann--Neumann case. We note that our family of scalings of the preconditioner was apparently first introduced by Sarkis [30, 31]; see also =-=[7]-=-. The other scaling a#ects the choice of the projection which is used in each step of the FETI iteration, whether preconditioned or not. We will show that, for a certain choice of the two scalings, ou...

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...r quite general coeÆcients such as those that model highly heterogeneous materials. Our work has been inspired by that of Mandel and Tezaur and it is also based on our own earlier work, in particular =-=[5]-=-, [6], and [13]. It is well known that domain decomposition algorithms cannot be scalable, i.e., have a rate of convergence which is independent of the number of subregions, unless a coarse space comp...

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...s the space of continuous piecewise linear functions on the triangulation T # ( # 1 ). We denote by # # 1 the set of nodal points of T # ( # 1 ) belonging tos# 1 . Following Dryja, Sarkis and Widlund =-=[18]-=-, we define an interpolation operator I M # : V h1(# 1 ) # V #(# 1 ) as follows. Definition 1. Given w # V h1(# 1 ), define w # = I M # w # V #(# 1 ) by the values of w # at two types of nodes of T # ...

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...arrangements of the highly conductive regions with respect to the coarse cells. The construction of preconditioners for these problems has been extensively studied in the last three decades (see e.g. =-=[7, 14, 19, 20, 28]-=-). In the context of domain decomposition methods, one can consider overlapping and nonoverlapping methods. It was shown that nonoverlapping domain decomposition methods converge independently of the ...

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... on overlapping subdomains. This choice of the coarse component will allow us to prove results which are independent of coefficient jumps. We note that there are earlier studies on multilevel methods =-=[9, 19]-=- in which the coarsest components are similarly borrowed from iterative substructuring algorithms. We will use nonoverlapping subdomains, and denote them by Ωi, i = 1, . . . , N, as well as overlappin...

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...n to assuming A ∈ L∞(Ω) with A∗(x) = A(x) > 0 a.e., we assume that A ρ(A)id, uniformly over the domain (isotropic diffusion), and that A is piecewise constant with respect to P0. Further, following =-=[DSW96]-=-, we assume that ρ = ρ(A) is quasi-monotone with respect to P0. That is, defining for v ∈ VP0, P0(v) = {s∈ P0 : v ∈s} ands(v) = argmax{ρ| :s∈ P0(v)}, we assume that for some absolute constant c > 0, ...