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Numerical experiments have shown that two-level Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables… (More)

Several domain decomposition methods of Neumann-Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by nite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that… (More)

Domain decomposition methods without overlapping for the approximation of parabolic problems are considered. Two kinds of methods are discussed. In the rst method systems of algebraic equations resulting from the approximation on each time level are solved iteratively with a Neumann-Dirichlet preconditioner. The second method is direct and similar to… (More)

Multilevel Schwarz methods are developed for a conforming nite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coeecients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the… (More)

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using… (More)

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… (More)

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2-D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ω i of size O(H i). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ω i , a conforming finite… (More)

A BDDC (balancing domain decomposition by constraints) algorithm is developed for elliptic problems with mortar discretizations for geometrically non-conforming partitions in both two and three spatial dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face which is an intersection of the… (More)

— A second order elliptic problem with highly discontinuous coefficients has been considered. The problem is discretized by two methods: 1) continuous finite element method (FEM) and 2) composite discretization given by a continuous FEM inside the substructures and a discontinuous Galerkin method (DG) across the boundaries of these substructures. The main… (More)