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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)

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 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)

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)

A restricted additive Schwarz (RAS) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems. In this paper, we provide one-level and two-level extensions of RAS for symmetric positive definite problems using the so-called harmonic overlaps (RASHO). Both RAS and RASHO outperform their counterparts of the… (More)

In this paper, we study several overlapping domain decomposition based iterative algorithms for the numerical solution of some non-linear strongly elliptic equations discretized by the nite element methods. In particular, we consider additive S c hwarz algorithms used together with the classical inexact Newton methods. We s h o w that the algorithms… (More)

Two variants of the additive Schwarz method for solving linear systems arising from the mortar finite element discretization on non-matching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed. The methods are defined on subdo-mains without overlap, and they use special coarse spaces, resulting in algorithms… (More)

We consider two-dimensional elliptic problems with discontinuous coefficients dis-cretized by the finite element method on geometrically conforming nonmatching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a dual-primal FETI method. In this paper we introduce and analyze a preconditioner with a… (More)