Maksymilian Dryja

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Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in nite element or nite diierence approximations of partial diierential equations. The preconditioners are constructed from exact or approximate solvers for the same partial diierential equation restricted to a set(More)
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)
In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual{ primal FETI methods which have recently been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is(More)
Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the(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)
In this paper we study several overlapping domain decompo sition 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 Schwarz algorithms used together with the classical inexact Newton methods We show that the algorithms con verge and(More)
1.1 Introduction This chapter develops two additive Schwarz methods with new coarse spaces. The methods are designed for elliptic problems in 2 and 3 dimensions with discontinuous coeecients. The methods use no explicit overlap of the subdomains, subdomain interaction is via the coarse space. The rst method has a rate of convergence proportional to (H=h)(More)
Second order elliptic problems with discontinuous coefficients are considered. The problem is discretized by the finite element method on geometrically conforming non-matching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a FETI-DP method. We prove that the method is convergent and its rate of(More)