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 interior to the subregions have been eliminated. In this paper, a supporting theory is developed.

. 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 finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi-level methods. The Neumann-Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. However, in its original form, the algorithm lacks a mechanism for global transportation of informatio...

The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in H −1 (Ω) (the Sobolev space of order minus one on Ω). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.

. In this paper, we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent Stokes equations using a mixed finite element spatial discretization. At each time step, a two by two block system corresponding to a perturbed Stokes problem must be solved. Although there are a number of techniques for iteratively solving this type of block system, to be effective, they require a good preconditioner for the resulting pressure operator (Schur complement). In contrast to the time independent Stokes equations where the pressure operator is well conditioned, the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced (and/or the Reynolds number is increased). In this paper, we shall describe and analyze preconditioners for the resulting pressure systems. These preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as t...

Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.

Abstract. In the Dual-Primal FETI method, introduced by Farhat et al. [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by C(1 + log2 (H/h)) for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model. 1. Introduction. This

. The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second-order elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuring--type methods and the Neumann--Neumann-type methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include local-global and global-local techniques. The analyses for both two- and three-dimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived. Key words. nonoverlapping domain decomposition, Schur complement, local-global and globallocal techniques, jumps in coe#cients, substructuring, Neumann--Neumann, balancing methods AMS subject classifications. 65N30, 65N55, 65F10 PII. S0036144596306800 1. Introduction. T...

Abstract. The classical Schwarz alternating method has recently been generalized in several directions. This e ort has resulted in a number of new powerful domain decomposition methods for elliptic problems, in new insight into multigrid methods and in the development ofavery useful framework for the analysis of a variety of iterative methods. Most of this work has focused on positive de nite, symmetric problems. In this paper a general framework is developed for multiplicativeSchwarz algorithms for nonsymmetric and inde nite problems. Several applications are then discussed including two- and multi-level Schwarz methods and iterative substructuring algorithms. Some new results on additive Schwarz methods are also presented. Key words. nonsymmetric elliptic problems, preconditioned conjugate gradient type methods, nite elements, multiplicative Schwarz algorithms AMS(MOS) subject classi cations. 65F10, 65N30, 65N55

. Two-level domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems. A bound is established for the condition number of these iterative methods, which grows only logarithmically with the number of degrees of freedom in each subregion. This bound holds for two and three dimensions and is independent of jumps in the value of the coefficients. Key words. domain decomposition, elliptic problems, preconditioned conjugate gradients, nonconforming finite elements, Schwarz methods AMS(MOS) subject classifications. 65F10, 65N30, 65N55 1. Introduction. The purpose of this paper is to develop a domain decomposition methods for second order elliptic partial differential equations approximated by a simple nonconforming finite element method, the nonconforming P 1 elements. We consider a variant of a two-level additive Schwarz method introduced in 1987 by Dryja and Widlund [5] for a conforming case. In these methods, a preconditioner i...

Part I. The aim of this paper is to study the tangential trace and tangential components of fields which belong to the space H(curl ;\Omega\Gamma2 when\Omega is a polyhedron with Lipschitz continuous boundary. The appropriate functional setting is developed in order to suitably define these traces on the whole boundary and on a part of it (for partially vanishing fields and general ones.) In both cases it is possible to define ad hoc dualities among tangential trace and tangential components and the validity of two related integration by parts formulae is provided. Part II. Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L 2 tangential fields and then the attention is focused on some particular Sobolev spaces of order \Gamma1=2. In order to reach this goal, it is required to properly define the first order differential operators and to study their properties. When the manifold \Gamma is the boundar...