Lower bounds for the condition numbers of the preconditioned systems are obtained for two-level additive Schwarz preconditioners for both second order and fourth order problems. They show that the known upper bounds are sharp in the case of a small overlap.

L’œuvre commune ou individuelle d’André et de Simone Schwarz-Bart constitue un champ d’expérimentation réflexion particulièrement riche sur la question l’intertextualité. Cette notion, délestée toute assignation théorique trop contraignante, est envisagée dans ce dossier thématique sous l’angle du dynamisme relationnel qu’elle engage. Elle invite à une collective les liens que l’œuvre schwarz-b...

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers several Schwarz alternating methods for nonlinear elliptic p...

Overlapping balancing domain decomposition methods and their combination with restricted additive Schwarz methods are proposed for the Helmholtz equation. These new methods also extend previous work on non-overlapping balancing domain decomposition methods toward simplifying their coarse problems and local solvers. They also extend restricted Schwarz methods, originally designed to overlapping ...

The Schwarz map of the hypergeometric differential equation is studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is the hyperbolic 3-space. This map can be considered to be a lifting to the 3-space of the Schwarz map. This paper studies the singularities of this map, and visual...

Tissue engineering scaffolds provide temporary mechanical support for tissue regeneration. To regenerate tissues more efficiently, an ideal structure of scaffolds should have appropriate porosity and pore structure. In this paper, we generate the Schwarz primitive P surface with various volume fractions using a phase-field model. The phase-field model enables us to design various surface-to-vol...

In this paper, we consider the well known Morley nonconforming element approximation of a nonlinear biharmonic equation which is related to the well-known two-dimensional Navier–Stokes equations. Firstly, optimal energy and H1-norm estimates are obtained. Secondly, a two-level additive Schwarz method is presented for the discrete nonlinear algebraic system. It is shown that if the Reynolds numb...

In recent years, an algebraic framework was introduced for the analysis of convergence of Schwarz methods for the solution of linear systems of the form Ax = b. Within this framework, additive and multiplicative Schwarz were shown to converge when the coefficient matrix A is a nonsingular M -matrix, or a symmetric positive definite matrix. In this paper, these results are extended to the case o...

The Schwarz Alternating Method was devised by H. A. Schwarz more than one hundred years ago to solve linear boundary value problems. It has garnered interest recently because of its potential as an efficient algorithm for parallel computers. See [Lio88], and [Lio89], the recent reviews [CM94], [LT94], and [XZ98], and the books [SBG96] and [QV99]. The literature for nonlinear problems is rather ...

We show that N = 8 spontaneously broken supergravity in four dimensions obtained by Scherk–Schwarz generalized dimensional reduction can be obtained from a pure four dimensional perspective by gauging a suitable electric subgroup of E7,7. Owing to the fact that there are non isomorphic choices of maximal electric subgroups of the U-duality group their gaugings give rise to inequivalent theories...