In recent years competitive domain decomposed preconditioned iterative tech niques have been developed for nonsymmetric elliptic problems In these tech niques a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow e ective solution on parallel machines A central question is how to choose these small problems and how to arrange the or...

The notions of N = 1 Neveu-Schwarz vertex operator superal-gebra over a Grassmann algebra and with odd formal variables and of N = 1 Neveu-Schwarz vertex operator superalgebra over a Grass-mann algebra and without odd formal variables are introduced, and we show that the respective categories of such objects are isomorphic. The weak supercommutativity and weak associativity properties for an N ...

The study of optimized Schwarz methods for Maxwell’s equations started with the Helmholtz equation, see [2, 3, 4, 11]. For the rot-rot formulation of Maxwell’s equations, optimized Schwarz methods were developed in [1], and for the more general form in [9, 10]. An entire hierarchy of families of optimized Schwarz methods was analyzed in [8], see also [5] for discontinuous Galerkin discretizatio...

In this paper, we introduce a new Schwarz preconditioner and with a new coarse space. We construct the preconditioner by grouping together, in one preconditioner, features from the additive overlapping Schwarz methods, from iterative substructuring methods and from a class of restricted additive Schwarz methods. The preconditioner is symmetric and considered as a symmetrized version of restrict...

Newton Krylov methods and Krylov Schwarz domain decomposition methods have begun to become established in computational uid dynamics CFD over the past decade The former employ a Krylov method inside of Newton s method in a Jacobian free manner through directional di erencing The latter employ an overlapping Schwarz domain decomposition to derive a preconditioner for the Krylov accelerator that ...

This paper begins with an introduction to additive and multiplicative Schwarz methods. A two-level method is then reviewed and a new result on its rate of convergence is established for the case when the overlap is small. Recent results by Xuejun Zhang, on multi-level Schwarz methods, are formulated and discussed. The paper is concluded with a discussion of recent joint results with Xiao-Chuan ...