Generalized Neumann - Neumann 485

INTRODUCTION In the past decade signiicant research eeorts have been applied to develop robust algorithms using domain decomposition methods. From both a theoretical and a numerical point of view these methods can now be considered mature. For widespread use, these methods must be accessible to the non-specialist and thus must be available as black boxes. This paper addresses the issues faced when trying to maintain an open software platform suitable for ongoing research while simultaneously meeting commercial code requirements as black boxes. The characteristics of the generalized Neumann-Neumann algorithm, a good black box domain decomposition algorithm candidate, are deened and illustrated with speciic examples. First the general methodology used to practically solve a given problem is described. The next sections review the main characteristics of the Neumann-Neumann preconditioner and discusses the key points of its practical implementation and use. This algorithm is implemented whitin the framework of the general purpose nite element library Modulef. For message passing the PVM library is used. Thus the software is portable and can be run on diierent platforms including standard parallel computers and clusters of workstations. To achieve eeciency and reliability modern software must be open and modular. An additional important feature is that it should be easlly reused in diierent situations. On the other hand, in order to promote the use of numerical methods for non-specialists, it is important to develop black boxes. A black box is a specialized software which is easy to use and minimizes the user's decisions. In addition, it has to be robust and eecient. The question is how to fullll the apparent connicting requirements of both modern software and black boxes. For numerical analysts who use an open software, the solution of a problem is built using a selection of appropriate modules. In general several solution methods are available and it is the user's responsibility to choose the best ones depending, for instance, on the problem type or size. Modules for direct or iterative solutions of linear systems are typical examples of software used in all such constructions. On the contrary, a black box contains all steps involved in the modeling process and corresponds to a particular choice at each step. The approach proposed to derive a black box from an open software for a given problem is the following: Select the optimal solution method, i.e. one whith serious theoretical background and proved to be robust and reliable. …