Collaborating PDE solvers

Collaborating PDE solvers refers to a methodology for solving sets of partial differential equations (PDEs) by iteratively solving one PDE on one domain at a time. The set of PDEs are related through interface conditions which, in their simplest form, are shared boundary conditions. For example, two rectangles with a common edge might have the requirement that the solutions across the edge be continuous and have a continuous first derivative. Schwartz splitting is a classic method of this nature and in recent years other instances have been found which are effective. In this paper we explore the possibilities that (a) the methodology might be effective for a wide range of PDE problems, (b) it might be one of the more effective methods for the enormously complex PDE problems (e.g., complete simulation of a vehicle) that will be attempted in this decade. The paper consists of three parts: a brief discussion of potential applications, a discussion of the interface relaxation problem, and description of the RELAX system for experimenting with this methodology.