Solving Composite Problems with Interface Relaxation

Abstract

This paper deals with a solution approach suitable for composite partial differential equations (PDEs) with interface conditions. We present a general framework based on interface relaxation which provides a uniform platform for building problem solving environments through efficient software integration and for implementing various relaxation schemes. Mathematically, this framework contains many existing domain decomposition methods and also allows the extension to a variety of new relaxers. In particular, we describe a class of relaxers which are suitable for general and complicated interface conditions. Interface relaxation is more general than the traditional domain decomposition methods in that it allows unrelated PDE problems on different subdomains. Convergence analysis, error estimates, and preconditioning strategies are presented which show that these relaxers are competitive with existing domain decomposition methods for model problems involving a single PDE. We present experimental results which demonstrate the wide applicability of this approach. Differences between this approach and other domain decomposition methods are also discussed.