The Development of Coarse Spaces for Domain Decomposition Algorithms

The importance of using coarse components, and thus at least one additional level, in the design of domain decomposition methods has been understood for at least twenty years. For many problems of interest, such a device, which provides at least a minimal amount of global transfer of information in each step, is necessary in order to obtain convergence rates which are independent of the number of subdomains. An historical overview, colored by the scientific history of its author, is given of the development of such coarse components of the domain decomposition algorithms. These algorithms are all preconditioned conjugate gradient methods or they are accelerated by using some alternative Krylov space method. The preconditioners are built from solvers of the given problem restricted to subdomains and a coarse approximation which often can be quite exotic.