Multi-Level Asynchronous Iteration for PDEs

Consider solving a PDE on a domain D using a domain decomposition or Schwarz splitting approach. We decompose D several times, with each level having a finer (more accurate) decomposition than the preceding one. Iterative methods are initiated on each level so long as the higher level iterations have not converged. This is implemented in a parallel computing environment (such as a hypercube machine). The passing of information between levels is asynchronous. Parameters of such a method are (1) number of levels and number of domains per level, (2) number of processors per level, (3) choice of method on the subdomains of each level (and their convergence rates), and (4) the tolerances used to terminate higher level iterations. We report on both analytic and experimental studies of the effectiveness of such methods and of good values for the parameters.