Parallel Computing 12 (1989) 131-144 131 North-Holland Asynchronous multilevel adaptive methods for solving partial differential equations on multiprocessors: Basic ideas * L. HART and S. McCORMICK Center for Applied Parallel Processing and Computational Mathematics Group, The University of Colorado at Denver, 1200 Larimer Street, Denver, CO 80204, U~S.A. Received October 1987 Revised December 1988 Abstract. Several mesh refinement methods exist for solving partial differential equations that make efficient use of local grids on scalar computers. On distributed memory multiprocessors, such methods benefit from their tendency to create multiple refinement regions, yet they suffer from the sequential way that the levels of refinement are treated. The asynchronous fast adaptive composite grid method (AFAC) is developed here as a method that can process refinement levels in parallel while maintaining full multilevel convergence speeds. In the present paper, we develop a simple two-level AFAC theory and provide estimates of its asymptotic convergence factors as it applies to very large scale examples. In a companion paper, we report on extensive timing results for AFAC, implemented on an Intel iPSC hypercube. Keywords. Fast adaptive composite grid method (FAC), partial differential equations, asynchronous FAC, distributed memory multiprocessors. 1. Introduction Developed originally in 1983 [5], the fast adaptive composite grid method (FAC) is a multilevel scheme that nominally uses global and local uniform grids for adaptive solution of partial differential equations. As with other such methods, it provides some parallelism by typically producing several independent refinement regions, but is hampered by the need to handle the refinement levels sequentially. In this paper, we introduce an asynchronous version of FAC, called AFAC, that overcomes this difficulty; AFAC allows for processing of the refinement levels in a parallel mode (simultaneous or asynchronous) without significant loss of performance. The development of AFAC will be based first on viewing FAC as a block Gauss-Seidel method applied to a particular singular system of equations; AFAC will then be introduced as a block Jacobi scheme applied to this system that has been modified to remove the singularity. This version of AFAC is actually the simultaneous one, which is easiest to analyze and most suitable for current applications; the asynchronous version corresponds to chaotic block relaxation of the modified system. * This work was supported by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the Department of Energy under grant number DE-AC03-84ER. 0167-8191/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)