Asynchronous multilevel adaptive methods for solving partial differential equations on multiprocessors: Basic ideas

Several mesh refinement methods exists 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.