Parallel elliptic preconditioners: Fourier analysis and performance on the connection machine

We study the performance of several widely used preconditioners for 2D elliptic partial differential equations (SSOR, ILU, MILU and polynomial preconditioners) with the natural and red-black orderings implemented on the Connection Machine (CM). Their performance is primarily influenced by two factors: the rate of convergence and the ease of parallelization. The convergence rate is analyzed by Fourier analysis and confirmed with experimental results. Although the naturally ordered SSOR and MILU preconditioners have convergence rates one order faster than the other preconditioners, the experiments show that the red-black ordered SSOR, ILU, MILU, polynomial preconditioners takes less execution time than their naturally ordered counterparts. This is partially due to the fact that the red-blavk ordering provides more parallelism than the natural ordering.