Incomplete block LU preconditioners on slightly overlapping subdomains for a massively parallel computer

Abstract

The ILU (Meijerink and van der Vorst, 1977; 1981) and MILU (Gustafsson, 1978) preconditioners have become more or less the standard for preconditioning. On parallel computers, however, the inherent sequentiality of these preconditioners precludes efficient implementation. Replacing (M)ILU by blocked variants may seem a good way to improve the parallelism, but generally these blocked variants come with a penalty in the form of more iterations. We will consider possibilities to improve the convergence of the block preconditioners by adapting ideas from Radicati and Robert (1989) and Tang (1992). We will use as preconditioner(s) the incomplete factorizations of the local systems of equations corresponding to slightly overlapping subdomains with certain parameterized algebraic boundary conditions. Although the selection of the optimal parameters is still an open problem, numerical experiments suggest that the iteration count can be almost constant when going from the sequential case to as much as 400 subdomains. We will also give details on the performance on a 400-processor parallel computer.