A Note on the Efficiency of Domain Decomposed Incomplete Factorizations

In the past several years, domain decomposition has been a very popular topic, motivated by the ease of parallelization. However, the question of whether it is better than parallelizing some standard sequential methods has seldom been directly addressed. In this paper it is shown, with some numerical examples, that the answer is affirmative in the case of iterative solutions of elliptic problems by preconditioned conjugate gradient iteration. Specifically shown is how to construct effective incomplete factorization preconditioners based on the domain decomposition principle. In addition to having all the advantages of domain decomposition, this also results in better convergence rates than the analogous preconditioners on the whole domain.