Vectorizable preconditioners for elliptic difference equations in three space dimensions

摘要

Until now, research in vectorizable preconditioners has mainly focused on problems in two space dimensions. Employing all degrees of freedom, we here present preconditioners for 3D-matrices based on point, line and plane factorizations, both in vectorizable and (for reasons of comparisons) in recursive versions. By a limit analysis and accompanying numerical tests of elliptic difference equations with anisotropy we find that methods based on line-block factorizations are to be preferred in general; among the vectorizable methods a line method where Euler expansion of the factors of the pivot blocks is used, is the most robust with respect to the problem parameters. Furthermore, this method turns out to approach the best of the recursive methods in efficiency in terms of operation counts; it has the added bonus of almost maximal vectorlength.