Dynamically Relaxed Block Incomplete Factorizations for Solving Two- and Three-Dimensional Problems

Abstract. To efficiently solve second-order discrete elliptic PDEs by Krylov subspace-like methods, one needs to use some robust preconditioning techniques. Relaxed incomplete factorizations (RILU) are powerful candidates. Unfortunately, their efficiency critically depends on the choice of the relaxation parameter ω whose “optimal” value is not only hard to estimate but also strongly varies from one problem to another. These methods interpolate between the popular ILU and its modified variant (MILU). Concerning the pointwise schemes, a new variant of RILU that dynamically computes variable ω = ωi has been introduced recently. Like its ancestor RILU and unlike standard methods, it is robust with respect to both existence and performance. On top of that, it breaks the problem-dependence of “ωopt.” A block version of this dynamically relaxed method is proposed and compared with classical pointwise and blockwise methods as well as with some existing “dynamic” variants, showing that with the new blockwise preconditioning technique, anisotropies are handled more effectively.