Taking Advantage of the Potentialities of Dynamically Modified Block Incomplete Factorizations

Recently, modified block incomplete factorizations with dynamic diagonal perturbations have been introduced as preconditioning techniques to solve large linear systems, and were successfully tested on isotropic and moderately anisotropic two-dimensional partial differential equations (PDEs). An analytic study is performed on the basis of improved versions of results published elsewhere, displaying why dynamic methods should be preferred to standard block methods; in particular the empirical observation that the optimal choice of the involved parameter does not significantly vary from one problem to another is theoretically confirmed. It is also explained why, in the case of strong anisotropy, lack of attention when adding diagonal perturbations may result in very poor performance. Rules to surmount this inconvenience are discussed and tested on discretized two-dimensional PDEs.

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