Computing 54, 167-183 (1995) ~ 1 ~ 9 Springer-Verlag 1995 Printed in Austria Explicit P r e c o n d i t i o n e d Iterative M e t h o d s for Solving Large U n s y m m e t r i c Finite E l e m e n t S y s t e m s E. A. Lipitakis and G. A. Grawanis, Athens Received November 25, 1993; revised May 30, 1994 Abstract-- Zusammenfassung Explicit Preconditioned lterative Methods for Solving Large Unsymmetric Finite Elemen A class of Generalized Approximate Inverse Matrix (GAIM) techniques, based on the concept LU-sparse factorization procedures, is introduced for computing explicitly approximate inverses large sparse unsymmetric matricesof irregular structure, without inverting the decomposition factors. Explicit preconditioned iterative methods, in conjunction with modified forms of the GAIM techniques, are presented for solving numerically initial/boundary value problems on multiprocessor systems. Application of the new methods on linear boundary-value problems is discussed and numerical results are given. AMS Subject Classification: 65F10 Key words." Approximateinverse matrix techniques, explicit precouditioners, parallel iterative methods, unsymmetric finite element systems, initial/bounda13,-value problems. Explizite Prfikonditionierungsverfahren zur Liisung groBer unsymmetrischer Finite-Element- Systeme. Es wird eine Methode zur Approximation der verallgemeinerten inversen Matrix (GAIM) diskutiert, die auf demKonzept der schwachbesetzten LU-Faktorisierung basiert und explizite Inverse grol3er schwachbesctzter unsymmetrischer Matrizen auf irregul~iren Strukturen approxim ohne die Zerlegungsfaktoren zu invertieren. In Verbindungmit Modifikationen der GAIM- Technik werden explizite Pr~ikonditionierungsmethoden zur numerischenL~3sung yon An- fangsrandwertproblemen auf Multiprozessorsystemen vorgestellt. Anwendungen der neuen Methoden auf lineare Randwertaufgaben werden diskutiert und numerische Resultate pr~isent 1. Introduction In recent years researchefforts have been directed to many aspects of the production of numerical software,and in particular, on the development of efficient and accurate algorithmic procedures for solving computational prob- lems on available machines (uniprocessor or multiprocessor systems), on tech- niques for analysing these algorithms and on their implementations. Current research efforts are now focusing on the development, testing and analysis of new numerical algorithms that are needed to effectively exploit multiprocessor systems. There are severalprinciples involvedin the parallel algorithm design, i.e. vectorization, "divide and conquer" partitioning, recursive doubling, designing of complete new parallel algorithms, conversion of imp