Abstract. In this paper a multilevel-like ILU preconditioner is introduced. The ILU factorization generates its own ordering during the elimination process. Both ordering and dropping depend on the size of the entries. The method can handle structured and unstructured problems. Results are presented for some important classes of matrices and for several well-known test examples. The results illustrate the efficiency of the method and show in several cases near grid independent convergence. Key words. multilevel methods, preconditioning, ILU, dropping strategies, Krylov-subspace methods AMS subject classifications. 65F10, 65N06PII. S0895479897319301 1. Introduction. Solving large sparse systems of equations continues to be a major research area. This attention is caused by the fact that solving such equations forms the bottleneck in many practical problems. For really large systems direct methods become too expensive in CPU time and storage requirements, and therefore an iterative approach is needed. In particular the use of preconditioned CG-type methods has proved to be very competitive. It is also widely recognized that the quality of the preconditioner determines the success of the iterative method. With a proper preconditioner the choice of the CG-like accelerator is not that critical.The preconditioner presented in this paper is a special multilevel-like incomplete factorization. In this introduction we briefly describe the various incomplete decomposition approaches available today and their relation to the approach presented here.The history of ILU factorizations is amongst others described in [15]. Moreover, historical notes are to be found in the textbooks of Axelsson [1], Hackbusch [31], and Saad [50]. The first roots of the approach lie in the 1960s [13,42,43] and since then the method has become applicable to a wide class of problems. Furthermore, analyses for important classes of matrices could be made. Today, ILU factorizations are an important tool for solving large-scale problems.Classical ILU approach. The classical approach is to allow only fill entries in the L and U factors, where the original matrix A has nonzeros. This simple approach allows for a very efficient implementation by Eisenstat [25] and is still very popular.As observed by Dupont, Kendall, and Rachford [23], an important improvement in the convergence of the classical approach can be obtained by lumping the dropped elements onto the diagonal. With this modification, the factorization, called MILU, is made exact for a constant vector. For a more general matrix A Gustafsson [30] found a similar result. For many second-order elliptic problems, the preconditioning with the classical ILU gives asymptotically the same condition number as with diagonal scaling, i.e., O(h −2 ). After this simple modification this improves to O(h −1 ). For Mmatrices the existence of ILU factorizations can be proved [35], but this is not the case

In this paper, a fully implicit numerical model of the three-dimensional thermoha-

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On the occasion of the third centenary of the appointment of Johann Bernoulli at the University of Groningen, a number of linear systems solvers for some Laplace-like equations have been compared during a one-day workshop. CPU times of several advanced solvers measured on the same computer (an HP-755 workstation) are presented, which makes it possible to draw clear conclusions about the performance of these solvers. © 1997 Elsevier Science B.V.

SUMMARYThe numerical method of soluti0n Of van de Vooren and Dijkstra [-1] for the semi-infinite flat plate has been extended to the case of the parabolic cylinder. Results are presented for the skin friction, the friction drag, the pressure and the pressure drag. The drag coefficients have been checked by means of an application of the momentum theorem.