Abstract
In the present paper we introduce truncated incomplete decompositions (TrILU) for constant coefficient matrices. This new ILU variant saves most of the memory and work usually needed to compute and store the factorization. Further it improves the smoothing and preconditioning properties of standard ILU-decompositions. Besides describing the algorithm, we give theoretical results concerning stability and convergence as well as the smoothing property and robustness for TrILU smoothing in a multi-grid method. Further, we add numerical results of TrILU as smoother in a multi-grid method and as preconditioner in a pcg-method fully confirming the theoretical results.
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References
Axelsson, O.:A survey of preconditioned iterative methods for linear systems of algebraic equations. BIT 25 (1985), 166–187.
Axelsson, O., Barker, V. A.:Finite element solutions of boundary value problems. Theory and computation. Academic Press, New York, 1985.
Axelsson, O., Lindskog, G.:On the eigenvalue distribution of a class of preconditioning methods. Numer. Math. 48, 479–498 (1986).
Axelsson, O., Lindskog, G.:On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math. 48, 499–523 (1986).
Gustafsson, I.:A class of 1st order factorization methods. Report 77.04, Department of Computer Sciences, Chalmers University, Göteborg (1977).
Hackbusch, W.:Multi-Grid Methods and Applications. Springer, Berlin, Heidelberg (1985).
Hackbusch, W.:Theorie und Numerik elliptischer Differentialgleichungen. Teubner, Stuttgart (1986).
Kettler, R.:Incomplete factorizations as smoothers in multi-grid methods. Thesis, Dept. of Mathematics, TU Delft (1988).
Liebau, F.:Unvollständige LU-Zerlegungen mit angenäherten Koeffizienten als Glätter in Mehrgitterverfahren. Diplomarbeit, Institut für Informatik und Praktische Mathematik, Kiel (1987).
Meijerink, J. A., Van der Vorst, H. A.:An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp. 31 (1977), 148–162.
Meijerink, J. A., Van der Vorst, H. A.:Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems. J. Comp. Phys. 44 (1981), 134–155.
Varga, R. S.:Matrix Iterative Analysis. Prentice Hall (1962).
Wesseling, P.:Theoretical and practical aspects of a multigrid method. SIAM J. Sci. Statist. Comp. 3 (1982), 387–407.
Wittum, G.:Distributive Iterationen für indefinite Systeme. Kiel, 1986.
Wittum, G.:On the robustness of ILU-smoothing. SIAM Journal of Scientific and Statistical Computing, 10 (1989), pp. 699–717.
Wittum, G.:Multi-grid methods for Stokes and Navier-Stokes equations. Transforming smoothers-algorithms and numerical results. Numerische Mathematik, 54, 543–563 (1989).
Wittum, G.:Linear Iterations as Smoothers in Multi-Grid Methods. Impact of Computing in Science and Engineering, 1 (1989),pp. 180–212.
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This work was supported by Deutsche Forschungsgemeinschaft.
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Wittum, G., Liebau, F. On truncated incomplete decompositions. BIT 29, 719–740 (1989). https://doi.org/10.1007/BF01932742
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DOI: https://doi.org/10.1007/BF01932742