Incomplete Line LU as smoother and as preconditioner
Results are reported for Incomplete line LU (ILLU) in two different roles. One role is the role of smoother in a multigrid method for the solution of linear systems resulting from the 9-point discretization of a general linear second-order elliptic PDE in two dimensions. Together with features like matrix-dependent gridtransferoperators we obtain a blackbox multigrid solver (MGD9V). Another role for ILLU is as preconditioner in a stabilised bi-cg method (Bi-CGSTAB), as recently developed by Van der Vorst. In this role the preconditioner can easily be generalized for a discretized system of PDEs. A comparison is made between the two different roles.
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- P.W. Hemker and P.M. de Zeeuw, Some implementations of multigrid linear system solvers in: D.J. Paddon and H. Holstein (Eds.) Multigrid methods for integral and differential equations ,Inst. Math. Appl. Conf. Ser. New Ser. (Oxford Univ. Press, New York), pp. 85–116, 1985.Google Scholar
- R. Kettler, Analysis and comparison of relaxation schemes in robust multigrid and precon ditioned conjugate gradient methods ,in: W. Hackbusch and U. Trottenberg (Eds.), Lecture Notes in Math. 960 (Springer, Berlin), pp. 502–534, 1981.Google Scholar
- R. Kettler and J.A. Meijerink, A multigrid method and a combined multigrid-conjugate gradient method for elliptic problems with strongly discontinuous coefficients in general domainsShell Publ. 604, KSEPL, Rijswijk, The Netherlands1981.Google Scholar
- P. Sonneveld, P. Wesseling and P.M. de Zeeuw, Multigrid and conjugate gradient methods as convergence acceleration techniques ,in: D.J. Paddon and H. Holstein (Eds.) Multigrid methods for integral and differential equations ,Inst. Math. Appl. Conf. Ser. New Ser. (Oxford Univ. Press, New York), pp. 117–167, 1985.Google Scholar
- R.R. Underwood, An approximate factorization procedure based on the block Cholesky de composition and its use with the conjugate gradient method ,Report NEDO-11386, General Electric Co., Nuclear Energy Div., San Jose, California, 1976.Google Scholar
- H.A. Van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems ,SIAM J. Sci. Statist. Comput. 13(2), 1992.Google Scholar
- P. Wesseling, A robust and efficient multigrid method ,in: W. Hackbusch and U. Trottenberg (Eds.), Lecture Notes in Math. 960 (Springer, Berlin), pp. 614–630, 1981.Google Scholar
- P. Wesseling, An Introduction to Multigrid Methods ,John Wiley & Sons Ltd, Chichester, England, 1991.Google Scholar
- P. Wesseling, P. Sonneveld, Numerical experiments with a multiple grid and a preconditioned Lanczos type method ,in: R. Rautmann (ed.), Approximation methods for Navier-Stokes problems ,Proceedings, Paderborn, Germany 1979, Lecture Notes in Mathematics, Springer, Berlin-etc, 1980.Google Scholar
- P.M. de Zeeuw, Incomplete line LU for discretized coupled PDEs as preconditioner in Bi-CGSTAB ,Report, Centre for Mathematics and Computer Science, Amsterdam, to appear.Google Scholar