On the stability of relaxed incomplete factorizations

Authors:
A. M. Bruaset, A. Tveito and R. Winther

Journal:
Math. Comp. **54** (1990), 701-719

MSC:
Primary 65F10; Secondary 15A23, 65N20

MathSciNet review:
993924

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Abstract | References | Similar Articles | Additional Information

Abstract: When solving large linear systems of equations arising from the discretization of elliptic boundary value problems, a combination of iterative methods and preconditioners based on incomplete LU factorizations is frequently used. Given a model problem with variable coefficients, we investigate a class of incomplete LU factorizations depending on a relaxation parameter. We show that the associated preconditioner and the factorization itself both are numerically stable. The theoretical results are complemented by numerical experiments.

**[1]**O. Axelsson and V. A. Barker,*Finite element solution of boundary value problems*, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. Theory and computation. MR**758437****[2]**Owe Axelsson and Gunhild Lindskog,*On the eigenvalue distribution of a class of preconditioning methods*, Numer. Math.**48**(1986), no. 5, 479–498. MR**839613**, 10.1007/BF01389447**[3]**Owe Axelsson and Gunhild Lindskog,*On the rate of convergence of the preconditioned conjugate gradient method*, Numer. Math.**48**(1986), no. 5, 499–523. MR**839614**, 10.1007/BF01389448**[4]**Petter E. Bjørstad and Olof B. Widlund,*Iterative methods for the solution of elliptic problems on regions partitioned into substructures*, SIAM J. Numer. Anal.**23**(1986), no. 6, 1097–1120. MR**865945**, 10.1137/0723075**[5]**J. H. Bramble, J. E. Pasciak, and A. H. Schatz,*The construction of preconditioners for elliptic problems by substructuring. I*, Math. Comp.**47**(1986), no. 175, 103–134. MR**842125**, 10.1090/S0025-5718-1986-0842125-3**[6]**Todd Dupont, Richard P. Kendall, and H. H. Rachford Jr.,*An approximate factorization procedure for solving self-adjoint elliptic difference equations*, SIAM J. Numer. Anal.**5**(1968), 559–573. MR**0235748****[7]**Howard C. Elman,*A stability analysis of incomplete 𝐿𝑈 factorizations*, Math. Comp.**47**(1986), no. 175, 191–217. MR**842130**, 10.1090/S0025-5718-1986-0842130-7**[8]**Ivar Gustafsson,*A class of first order factorization methods*, BIT**18**(1978), no. 2, 142–156. MR**499230**, 10.1007/BF01931691**[9]**Ivar Gustafsson,*On modified incomplete Cholesky factorization methods for the solution of problems with mixed boundary conditions and problems with discontinuous material coeff*, Internat. J. Numer. Methods Engrg.**14**(1979), no. 8, 1127–1140. MR**539209**, 10.1002/nme.1620140803**[10]**-,*Stability and rate of convergence of modified incomplete Cholesky factorization methods*, Ph.D. thesis, Department of Computer Sciences, Chalmers University of Technology and the University of Göteborg, Sweden, 1979. Also available as Research Report 79.02R.**[11]**J. A. Meijerink and H. A. van der Vorst,*An iterative solution method for linear systems of which the coefficient matrix is a symmetric 𝑀-matrix*, Math. Comp.**31**(1977), no. 137, 148–162. MR**0438681**, 10.1090/S0025-5718-1977-0438681-4

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DOI:
https://doi.org/10.1090/S0025-5718-1990-0993924-X

Article copyright:
© Copyright 1990
American Mathematical Society