Abstract
We show that the alternating direction implicit algorithm of Peaceman—Rachford can be adapted to solve linear complementarity problems arising from free boundary problems. The alternating direction implicit algorithm is significantly faster than modified SOR algorithms. Some numerical results are given.
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References
Duvaut G and Lion JL (1976) Inequalities in mechanics and physics. Paris: Dunod
Cryer CW (1977) A bibliography of free boundary problems. Technical Summary Report No. 1793, Mathematics Research Center, University of Wisconsin, Madison
Cottle RW, Giannessi F and Lions JL (eds) (1980) Variational inequalities and complementarity problems. New York: John Wiley
Kinderlehrer G and Stampacchia G (1980) An introduction to variational inequalities and their applications. New York: Academic Press
Glowinski R, Lions JL, and Tremolieres R (1981) Numerical analysis of variational inequalities. Amsterdam: North-Holland
Balinski ML and Cottle RW (1978) Complementarity and fixed point problems. Amsterdam: North-Holland
Cryer CW (1971) The solution of quadratic programming problem using systematic overrelaxation. SIAM J Control 9:385–392
Cottle RW, Golub GH, and Sacher RS (1978) On the solution of large, structured linear complementarity problems: the block partitioned case. Appl Math Optim 4:347–363
Varga RS (1962) Matrix iterative analysis. Englewood Cliffs, Prentice-Hall
Brandt A and Cryer CW (1980) Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems. Technical Summary Report No. 2131, Mathematics Research Center, University of Wisconsin, Madison
Cottle RW and Sacher RS (1977) On the solution of large, structured linear complementarity problems: the tridiagonal case. Appl Math Optim 3:321–340
Cryer (1983) The efficient solution of linear complementarity problems for tridiagonal Minkowski matrices. ACM Transactions on Mathematical Software 9:199–214
Schechter S (1968) Relaxation methods for convex problems. SIAM J Numer Anal 5:601–602
Mangasarian OL (1977) Solution of symmetric linear complementarity problems by iterative methods. J Optim Theory Appl 22:465–485
Pang JS (1982) On the convergence of a basic iterative method for the implicit complementarity problem. J Optim Theory 37:149–162
Birkhoff G, Varga RS, and Young D (1962) Alternating direction implicit methods. Advances in Computers 3:189–273
Cottle RW (1974) Computational experience with large-scale linear complementarity problems. Technical Report Sol 74-13, Systems Opimization Laboratory, Department of Operation Research, Stanford University, Stanford
Bear J (1972) Dynamics of fluids in porous media. New York: Elsevier
Cryer CW (1976) A survey of steady state porous flow free boundary problems. Technical Summary Report No. 1657, Mathematics Research Center, University of Wisconsin, Madison
Baiocci C (1971) Sur un probleme a frontiere libre traduisant le filtrage de liquides a travers des milieux proeux. Comptes Rendus Acad Sci Paris A273:1215–1217
Bierlein J (1975) The journal bearing. Scientific American, 233:50–64
Cimmati G (1977) On a problem of the theory of lubrication governed by a variational inequality. Appl Math Optim 3:227–243
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Communicated by G. H. Golub
This work was partially supported by the National Science Foundation under grant number MCS 77-27632.
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Lin, Y., Cryer, C.W. An alternating direction implicit algorithm for the solution of linear complementarity problems arising from free boundary problems. Appl Math Optim 13, 1–17 (1985). https://doi.org/10.1007/BF01442196
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DOI: https://doi.org/10.1007/BF01442196