Summary
In this paper we prove some results concerning the convergence of the Peaceman-Rachford iterative method. The main result covers both the stationary and the instationary case. No use is made of the so called commutativity condition which often was used in the literature in the instationary case. In proving the results of this paper it is made use of the theory of regular splittings which was introduced by R.S. Varga. Finally it is demonstrated how the results can be applied to discrete versions of elliptic boundary value problems.
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Literatur
Birkoff, G., Varga, R. S.: Implicit Alternating Direction Methods. Trans. Amer. Math. Soc.92, 190–273 (1959)
Birkoff, G., Varga, R. S., Young, D. M.: Alternating Direction Implicit Methods. In: Advances in Computers, 3, New York: Academic Press 1962
Casper, J.: Applications of Alternating Direction Methods to Mildly Nonlinear Problems, Ph. D. Diss., Univ of Maryland, College Park, Maryland (1969)
Guilinger, W. H., Jr.: The Peaceman-Rachford Method for Small Mesh Increments. J. Math. Anal. Appl.11, 261–277 (1965)
Habetler, G. J.: Concerning the Implicit Alternating Direction Method. Report KAPL-2040, Knolls Atomic Power Laboratory, Schenectady, New York (1959)
More, J. M.: Global Convergence of Newton-Gauss-Seidel Methods. SIAM J. Numer. Anal.8, 325–336 (1971)
Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press 1970
Ostrowski, A. M.: Über die Determinanten mit überwiegender Hauptdiagonale. Comment. Math. Helv.10, 69–96 (1937)
Pearcy, C.: On Convergence of Alternating Direction Procedures. Numer. Math.4, 172–176 (1962)
Price, H., Varga, R. S.: Recent Numerical Experiments Comparing Successive Overrelaxation Interative Methods with Implicit Alternating Direction Methods. Report Nr. 91, Gulf Research and Development Co. Pittsburgh, Pennsylvania (1962)
Varga, R. S.: Matrix Iterative Analysis. Series in Automatic Computation, Engle-wood Cliffs, N. J.: Prentice Hall, (1962)
Wachspress, E.: Iterative Solution of Elliptic Systems. Englewood Cliffs, N.J.: Prentice Hall 1966
Widlund, O. B.: On the Rate of Convergence of an Alternating Direction Implicit Method in a Noncommutative Case. Math. Comp.20, 500–515 (1966)
Young, D. M.: Iterative Solution of Large Linear Systems. New York: Academic Press 1971
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Alefeld, G. Zur Konvergenz des Peaceman-Rachford-Verfahrens. Numer. Math. 26, 409–419 (1976). https://doi.org/10.1007/BF01409962
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DOI: https://doi.org/10.1007/BF01409962