Abstract
The convergence of the method is proved and it is shown that the objective function corresponding to the quadratic programming problem is monotonically decreasing. The results of numerical tests for an elasticity contact problem are presented.
Zusammenfassung
Eine Mehrgittermethode für Variationsungleichungen bei Kontaktproblemen wird untersucht. Es wird Konvergenz bewiesen und gezeigt, daß die der quadratischen Programmierungsaufgabe zugrundeliegende Zielfunktion monoton fällt. Numerische Testresultate werden für ein elastisches Kontaktproblem präsentiert.
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References
Künzi, H., Krelle, W.: Nichtlineare Programmierung. Berlin: Springer 1962.
Daugavet, V. A.: Modification of Wolfe method. Z. Vycisl. Mat. Mat. Fiz.21, 504–508 (1981).
Pchenithnyi, B. N.: A Linearization Method. Moscow: Nauka 1981.
Cryer, C. W.: The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control.9, 385–392 (1971).
Fedorenko, R. P.: A relaxation method for solving elliptic difference equations. Z. Vycisl. Mat. Mat. Fiz.1, 922–927, 1961 (U.S.S.R. Comp. Math. Math. Phys.1, 1092–1096, 1961).
Bachvalov, N. S.: On the convergence of a relaxation method with natural constraints on the elliptic operator. Z. Vycisl. Mat. Mat. Fiz.6/5, 861–883, 1966 (U.S.S.R. Comp. Math. Math. Phis.6, 101–135, 1966).
Astrachancev, G. P.: An iterative method of solving elliptic net problems. Z. Vycisl. Mat. Mat. Fiz.11/2, 439–448, 1971 (U.S.S.R. Comp. Math. Math. Phys.11, 171–182, 1971).
Mc Cormick, S.F.: Multigrid methods for variational problems: further results. SIAM J. Numer. Anal.21, 255–263 (1984).
Braess, D., Hackbusch, W.: A new convergence proof for the multigrid method including the V-cycle. SIAM J. Numer. Anal.20, 967–975 (1983).
Hackbusch, W., Reusken, A.: Analysis of a damped nonlinear multilevel method. Numer. Math.55, 225–246 (1989).
Mandel, J.: A multilevel iterative method for symmetric, positive definite linear complementarity problems. Appl. Math. Optim.11, 77–95 (1984).
Brandt, A., Cryer, C. W.: Multigrid algorithms for solution of linear complementarity problems arising from free boundary problems. SIAM J. Sci. Stat. Comput.4, 655–684 (1983).
Smoch, M.: Anmerkungen zu einem Mehrgitterverfahren für lineare Komplementaritätsprobleme. Numer. Math.57, 80–83 (1990).
Hackbusch, W., Mittelmann, H. D.: On multi-grid methods for variational inequalities. Numer. Math.42, 65–76 (1983).
Hoppe, R. H. W.: On the numerical solution of variational inequalities by multigrid techniques, pp. 59–87. Proc. Int. Symp. on Numerical Analysis, Ankara, Turkey, Sept. 1–4, 1987. New York: Plenum Press 1988.
Hoppe, R. H. W.: Multigrid algorithms for variational inequalities. SIAM J. Numer. Anal.24, 1046–1065 (1987).
Bloss, M., Hoppe, R. H. W.: Numerical computation of the value function of optimally controlled stochastic switching processes by multi-grid techniques. Numer. Funct. Anal. and Optimiz.10, 275–304 (1989).
Hoppe, R. H. W.: Numerical solution of multicomponent alloy solidification by multigrid techniques. IMPACT of Computing in Science and Engineering2, 125–156 (1990).
Hoppe, R. H. W.: Multi-grid methods for Hamilton-Jacobi-Bellman equations. Numer. Math.49, 239–254 (1986).
Hackbusch, W.: Convergence of multi-grid iterations applied to difference equations. Math. Comp.34, 425–440 (1980).
Pang, J. S.: On the convergence of a basic iterative method for the implicit complementarity problem. J. Optimization Theory and Applications37, 149–162 (1982).
Washizu, K.: Variational methods in elasticity and plasticity. Oxford, New York, Toronto, Sydney, Paris, Frankfurt: Pergamon Press 1982.
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Belsky, V. A multi-grid method for variational inequalities in contact problems. Computing 51, 293–311 (1993). https://doi.org/10.1007/BF02238537
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DOI: https://doi.org/10.1007/BF02238537