A duality-type method for the obstacle problem

Merlusca, Diana Rodica

doi:10.2478/auom-2013-0051

Abstract

Based on a duality property, we solve the obstacle problem on Sobolev spaces of higher order. We have considered a new type of approximate problem and with the help of the duality we reduce it to a quadratic optimization problem, which can be solved much easier.

References

[1] R. Adams. Sobolev spaces, Acad. Press, 1975.
Search in Google Scholar
[2] L. Badea. One- and Two-Level Domain Decomposition Methods for Nonlinear Problems. In First International Conference on Parallel, Distributed and Grid Computing for Engineering, Stirlingshire, Scotland, pp. 1 - 18, 2009.
Search in Google Scholar
[3] Baiocchi, C., Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl., 92, pp. 107 - 127, 1972.10.1007/BF02417940
Search in Google Scholar
[4] V. Barbu, T. Precupanu. Convexity and optimization in Banach spaces, Noordhoff, 1978.
Search in Google Scholar
[5] H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.10.1007/978-0-387-70914-7
Search in Google Scholar
[6] Burger, M., Matevosyan, N., Wolfram, M.T. A level set based shape optimization method for an elliptic obstacle problem. Math. Models Methods Appl. Sci., 21, no. 4, pp. 619 - 649, 2011.10.1142/S0218202511005155
Search in Google Scholar
[7] Duvaut, G., Lions, J.-L. Les inéquations en mécanique et en physique. Travaux et Recherches Mathematiques, No. 21. Dunod, Paris, 1972.
Search in Google Scholar
[8] G. Fichera: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Memorie (Cl. Sci. fis., mat. e nat.), serie 8, vol. 7, 1964, pp. 91 - 140.
Search in Google Scholar
[9] G. Fichera: Boundary value problems of elasticity with unilateral constraints. Handbuch der Physik, Band VI a/2, Mechanics of Solids II Springer, 1972, pp. 391 - 424.10.1007/978-3-642-69567-4_4
Search in Google Scholar
[10] Frehse, J. On the regularity of the solution of a second order variational inequality. Boll. Un. Mat. Ital.(4) 6 , pp. 312 - 315, 1972.
Search in Google Scholar
[11] Griesse, R., Kunisch, K. A semi-smooth Newton method for solving elliptic equations with gradient constraints. M2AN Math. Model. Numer. Anal. 43, no. 2, pp. 209 - 238, 2009.10.1051/m2an:2008049
Search in Google Scholar
[12] R. Glowinski. Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984.10.1007/978-3-662-12613-4
Search in Google Scholar
[13] Ito, K., Kunisch, K., SemiSmooth Newton Methods For Variational Inequalities Of The First Kind, Mathematical Modelling and Numerical Analysis, Vol. 37, No 1, pp. 41 - 62, 2003.10.1051/m2an:2003021
Search in Google Scholar
[14] Kinderlehrer, D., Stampacchia, G. An introduction to variational inequalities and their applications. Pure and Applied Mathematics, Academic Press, New York, 1980.
Search in Google Scholar
[15] Merlusca, D.R., A duality algorithm for the obstacle problem. (to appear) Annals of the Academy of Romanian Scientists, Vol. 5, 2013.
Search in Google Scholar
[16] Murea, C.M., Tiba, D. A direct algorithm in some free boundary problems, BCAM Publications, 2012.
Search in Google Scholar
[17] Perez, F.A., Cascon, J.M., Ferragut, L. A Numerical Adaptive Algorithm for the Obstacle Problem. 4th International Conference, Krakow, Poland, June 6-9, 2004, Proceedings, Part II, pp. 130 - 137, 2004.10.1007/978-3-540-24687-9_17
Search in Google Scholar
[18] Schaeffer, D.G. A stability theorem for the obstacle problem Advances in Mathematics, Volume 17, Issue 1, pp. 34 - 47, July 1975.10.1016/0001-8708(75)90085-7
Search in Google Scholar
[19] Wilmott, P., Howison, S., Dewynne, J. The mathematics of financial derivatives. A student introduction. Cambridge University Press, Cambridge, 1995. 10.1017/CBO9780511812545
Search in Google Scholar

A duality-type method for the obstacle problem

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