Accessible Requires Authentication Published by De Gruyter May 19, 2016

A direct algorithm in some free boundary problems

Cornel M. Murea and Dan Tiba

Abstract

In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like one- and two- phase Stefan problem and of obstacle type. Our approach enters the category of fixed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The approximating coincidence set is explicitly computed and it converges in the Hausdorff-Pompeiu sense to the searched geometry. In the numerical examples, the algorithm has a very fast convergence and the obtained solutions (including the free boundaries) are accurate.

MSC 2010: 35R35; 35J86; 35K85; 65N30; 65M12

Bibliography

[1] Y. Achdou, F. Hecht and D. Pommier, A posteriori error estimates for parabolic variational inequalities, J. Sci. Comput. 37 (2008), 336–366. Search in Google Scholar

[2] Y. Achdou and O. Pironneau, Computational Methods for Option Pricing, Frontiers in Applied Mathematics, 30. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2005. Search in Google Scholar

[3] C. Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica, Ann. Mat. Pura Appl., 92 (1972), 107–127. Search in Google Scholar

[4] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, 1984. Search in Google Scholar

[5] M. Burger, N. Matevosyan and M.T. Wolfram, A level set based shape optimization method for an elliptic obstacle problem,Math.ModelsMethods Appl. Sci. 21 (2011), 619–649. Search in Google Scholar

[6] H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9) 51 (1972), 1–168. Search in Google Scholar

[7] H. Brézis and M. Sibony, Équivalence de deux inéquations variationnelles et applications. (French) Arch. Rational Mech. Anal. 41 (1971), 254–265. Search in Google Scholar

[8] M. Broadie and J. Detemple, The valuation of American options on multiple assets, Math. Finance, 7 (1997), 241–286. Search in Google Scholar

[9] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, New York, 1976. Search in Google Scholar

[10] C.M. Elliott and J. R. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Research Notes in Mathematics, 59. Pitman, London, 1982. Search in Google Scholar

[11] M. Fortin, R. Glowinski, Augmented Lagrangian methods. Applications to the numerical solution of boundary value problems. Studies in Mathematics and its Applications, 15. North-Holland Publishing Co., Amsterdam, 1983 Search in Google Scholar

[12] R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, Studies inMathematics and its Applications, 8. North-Holland Publishing Co., Amsterdam-New York, 1981. Search in Google Scholar

[13] R. Griesse and K. Kunisch, A semi-smooth Newton method for solving elliptic equations with gradient constraints, M2AN Math. Model. Numer. Anal. 43 (2009), 209–238. Search in Google Scholar

[14] A. Halanay C.M. Murea and D. Tiba, Existence and approximation for a steady fluid-structure interaction problem using fictitious domain approach with penalization, Mathematics and its Applications, 5 (2013), 120–147. Search in Google Scholar

[15] F. Hecht, http://www.freefem.org Search in Google Scholar

[16] H. Huang, W. Han and J. Zhou, The regularization method for an obstacle problem, Numer. Math. 69 (1994), 155–166. Search in Google Scholar

[17] K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind, M2AN Math. Model. Numer. Anal. 37 (2003), 41–62. Search in Google Scholar

[18] K. Ito and K. Kunisch, Parabolic variational inequalities: The Lagrange multiplier approach, J. Math. Pures Appl. 85 (2006), 415–449. Search in Google Scholar

[19] S. L. Kamenomostskaja, On Stefan’s problem. (Russian) Mat. Sb. (N.S.)53 (95) (1961) 489–514. (in Russian) Search in Google Scholar

[20] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Reprint of the 1980 original. Classics in Applied Mathematics, 31, SIAM, 2000. Search in Google Scholar

[21] C. H. Li, A finite-element front-tracking enthalpy method for Stefan problems, IMA J. Numer. Anal. 3 (1983), 87–107. Search in Google Scholar

[22] E. Lindgren, On the penalized obstacle problem in the unit half ball, Electron. J. Differential Equations, 9 (2010) 12 pp. Search in Google Scholar

[23] M. Natori and H. Kawarada, An application of the integrated penalty method to free boundary problems of Laplace equation Numer. Funct. Anal. Optim. 3 (1981), 1–17. Search in Google Scholar

[24] Neittaanmaki, P., Pennanen, A., Tiba, D. Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions, Inverse Problems, 25 (2009), 1–18. Search in Google Scholar

[25] Neittaanmaki, P., Repin, S., Reliable methods for computer simulation. Error control and a posteriori estimates. Studies in Mathematics and its Applications, 33. Elsevier Science B.V., Amsterdam, 2004. Search in Google Scholar

[26] P. Neittaanmaki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Theory and Applications. Springer Monographs in Mathematics. Springer, New York, 2006. Search in Google Scholar

[27] P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms, and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 179. Marcel Dekker, Inc., New York, 1994. Search in Google Scholar

[28] P. Neittaanmaki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 1–35. Search in Google Scholar

[29] J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, London, 1970. Search in Google Scholar

[30] A. Petrosyan and H. Shahgholian, Parabolic Obstacle Problems Applied to Finance, in Recent developments in nonlinear partial differential equations (ed. D. Danielli), Contemp. Math., 439, Amer. Math. Soc., Providence, RI, (2007), 117–133. Search in Google Scholar

[31] O. Pironneau and F. Hecht, Mesh adaption for the Black and Scholes equations, East-West J. Numer. Math. 8 (2000), 25–35. Search in Google Scholar

[32] J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Publishing Co., Amsterdam, 1987. Search in Google Scholar

[33] D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer, Berlin, 1990. Search in Google Scholar

[34] F. Wang and X.-L. Cheng, An algorithm for solving the double obstacle problems, Appl. Math. Comput. 201 (2008), 221–228. Search in Google Scholar

[35] P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives. A Student Introduction. Cambridge University Press, Cambridge, 1995. Search in Google Scholar

[36] C. S. Zhang, Adaptive Finite Element Methods for Variational Inequalities: Theory and Applications in Finance, PhD thesis, University of Maryland, 2007. Search in Google Scholar

[37] K. Zhang, X. Q. Yang and K. L. Teo, Augmented Lagrangian method applied to American option pricing, Automatica J. IFAC42 (2006), 1407–1416. Search in Google Scholar

[38] K. Zhang, S. Wang, X. Q. Yang and K. L. Teo, A power penalty approach to numerical solutions of two-asset American options, Numer. Math. Theory Methods Appl. 2 (2009), 202–223. Search in Google Scholar

Received: 2015-4-24
Accepted: 2015-4-24
Published Online: 2016-5-19
Published in Print: 2016-12-1

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