Browse > Article
http://dx.doi.org/10.4134/JKMS.2012.49.5.893

A FAST NUMERICAL METHOD FOR SOLVING A REGULARIZED PROBLEM ASSOCIATED WITH OBSTACLE PROBLEMS  

Yuan, Daming (College of Mathematics and Information Science Nanchang Hangkong University, Department of Mathematics Zhejiang University)
Li, Xi (College of Mathematics and Information Science Nanchang Hangkong University)
Lei, Chengfeng (College of Mathematics and Information Science Nanchang Hangkong University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.5, 2012 , pp. 893-905 More about this Journal
Abstract
Kirsi Majava and Xue-Cheng Tai [12] proposed a modified level set method for solving a free boundary problem associated with unilateral obstacle problems. The proximal bundle method and gradient method were applied to solve the nonsmooth minimization problems and the regularized problem, respectively. In this paper, we extend this approach to solve the bilateral obstacle problems and employ Rung-Kutta method to solve the initial value problem derived from the regularized problem. Numerical experiments are presented to verify the efficiency of the methods.
Keywords
Rung-Kutta method; level set method; obstacle problem;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), no. 4, 1591-1596.   DOI   ScienceOn
2 J. A. Sethian and J. D. Strain, Crystal growth and dendritic solidification, J. Comput. Phys. 98 (1992), no. 2, 231-253.   DOI   ScienceOn
3 F. Wang and X. L. Cheng, An algorithm for solving the double obstacle problems, Appl. Math. Comput. 201 (2008), no. 1-2, 221-228.   DOI   ScienceOn
4 F. Wang, W. M. Han, and X. L. Cheng, Discontinuous Galerkin methods for solving elliptic variational inequalities, SIAM J. Numer. Anal. 48 (2010), no. 2, 708-733.   DOI   ScienceOn
5 L. Xue and X. L. Cheng, An algorithm for solving the obstacle problems, Comput. Math. Appl. 48 (2004), no. 10-11, 1651-1657.   DOI   ScienceOn
6 Y. Zhang, Multilevel projection algorithm for solving obstacle problems, Comput. Math. Appl. 41 (2001), no. 12, 1505-1513.   DOI   ScienceOn
7 R. Hoppe, Multigrid algorithms for variational inequalities, SIAM J. Numer. Anal. 24 (1987), 1046-1065.   DOI   ScienceOn
8 R. Hoppe and R. Kornhuber, Adaptive multilevel methods for obstacle problems, SIAM J. Numer. Anal. 31 (1994), no. 2, 301-323.   DOI   ScienceOn
9 S. Howison, F. Wilmott, and J. Dewynne, The Mathematics of Financial Derivative, Cambridge University Press, Cambridge, 1995.
10 T. Karkkainen, K. Kunisch, and P. Tarvainen, Augmented Lagrangian active set methods for obstacle problems, J. Optim. Theory Appl. 119 (2003), no. 3, 499-533.   DOI   ScienceOn
11 S. Osher and R. Fedkiw, Level Set Method and Dynamic Implicit Surfaces, Springer, NewYork, 2000.
12 R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities I, Numer. Math. 69 (1994), no. 2, 167-184.   DOI
13 R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities II, Numer. Math. 72 (1996), no. 4, 481-499.   DOI
14 K. Majava and X.-C. Tai, A level set method for solving free boundary problems associated with obstacles, Int. J. Numer. Anal. Model. 1 (2004), no. 2, 157-171.
15 S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Al- gorithms based on Hamilton-Jacobi formulation, J. Comput. Phys. 79 (1988), no. 1, 12-49.   DOI   ScienceOn
16 J. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier Science 1987.
17 J. A. Sethian, Level Set Methods, Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, Cambridge University Press, Cambridge, 1996.
18 J. A. Sethian, Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws, J. Differential Geom. 31 (1990), no. 1, 131-161.   DOI
19 L. Brugnano and V. Casulli, Iterative solution of piecewise linear systems, SIAM J. Sci. Comput. 30 (2008), no. 1, 463-472.   DOI   ScienceOn
20 D. Adalsteinsson and J. A. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys. 148 (1999), no. 1, 2-22.   DOI   ScienceOn
21 R. Courant, K. Friedrichs, and H. Lewy, On the partial difference equaton of mathematical physics, IBM J. Res. Dev. 11 (1928), no. 2, 215-234.
22 G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, Germany, 1976.
23 R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984.