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New approximate method for the Allen–Cahn equation with double-obstacle constraint and stability criteria for numerical simulations
1 Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686, Japan
2 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan
Received: , Accepted: , Published:
Special Issue: Nonlinear Evolution PDEs, Interfaces and Applications
1. S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall., 27 (1979), 1084-1095.
2. H. Attouch, Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984.
3. L. Blank, H. Garcke, and L. Sarbu, Primal-dual active set methods for Allen-Cahn variational inequalities with nonlocal constraints. Numer. Methods Partial Di erential Equations, 29 (2013), 999-1030.
4. H. Br´ezis, Op´erateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam, 1973.
5. H. Br´ezis, M. G. Crandall, and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space. Comm. Pure Appl. Math., 23 (1970), 123-144.
6. L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J. Differential Equations, 90 (1991), 211-237.
7. X Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem. Proc. Roy. Soc. London Ser. A, 444 (1994), 429-445.
8. M. H. Farshbaf-Shaker, T. Fukao, and N. Yamazaki, Singular limit of Allen-Cahn equation with constraints and its Lagrange multiplier. Discrete Contin. Dyn. Syst., AIMS Proceedings (2015), 418-427.
9. M. H. Farshbaf-Shaker, T. Fukao, and N. Yamazaki, (In Press) Lagrange multiplier and singular limit of double-obstacle problems for the Allen-Cahn equation with constraint. Math. Methods Appl. Sci..
10. X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math., 94 (2003), 33-65.
11. X. Feng, H. Song, and T. Tang, Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Probl. Imaging, 7 (2013), 679-695.
12. P. C. Fife, Dynamics of internal layers and diffusive interfaces. CBMS-NSF Regional Conf. Ser. in: Appl. Math., 53 (1988), SIAM, Philadelphia.
13. A. Friedman, Partial Di erential Equations of Parabolic Type, Prentice-Hall, INC., Englewood Cliffs, N. J., 1964.
14. Y. Giga, Y. Kashima, and N. Yamazaki, Local solvability of a constrained gradient system of total variation. Abstr. Appl. Anal., 2004 (2004), 651-682.
15. A. Ito, N. Yamazaki, and N. Kenmochi, Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces. Dynamical systems and differential equations, Vol. I (Springfield, MO, 1996), Discrete Contin. Dynam. Systems 1998, Added Volume I, 327-349.
16. N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.
17. N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities. Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 4 (2007) ed. M. Chipot, Chapter 4, 203-298, North Holland, Amsterdam.
18. N. Kenmochi and M. Niezg´odka, Systems of nonlinear parabolic equations for phase change problems. Adv. Math. Sci. Appl., 3 (1993/94), Special Issue, 89-117.
19. U. Mosco, Convergence of convex sets and of solutions variational inequalities. Advances Math., 3 (1969), 510-585.
20. T. Ohtsuka, Motion of interfaces by an Allen-Cahn type equation with multiple-well potentials. Asymptot. Anal., 56 (2008), 87-123.
21. T. Ohtsuka, K. Shirakawa, and N. Yamazaki, Optimal control problems of singular diffusion equation with constraint. Adv. Math. Sci. Appl., 18 (2008), 1-28.
22. T. Ohtsuka, K. Shirakawa, and N. Yamazaki, Convergence of numerical algorithm for optimal control problem of Allen-Cahn type equation with constraint. Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation-Mathematical Analysis, Modelling and Simulation, GAKUTO Intern. Ser. Math. Appl., Vol. 29 (2008), 441-462.
23. T. Ohtsuka, K. Shirakawa, and N. Yamazaki, Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 159-181.
24. J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discret. Contin. Dyn. Syst. Ser. A, 28 (2010), 1669-1691.
25. T. Suzuki, K. Takasao, and N. Yamazaki, Remarks on numerical experiments of Allen-Cahn equations with constraint via Yosida approximation. Adv. Numer. Anal., 2016, Article ID 1492812, 16 pages.
26. Y. Tonegawa, Integrality of varifolds in the singular limit of reaction-di usion equations. Hiroshima Math. J., 33 (2003), 323-341.
27. X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070.
28. J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit. SIAM J. Sci. Comput., 31 (2009), 3042-3063.
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