A convergent scheme for a non-local coupled system modelling dislocations densities dynamics

El Hajj, A.; Forcadel, N.

doi:10.1090/S0025-5718-07-02038-8

A convergent scheme for a non-local coupled system modelling dislocations densities dynamics
HTML articles powered by AMS MathViewer

by A. El Hajj and N. Forcadel PDF

Math. Comp. 77 (2008), 789-812 Request permission

Abstract:

In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Within the framework of viscosity solutions, we prove a long time existence and uniqueness result for the solution of this model. We also propose a convergent numerical scheme and we prove a Crandall-Lions type error estimate between the continuous solution and the numerical one. As far as we know, this is the first error estimate of Crandall-Lions type for Hamilton-Jacobi systems. We also provide some numerical simulations.

References

O. Alvarez, P. Cardaliaguet, and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces Free Bound. 7 (2005), no. 4, 415–434. MR 2191694, DOI 10.4171/IFB/131
O. Alvarez, E. Carlini, R. Monneau, and E. Rouy, Convergence of a first order scheme for a non-local eikonal equation, Appl. Numer. Math. 56 (2006), no. 9, 1136–1146. MR 2244967, DOI 10.1016/j.apnum.2006.03.002
O. Alvarez, E. Carlini, R. Monneau, and E. Rouy, A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics, Numer. Math. 104 (2006), no. 4, 413–444. MR 2249672, DOI 10.1007/s00211-006-0030-5
Olivier Alvarez, Philippe Hoch, Yann Le Bouar, and Régis Monneau, Résolution en temps court d’une équation de Hamilton-Jacobi non locale décrivant la dynamique d’une dislocation, C. R. Math. Acad. Sci. Paris 338 (2004), no. 9, 679–684 (French, with English and French summaries). MR 2065373, DOI 10.1016/j.crma.2004.03.007
—, Dislocation dynamics: short time existence and uniqueness of the solution, Archive for Rational Mechanics and Analysis 85 (2006), no. 3, 371–414.
Martino Bardi and Italo Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. MR 1484411, DOI 10.1007/978-0-8176-4755-1
Guy Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 17, Springer-Verlag, Paris, 1994 (French, with French summary). MR 1613876
Guy Barles and Olivier Ley, Nonlocal first-order Hamilton-Jacobi equations modelling dislocations dynamics, Comm. Partial Differential Equations 31 (2006), no. 7-9, 1191–1208. MR 2254611, DOI 10.1080/03605300500361446
Ariela Briani et al., Homogenization in time for a model with dislocations densities, In preparation (2006).
M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43 (1984), no. 167, 1–19. MR 744921, DOI 10.1090/S0025-5718-1984-0744921-8
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Hans Engler and Suzanne M. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations, Proc. London Math. Soc. (3) 63 (1991), no. 1, 212–240. MR 1105722, DOI 10.1112/plms/s3-63.1.212
Nicolas Forcadel, Dislocations dynamics with a mean curvature term: short time existence and uniqueness, Preprint (2005).
I. Groma, Link between the microscopic and mesoscopic length-scale description of the collective behaviour of dislocations, Phys. Rev. B 56 (1997), 5807.
I. Groma and P. Balogh, Link between the individual and continuum approaches of the description of the collective behavior of dislocations, Mat. Sci. Eng. A234-236 (1997), 249–252.
—, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta mater 47 (1999), no. 13, 3647–3654.
John Price Hirth and Jens Lothe, Theory of dislocations, Second edition, Krieger, Malabar, Florida, 1992.
Hitoshi Ishii, Perron’s method for monotone systems of second-order elliptic partial differential equations, Differential Integral Equations 5 (1992), no. 1, 1–24. MR 1141724
Hitoshi Ishii and Shigeaki Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Comm. Partial Differential Equations 16 (1991), no. 6-7, 1095–1128. MR 1116855, DOI 10.1080/03605309108820791
Hitoshi Ishii and Shigeaki Koike, Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games, Funkcial. Ekvac. 34 (1991), no. 1, 143–155. MR 1116886
E. R. Jakobsen and K. H. Karlsen, Convergence rates for semi-discrete splitting approximations for degenerate parabolic equations with source terms, BIT 45 (2005), no. 1, 37–67. MR 2164225, DOI 10.1007/s10543-005-2641-0
Espen Robstad Jakobsen, Kenneth Hvistendahl Karlsen, and Nils Henrik Risebro, On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms, SIAM J. Numer. Anal. 39 (2001), no. 2, 499–518. MR 1860266, DOI 10.1137/S003614290036823X
Suzanne M. Lenhart, Viscosity solutions for weakly coupled systems of first-order partial differential equations, J. Math. Anal. Appl. 131 (1988), no. 1, 180–193. MR 934440, DOI 10.1016/0022-247X(88)90199-0
Suzanne M. Lenhart and Stavros A. Belbas, A system of nonlinear partial differential equations arising in the optimal control of stochastic systems with switching costs, SIAM J. Appl. Math. 43 (1983), no. 3, 465–475. MR 700525, DOI 10.1137/0143030
Suzanne M. Lenhart and Naoki Yamada, Viscosity solutions associated with switching game for piecewise-deterministic processes, Stochastics Stochastics Rep. 38 (1992), no. 1, 27–47. MR 1274894, DOI 10.1080/17442509208833744
Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. MR 965860, DOI 10.1016/0021-9991(88)90002-2
Elisabeth Rouy and Agnès Tourin, A viscosity solutions approach to shape-from-shading, SIAM J. Numer. Anal. 29 (1992), no. 3, 867–884. MR 1163361, DOI 10.1137/0729053
Naoki Yamada, Viscosity solutions for a system of elliptic inequalities with bilateral obstacles, Funkcial. Ekvac. 30 (1987), no. 2-3, 417–425. MR 927191