A posteriori error estimates for the effective Hamiltonian of dislocation dynamics

Numerische Mathematik

Volumn 121, Issue 2, 2012, Pages 281-335

  Cacace, S ^a   Chambolle, A ^b   Monneau, R ^c  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^c UNIVERSITÉ PARIS EST   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84861199320     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-011-0430-z     Document Type: Article

References (25)

1. Achdou Y., Camilli F., Capuzzo Dolcetta I.: Homogenization of Hamilton-Jacobi equations: numerical methods. Math. Models Methods Appl. Sci. 18(7), 1115-1143 (2008).
2. Albert S., Cockburn B., French D. A., Peterson T. E.: A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: the steady state case. Math. Comp. 71, 49-76 (2001).
3. Albert, S., Cockburn, B., French, D. A., Peterson, T. E.: A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part II: the time-dependent case, finite volumes for complex applications, III (Porquerolles, 2002), 3-10, Hermes Science Publications, Paris (2002).
4. Alvarez O., Carlini E., Monneau R., Rouy E.: A convergent scheme for a non local Hamilton-Jacobi equation modelling dislocation dynamics. Numer. Mathematik 104(4), 413-444 (2006).
5. Alvarez O., Carlini E., Monneau R., Rouy E.: Convergence of a first order scheme for a non local eikonal equation. IMACS J. Appl. Numer. Math. 56, 1136-1146 (2006).
6. Alvarez O., Hoch P., Le Bouar Y., Monneau R.: Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Ration. Mech. Anal. 181(3), 449-504 (2006).
7. Barles G., Cardaliaguet P., Ley O., Monneau R.: General existence results and uniqueness for dislocation equations. SIAM J. Math. Anal. 40(1), 44-69 (2008).
8. Camilli F., Capuzzo Dolcetta I., Gomes D. A.: Error estimates for the approximation of the effective Hamiltonian. Appl. Math. Optim. 57, 30-57 (2008).
9. Carlini, E., Forcadel, N., Monneau, R.: A Generalized fast marching method for dislocation dynamics (work in progress).
10. Crandall M. G., Lions P. L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp. 43, 1-19 (1984).
11. Da Lio F., Forcadel N., Monneau R.: Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics. J. Eur. Math. Soc. 10(4), 1061-1104 (2008).
12. El Hajj A., Forcadel N.: A convergent scheme for a non-local coupled system modelling dislocations densities dynamics. Math. Comp. 77, 789-812 (2008).
13. El Hajj, A., Ibrahim, H., Monneau, R.: Homogenization of dislocation dynamics. Accepted to the Proceedings of the Conference Dislocations, IOP Conferences Series: Materials Science and Engineering (2008).
14. Forcadel N.: An error estimate for a new scheme for mean curvature motion. SIAM J. Numer. Anal. 46(5), 2715-2741 (2008).
15. Forcadel N., Imbert C., Monneau R.: Homogenization of the dislocation dynamics and of some particle systems with two-body interactions. Discret. Contin. Dyn. Syst. A 23(3), 785-826 (2009).
16. Ghorbel, A., Hoch, P., Monneau, R.: A numerical study for the homogeneization of one-dimensional models describing the motion of dislocations. Int. J. Comput. Sci. Math. (accepted).
17. Gomes D. A., Oberman A. M.: Computing the effective Hamiltonian using a variational approach. SIAM J. Control Optim. 43(3), 792-812 (2004).
18. Hirth J. P., Lothe J.: Theory of dislocations, second edition. Krieger, Malabar (1982).
19. Imbert C., Rouy E., Monneau R.: Homogenization of first order equations with u/epsilon-periodic Hamiltonians. Part II: application to dislocations dynamics. Commun. Partial Differen. Equ. 33(1-3), 479-516 (2008).
20. Ishii, H., Matsumura, Y.: Non-local Hamilton-Jacobi equations arising in dislocation dynamics. preprint.
21. Osher S., Sethian J. A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi. J. Comput. Phys. 79, 12-49 (1988).
22. Qian, J.: Two approximations for effective Hamiltonians arising from homogenization of Hamilton-Jacobi equations, UCLA, CAM reports, 03-39 (2003).
23. Rorro M.: An approximation scheme for the effective Hamiltonian and applications. Appl. Numer. Math 56, 1238-1254 (2006).
24. Rouy E., Tourin A.: A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29(3), 867-884 (1992).
25. Slepčev D.: Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions. Nonlinear Anal. 52, 79-115 (2003).