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Convergence of approximation schemes for nonlocal front propagation equations


Author: Aurélien Monteillet
Journal: Math. Comp. 79 (2010), 125-146
MSC (2000): Primary 65M12, 53C44, 35K65, 70H20, 49L25
DOI: https://doi.org/10.1090/S0025-5718-09-02270-4
Published electronically: June 8, 2009
MathSciNet review: 2552220
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Abstract: We provide a convergence result for numerical schemes approximating nonlocal front propagation equations. Our schemes are based on a recently investigated notion of a weak solution for these equations. We also give examples of such schemes, for a dislocation dynamics equation, and for a FitzHugh-Nagumo type system.


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  • 1. 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 (2008f:74041)
  • 2. O. Alvarez, P. Hoch, Y. Le Bouar, and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal. 181 (2006), no. 3, 449-504. MR 2231781
  • 3. G. Barles, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time, C. R. Math. Acad. Sci. Paris 343 (2006), no. 3, 173-178. MR 2246335
  • 4. G. Barles, P. Cardaliaguet, O. Ley, and R. Monneau, Global existence results and uniqueness for dislocation equations, SIAM J. Math. Anal. 40 (2008), no. 1, 44-69. MR 2403312 (2009d:49049)
  • 5. G. Barles, P. Cardaliaguet, O. Ley, and A. Monteillet, Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations, to appear in Nonlinear Analysis Series A: Theory, Methods & Applications.
  • 6. -, Uniqueness results for nonlocal Hamilton-Jacobi equations, Preprint (2008).
  • 7. G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal. 32 (1995), no. 2, 484-500. MR 1324298 (96c:65140)
  • 8. G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no. 3, 271-283. MR 1115933 (92d:35137)
  • 9. M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with $ L\sp 1$ dependence in time and Neumann boundary conditions, Discrete Contin. Dyn. Syst. 21 (2008), no. 3, 763-800. MR 2399437
  • 10. -, Viscosity solutions of fully nonlinear second order parabolic equations with $ L\sp 1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach, Discrete Contin. Dyn. Syst. 21 (2008), no. 4, 1047-1069. MR 2399449
  • 11. P. Cardaliaguet and D. Pasquignon, On the approximation of front propagation problems with nonlocal terms, M2AN Math. Model. Numer. Anal. 35 (2001), no. 3, 437-462. MR 1837079 (2002d:65108)
  • 12. Y. Giga, Surface evolution equations. A level set approach, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. MR 2238463
  • 13. Y. Giga, S. Goto, and H. Ishii, Global existence of weak solutions for interface equations coupled with diffusion equations, SIAM J. Math. Anal. 23 (1992), no. 4, 821-835. MR 1166559 (93g:35068)
  • 14. H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ. 28 (1985), 33-77. MR 845397 (87k:35055)
  • 15. N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence, RI, 2008. MR 2435520
  • 16. D. Nunziante, Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence, Differential Integral Equations 3 (1990), no. 1, 77-91. MR 1014727 (90i:35135)
  • 17. -, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Anal. 18 (1992), no. 11, 1033-1062. MR 1167420 (93f:35125)
  • 18. S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, vol. 153, Springer-Verlag, New York, 2003. MR 1939127 (2003j:65002)
  • 19. S. Osher and J. 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 (89h:80012)
  • 20. D. Rodney, Y. Le Bouar, and A. Finel, Phase field methods and dislocations, Acta Materialia 51 (2003), 17-30.
  • 21. D. Slepčev, Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Anal. 52 (2003), no. 1, 79-115. MR 1938652 (2003k:65096)
  • 22. P. Soravia and P. E. Souganidis, Phase-field theory for FitzHugh-Nagumo-type systems, SIAM J. Math. Anal. 27 (1996), no. 5, 1341-1359. MR 1402444 (97e:35013)

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Additional Information

Aurélien Monteillet
Affiliation: Université de Bretagne Occidentale, UFR Sciences et Techniques, 6 av. Le Gorgeu, BP 809, 29285 Brest, France
Email: aurelien.monteillet@univ-brest.fr

DOI: https://doi.org/10.1090/S0025-5718-09-02270-4
Keywords: Approximation schemes, front propagations, level-set approach, nonlocal Hamilton-Jacobi equations, second-order equations, viscosity solutions, $L^1$ dependence in time, dislocation dynamics, FitzHugh-Nagumo system.
Received by editor(s): September 22, 2008
Received by editor(s) in revised form: February 13, 2009
Published electronically: June 8, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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