Convergence of approximation schemes for nonlocal front propagation equations

Monteillet, Aurélien

doi:10.1090/S0025-5718-09-02270-4

Convergence of approximation schemes for nonlocal front propagation equations
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Math. Comp. 79 (2010), 125-146 Request permission

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.

References

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, Dislocation dynamics: short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal. 181 (2006), no. 3, 449–504. MR 2231781, DOI 10.1007/s00205-006-0418-5
Guy 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 (English, with English and French summaries). MR 2246335, DOI 10.1016/j.crma.2006.06.022
Guy Barles, Pierre Cardaliaguet, Olivier Ley, and Régis Monneau, Global existence results and uniqueness for dislocation equations, SIAM J. Math. Anal. 40 (2008), no. 1, 44–69. MR 2403312, DOI 10.1137/070682083
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.
—, Uniqueness results for nonlocal Hamilton-Jacobi equations, Preprint (2008).
Guy Barles and Christine 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, DOI 10.1137/0732020
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
Mariane Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions, Discrete Contin. Dyn. Syst. 21 (2008), no. 3, 763–800. MR 2399437, DOI 10.3934/dcds.2008.21.763
Mariane Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ 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, DOI 10.3934/dcds.2008.21.1047
Pierre Cardaliaguet and Denis Pasquignon, On the approximation of front propagation problems with nonlocal terms, M2AN Math. Model. Numer. Anal. 35 (2001), no. 3, 437–462. MR 1837079, DOI 10.1051/m2an:2001120
Yoshikazu Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. A level set approach. MR 2238463
Yoshikazu Giga, Shun’ichi Goto, and Hitoshi 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, DOI 10.1137/0523043
Hitoshi Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ. 28 (1985), 33–77. MR 845397
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, DOI 10.1090/gsm/096
Diana 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
Diana Nunziante, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Anal. 18 (1992), no. 11, 1033–1062. MR 1167420, DOI 10.1016/0362-546X(92)90194-J
Stanley Osher and Ronald Fedkiw, Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, vol. 153, Springer-Verlag, New York, 2003. MR 1939127, DOI 10.1007/b98879
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
D. Rodney, Y. Le Bouar, and A. Finel, Phase field methods and dislocations, Acta Materialia 51 (2003), 17–30.
Dejan Slepčev, Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Anal. 52 (2003), no. 1, 79–115. MR 1938652, DOI 10.1016/S0362-546X(02)00098-6
Pierpaolo Soravia and Panagiotis E. Souganidis, Phase-field theory for FitzHugh-Nagumo-type systems, SIAM J. Math. Anal. 27 (1996), no. 5, 1341–1359. MR 1402444, DOI 10.1137/S0036141094261318