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


Authors: A. El Hajj and N. Forcadel
Journal: Math. Comp. 77 (2008), 789-812
MSC (2000): Primary 35Q72, 49L25, 35F25, 35L40, 65M06, 65M12, 65M15, 74H20, 74H25
DOI: https://doi.org/10.1090/S0025-5718-07-02038-8
Published electronically: November 8, 2007
MathSciNet review: 2373180
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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.


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Additional Information

A. El Hajj
Affiliation: Cermics, Ecole des Ponts, ParisTech 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2

N. Forcadel
Affiliation: Cermics, Ecole des Ponts, ParisTech 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2

DOI: https://doi.org/10.1090/S0025-5718-07-02038-8
Keywords: Hamilton Jacobi equations, viscosity solutions, dislocations densities dynamics, numerical scheme, error estimate, system.
Received by editor(s): June 15, 2006
Received by editor(s) in revised form: January 26, 2007
Published electronically: November 8, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.