[Méthode numériques pour la résolution d'un système d'équations eiconales avec conditions aux limites de Dirichlet]
Dans cette Note, on étudie la résolution numérique d'un système d'équations eiconales avec conditions aux limites du type Dirichlet. Dans la mesure, où le problème considéré a une infinité de solutions on recherche celles qui sont non-négatives et de norme L1 maximale (on presque maximale). La méthodologie numérique combine pénalité, régularisation biharmonique, décomposition d'opérateurs, et approximations par éléments finis. Son implémentation demande essentiellement la résolution d'équations à une variable du troisième degré et de problèmes linéaires elliptiques discrets pour le Laplacien et l'opérateur d'Helmholtz. Comme prévu, quand le domaine spatial est un carré de côtés parallèles aux axes de coordonnées les solutions calculées montrent un comportement fractal au voisinage de la frontière et plus particulièrement des coins.
In this Note, we discuss the numerical solution of a system of Eikonal equations with Dirichlet boundary conditions. Since the problem under consideration has infinitely many solutions, we look for those solutions which are nonnegative and of maximal (or nearly maximal) L1-norm. The computational methodology combines penalty, biharmonic regularization, operator splitting, and finite element approximations. Its practical implementation requires essentially the solution of cubic equations in one variable and of discrete linear elliptic problems of the Poisson and Helmholtz type. As expected, when the spatial domain is a square whose sides are parallel to the coordinate axes, and when the Dirichlet data vanishes at the boundary, the computed solutions show a fractal behavior near the boundary, and particularly, close to the corners.
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@article{CRMATH_2003__336_6_511_0,
author = {Dacorogna, Bernard and Glowinski, Roland and Pan, Tsorng-Whay},
title = {Numerical methods for the solution of a system of {Eikonal} equations with {Dirichlet} boundary conditions},
journal = {Comptes Rendus. Math\'ematique},
pages = {511--518},
publisher = {Elsevier},
volume = {336},
number = {6},
year = {2003},
doi = {10.1016/S1631-073X(03)00024-4},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00024-4/}
}
TY - JOUR AU - Dacorogna, Bernard AU - Glowinski, Roland AU - Pan, Tsorng-Whay TI - Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions JO - Comptes Rendus. Mathématique PY - 2003 SP - 511 EP - 518 VL - 336 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00024-4/ DO - 10.1016/S1631-073X(03)00024-4 LA - en ID - CRMATH_2003__336_6_511_0 ER -
%0 Journal Article %A Dacorogna, Bernard %A Glowinski, Roland %A Pan, Tsorng-Whay %T Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions %J Comptes Rendus. Mathématique %D 2003 %P 511-518 %V 336 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(03)00024-4/ %R 10.1016/S1631-073X(03)00024-4 %G en %F CRMATH_2003__336_6_511_0
Dacorogna, Bernard; Glowinski, Roland; Pan, Tsorng-Whay. Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions. Comptes Rendus. Mathématique, Tome 336 (2003) no. 6, pp. 511-518. doi : 10.1016/S1631-073X(03)00024-4. http://www.numdam.org/articles/10.1016/S1631-073X(03)00024-4/
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