[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.
Accepté le :
Publié le :
Bernard Dacorogna 1 ; Roland Glowinski 2 ; Tsorng-Whay Pan 2
@article{CRMATH_2003__336_6_511_0,
author = {Bernard Dacorogna and Roland Glowinski and Tsorng-Whay Pan},
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},
}
TY - JOUR AU - Bernard Dacorogna AU - Roland Glowinski AU - Tsorng-Whay Pan 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 DO - 10.1016/S1631-073X(03)00024-4 LA - en ID - CRMATH_2003__336_6_511_0 ER -
%0 Journal Article %A Bernard Dacorogna %A Roland Glowinski %A Tsorng-Whay Pan %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 %R 10.1016/S1631-073X(03)00024-4 %G en %F CRMATH_2003__336_6_511_0
Bernard Dacorogna; Roland Glowinski; Tsorng-Whay Pan. Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions. Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 511-518. doi : 10.1016/S1631-073X(03)00024-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00024-4/
[1] Implicit Partial Differential Equations, Birkhäuser, Basel, 1999
[2] Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978
[3] Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984
[4] Finite element methods for incompressible viscous flow (P.G. Ciarlet; J.-L. Lions, eds.), Handbook of Numerical Analysis, IX, North-Holland, Amsterdam, 2003
[5] R. Glowinski, Y.A. Kuznetsov, T.-W. Pan, On a penalty/Newton/conjugate gradient method for the solution of obstacle problems, C. R. Acad. Sci. Paris, Ser. I 336 (2003), in press
[6] Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989
[7] A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: Description of the numerical methods, Japan J. Appl. Math., Volume 7 (1990), pp. 1-76
[8] A Lagrange multiplier/fictitious domain method for the Dirichlet problem. Generalization to some flow problems, Japan J. Indust. Appl. Math., Volume 12 (1995), pp. 87-108
[9] Computation of nonclassical solutions to Hamilton–Jacobi problems, SIAM J. Sci. Comput., Volume 21 (1999) no. 2, pp. 502-521
- Hybrid of differential quadrature and sub-gradients methods for solving the system of Eikonal equations, Nonlinear Engineering, Volume 10 (2021) no. 1, p. 436 | DOI:10.1515/nleng-2021-0035
- Analytic solution of the eikonal for a stigmatic singlet lens, Physica Scripta, Volume 95 (2020) no. 8, p. 085201 | DOI:10.1088/1402-4896/ab99fa
- Numerical approximation of orthogonal maps, SIAM Journal on Scientific Computing, Volume 41 (2019) no. 6, p. b1341-b1367 | DOI:10.1137/19m1243683 | Zbl:1435.65195
- A penalty-regularization-operator splitting method for the numerical solution of a scalar eikonal equation, Chinese Annals of Mathematics. Series B, Volume 36 (2015) no. 5, pp. 659-688 | DOI:10.1007/s11401-015-0930-8 | Zbl:1325.65089
- On the numerical solution of some eikonal equations: an elliptic solver approach, Chinese Annals of Mathematics. Series B, Volume 36 (2015) no. 5, pp. 689-702 | DOI:10.1007/s11401-015-0971-z | Zbl:1326.65075
- A selection criterion of solutions to a system of eikonal equations, Advances in Calculus of Variations, Volume 4 (2011) no. 3 | DOI:10.1515/acv.2011.003
- Calculus of variations, implicit partial differential equations and microstructure, GAMM-Mitteilungen, Volume 29 (2006) no. 2, pp. 150-171 | DOI:10.1002/gamm.201490028 | Zbl:1157.49003
- Viscosity solutions, almost everywhere solutions and explicit formulas, Transactions of the American Mathematical Society, Volume 356 (2004) no. 11, pp. 4643-4653 | DOI:10.1090/s0002-9947-04-03506-8 | Zbl:1069.34007
- A penalty/Newton/conjugate gradient method for the solution of obstacle problems, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 336 (2003) no. 5, pp. 435-440 | DOI:10.1016/s1631-073x(03)00025-6 | Zbl:1028.65073
Cité par 9 documents. Sources : Crossref, zbMATH
Commentaires - Politique