Comptes Rendus
no. 5
Numerical Analysis/Calculus of Variations
A penalty/Newton/conjugate gradient method for the solution of obstacle problems
[Sur une méthode de pénalité/Newton et gradient conjugué pour la résolution de problèmes d'obstacles]

Présenté par : Philippe G. Ciarlet

Roland Glowinski1 ; Yuri A. Kuznetsov1 ; Tsorng-Whay Pan1
1 University of Houston, Department of Mathematics, Houston, TX 77204-3476, USA
Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 435-440.

Motivé par la recherche des solutions non négatives d'un système d'équations eiconales, avec conditions aux limites de Dirichlet, on étudie dans cette Note une méthode pour la résolution numérique de problèmes d'inéquations variationnelles paraboliques pour des ensembles convexes du type K={v∣v∈H01(Ω), v⩾ψ p.p. sur Ω}. La méthode numérique combine pénalité et algorithme de Newton, les problèmes linéarisés étant résolus par un algorithme de gradient conjugué qui demande à chaque iteration la résolution d'un problème linéaire pour un analogue discret de l'opérateur elliptique I−μΔ avec μ>0. Les essais numériques montrent que la méthode ainsi obtenue a de bonnes propriétés de convergence, même pour des petites valeurs du paramètre de pénalité.

Motivated by the search for non-negative solutions of a system of Eikonal equations with Dirichlet boundary conditions, we discuss in this Note a method for the numerical solution of parabolic variational inequality problems for convex sets such as K={v∣v∈H01(Ω), v⩾ψ a.e. on Ω}. The numerical methodology combines penalty and Newton's method, the linearized problems being solved by a conjugate gradient algorithm requiring at each iteration the solution of a linear problem for a discrete analogue of the elliptic operator I−μΔ. Numerical experiments show that the resulting method has good convergence properties, even for small values of the penalty parameter.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00025-6

Roland Glowinski 1 ; Yuri A. Kuznetsov 1 ; Tsorng-Whay Pan 1

1 University of Houston, Department of Mathematics, Houston, TX 77204-3476, USA 
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Roland Glowinski; Yuri A. Kuznetsov; Tsorng-Whay Pan. A penalty/Newton/conjugate gradient method for the solution of obstacle problems. Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 435-440. doi : 10.1016/S1631-073X(03)00025-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00025-6/

[1] H. Carlsson; R. Glowinski Vibrations of Euler–Bernoulli beams with pointwise obstacles (R. Gatignol; Soubbaramayer, eds.), Advances in Kinetic Theory and Continuum Mechanics, Springer-Verlag, Berlin, 1991, pp. 261-275

[2] P.G. Ciarlet The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978

[3] P.G. Ciarlet Basic error estimates for elliptic problems (P.G. Ciarlet; J.-L. Lions, eds.), Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 17-352

[4] B. Dacorogna; R. Glowinski; T.-W. Pan Numerical solution of a system of Eikonal equations, C. R. Acad. Sci. Paris, Sér. I, Volume 336 (2003)

[5] E.J. Dean; R. Glowinski; Y.M. Kuo; M.G. Nasser On the discretization of some second order in time differential equations. Applications to nonlinear wave problems (A.V. Balakrishnan, ed.), Computational Techniques in Identification and Control of Flexible Flight Structures, Optimization Software Inc., Los Angeles, 1990, pp. 199-246

[6] G. Duvaut; J.-L. Lions Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976

[7] R. Glowinski Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984

[8] R. Glowinski; J.-L. Lions; R. Tremolières Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981

  • Yarui Duan; Song Wang; Yuying Zhou A power penalty approach to a mixed quasilinear elliptic complementarity problem, Journal of Global Optimization, Volume 81 (2021) no. 4, p. 901 | DOI:10.1007/s10898-021-01000-7
  • Alexandre Caboussat; Roland Glowinski; Dimitrios Gourzoulidis; Marco Picasso Numerical Approximation of Orthogonal Maps, SIAM Journal on Scientific Computing, Volume 41 (2019) no. 6, p. B1341 | DOI:10.1137/19m1243683
  • Cheng Fang; Yuan Li Fully Discrete Finite Element Methods for Two-Dimensional Bingham Flows, Mathematical Problems in Engineering, Volume 2018 (2018), p. 1 | DOI:10.1155/2018/4865849
  • Alexandre Caboussat; Roland Glowinski 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, p. 659 | DOI:10.1007/s11401-015-0930-8
  • Alexandre Caboussat; Roland Glowinski; Tsorng-Whay Pan On the numerical solution of some Eikonal equations: An elliptic solver approach, Chinese Annals of Mathematics, Series B, Volume 36 (2015) no. 5, p. 689 | DOI:10.1007/s11401-015-0971-z
  • Y. Y. Zhou; S. Wang; X. Q. Yang A penalty approximation method for a semilinear parabolic double obstacle problem, Journal of Global Optimization, Volume 60 (2014) no. 3, p. 531 | DOI:10.1007/s10898-013-0122-6
  • Roland Glowinski; Anthony Wachs On the Numerical Simulation of Viscoplastic Fluid Flow, Numerical Methods for Non-Newtonian Fluids, Volume 16 (2011), p. 483 | DOI:10.1016/b978-0-444-53047-9.00006-x
  • Edward J. Dean; Roland Glowinski; Giovanna Guidoboni On the numerical simulation of Bingham visco-plastic flow: Old and new results, Journal of Non-Newtonian Fluid Mechanics, Volume 142 (2007) no. 1-3, p. 36 | DOI:10.1016/j.jnnfm.2006.09.002
  • Roland Glowinski; LieJune Shiau; Ying Ming Kuo; George Nasser On the numerical simulation of friction constrained motions, Nonlinearity, Volume 19 (2006) no. 1, p. 195 | DOI:10.1088/0951-7715/19/1/010
  • A. Caboussat; R. Glowinski A two-grids/projection algorithm for obstacle problems, Computers Mathematics with Applications, Volume 50 (2005) no. 1-2, p. 171 | DOI:10.1016/j.camwa.2004.11.016
  • 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, p. 511 | DOI:10.1016/s1631-073x(03)00024-4

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