Advertisement

Computational Optimization and Applications

, Volume 66, Issue 1, pp 123–162 | Cite as

A preconditioned descent algorithm for variational inequalities of the second kind involving the p-Laplacian operator

  • Sergio González-AndradeEmail author
Article

Abstract

This paper is concerned with the numerical solution of a class of variational inequalities of the second kind, involving the p-Laplacian operator. This kind of problems arise, for instance, in the mathematical modelling of non-Newtonian fluids. We study these problems by using a regularization approach, based on a Huber smoothing process. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to the solution of the original problem is verified. We propose a preconditioned descent method for the numerical solution of these problems and analyze the convergence of this method in function spaces. The existence of admissible descent directions is established by variational methods and admissible steps are obtained by a backtracking algorithm which approximates the objective functional by polynomial models. Finally, several numerical experiments are carried out to show the efficiency of the methodology here introduced.

Keywords

Variational inequalities p-Laplacian Optimization and variational techniques Herschel–Bulkley model 

Mathematics Subject Classification

47J20 65K10 65K15 65N30 76A05 

Notes

Acknowledgments

I would like to thank Prof. Dr. Juan Carlos De los Reyes (ModeMat-Quito) and Prof. Dr. Eduardo Casas (Univ. de Cantabria-Spain) for all the helpful discussions and good insights in the problem. I also would like to thank the anonymous referees for many helpful comments which lead to a significant improvement of the article. Finally, thanks to Prof. Dr. Michael Hinze, Prof. Dr. Winniefred Wollner and Prof. Dr. Ingenuin Gasser for the kind hospitality and interesting discussions during my stay in Hamburg Universität. Supported in part by the Ecuadorian Secretary of Higher Education, Science, Technology and Innovation, SENESCYT, under the project PIC-13-EPN-001 “Numerical Simulation of Cardiac and Circulatory Systems”, the Escuela Politécnica Nacional, under the project PIMI 14-12 “Numerical Simulation of Viscoplastic Fluids in Food Industry” and the MATH-AmSud Project “SOCDE-Sparse Optimal Control of Differential Equations: Algorithms and Applications”.

References

  1. 1.
    Alberty, J., Carstensen, C., Funken, S.A.: Remarks around 50 lines of Matlab: short finite element implementation. Numer Algorithms 20, 117–137 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antontsev, S.N., Díaz, J.I., Shmarev, S.: Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics. Birkhäuser, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Barrett, J.W., Liu, W.B.: Finite element approximation of the \(p\)-Laplacian. Math. Comput. 61, 523–537 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bermejo, R., Infante, J.A.: A multigrid algorithm for the \(p\)-Laplacian. SIAM J. Sci. Comput. 21, 1774–1789 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)zbMATHGoogle Scholar
  6. 6.
    Casas, E., Fernández, L.A.: Distributed control of systems governed by a general class of quasilinear elliptic equations. J. Differ. Equ. 104, 20–47 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chhabra, R.P., Richardson, J.F.: Non-Newtonian Flow and Applied Rheology. Elsevier, Budapest (2008)Google Scholar
  9. 9.
    Coffman, C.V., Duffin, V., Mizel, V.J.: Positivity of weak solutions of non uniformly elliptic equations. Ann. Mat. Pura Appl. 104, 209–238 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 2: Functional Analysis and Variational Methods. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    De los Reyes, J.C.: Numerical PDE-Constrained Optimization. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  12. 12.
    De los Reyes, J.C., González Andrade, S.: Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods. J. Comput. Appl. Math. 235, 11–32 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    De los Reyes, J.C., González Andrade, S.: A combined BDF-semismooth Newton approach for time-dependent Bingham flow. Numer. Methods Partial Differ. Equ. 28, 834–860 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    De los Reyes, J.C., González, S.: Path following methods for steady laminar bingham flow in cylindrical pipes. Math. Modell. Numer. Anal. 43, 81–117 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    De los Reyes, J.C., Hintermüller, M.: A duality based semismooth Newton framework for solving variational inequalities of the second kind. Interfaces Free Bound. 13, 437–462 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Company, The Netherlands (1976)zbMATHGoogle Scholar
  18. 18.
    Geiger, C., Kanzow, C.: Numerische Verfahren zur Lösung unrestringierter Optimierungsaufgaben. Springer, Berlin (1999)CrossRefGoogle Scholar
  19. 19.
    Glowinski, R., Marroco, A.: Sur L’Approximation par Elements Finisd’Ordre Un, et la Resolution, par Penalisation-Dualite, d’une Classede Problemes de Dirichlet non Lineaires. R.A.I.R.O 9, 41–76 (1975)zbMATHGoogle Scholar
  20. 20.
    Gröger, K.: A \(W^{1, p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math Ann 283, 679–687 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13, 865–888 (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hintermüller, M., Rautenberg, C.: A sequential minimization technique for elliptic quasi-variational inequalities with gradient constraints. SIAM J. Optim. 22, 1224–1257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hintermüller, M., Rautenberg, C.: Parabolic quasi-variational inequalities with gradient-type constraints. SIAM J. Optim. 23, 2090–2123 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Secaucus (2009)zbMATHGoogle Scholar
  25. 25.
    Huang, Y.Q., Li, R., Liu, W.: Preconditioned descent algorithms for p-Laplacian. J. Sci. Comput. 32, 343–371 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Huilgol, R.R., You, Z.: Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel-Bulkley Fluids. J. Non-Newton. Fluid Mech. 128, 126–143 (2005)CrossRefzbMATHGoogle Scholar
  27. 27.
    Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (2007)zbMATHGoogle Scholar
  28. 28.
    Jouvet, G., Bueler, E.: Steady, shallow ice sheets as obstacle problems: well-posedness and finite element approximation. SIAM J. Optim. 23, 2090–2123 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  30. 30.
    Lieb, E.H., Loss, M.: Analysis. AMS, Providence (2001)CrossRefzbMATHGoogle Scholar
  31. 31.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  32. 32.
    Liu, W.B., Barret, J.W.: Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities. ESAIM: Math. Modell. Numeri. Anal. 28, 725–744 (1994)MathSciNetGoogle Scholar
  33. 33.
    Nocedal, J.: Theory of algorithms for unconstrained optimization. Acta Numer. 1, 199–242 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Quarteroni, A., Tuveri, M., Veneziani, A.: Computational vascular fluid dynamics: problems, models and methods. Comput. Vis. Sci. 2, 163–197 (2000)CrossRefzbMATHGoogle Scholar
  35. 35.
    Sankar, D.S., Lee, Usik: Two-fluid Herschel-Bulkley model for blood flow in catheterized arteries. J. Mech. Sci. Technol. 22, 1008–1018 (2008)CrossRefGoogle Scholar
  36. 36.
    Shah, S.R.: An innovative study for non-Newtonian behaviour of blood flow in stenosed artery using Herschel-Bulkley fluid model. Int. J. Bio-Sci. Bio-Technol. 5, 233–240 (2013)CrossRefGoogle Scholar
  37. 37.
    Simader, C.G.: On Dirichlet’s Boundary Value Problems. Lecture Notes in Mathematics, No. 268. Springer, Berlin (1972)Google Scholar
  38. 38.
    Simon, J.: 1978. Regularité de la Solution d’une Equation nonLineaire dans \({\mathbb{R}}^N\). In: Benilan, P. (ed) Lecture Notes in Mathematics, No. 665. Springer, pp. 205–227Google Scholar
  39. 39.
    Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Berlin (2008)zbMATHGoogle Scholar
  40. 40.
    Sun, W., Yuan, Y.-X.: Optimization Theory and Methods. Nonlinear Programming. Springer, New York (2006)zbMATHGoogle Scholar
  41. 41.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, vol. GDR. North Holland Publishing Company, Amsterdam (1978)zbMATHGoogle Scholar
  42. 42.
    Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa 27, 265–308 (1973)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13, 805–841 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Research Center on Mathematical Modeling (ModeMat) and Departamento de MatemáticaEscuela Politécnica Nacional, QuitoQuitoEcuador

Personalised recommendations