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

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

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)

    MathSciNet  Article  MATH  Google 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)

    Google Scholar 

  3. 3.

    Barrett, J.W., Liu, W.B.: Finite element approximation of the \(p\)-Laplacian. Math. Comput. 61, 523–537 (1993)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bermejo, R., Infante, J.A.: A multigrid algorithm for the \(p\)-Laplacian. SIAM J. Sci. Comput. 21, 1774–1789 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    Google Scholar 

  11. 11.

    De los Reyes, J.C.: Numerical PDE-Constrained Optimization. Springer, Berlin (2015)

    Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)

    Google Scholar 

  17. 17.

    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Company, The Netherlands (1976)

    Google Scholar 

  18. 18.

    Geiger, C., Kanzow, C.: Numerische Verfahren zur Lösung unrestringierter Optimierungsaufgaben. Springer, Berlin (1999)

    Google 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)

    MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Hintermüller, M., Rautenberg, C.: Parabolic quasi-variational inequalities with gradient-type constraints. SIAM J. Optim. 23, 2090–2123 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Secaucus (2009)

    Google Scholar 

  25. 25.

    Huang, Y.Q., Li, R., Liu, W.: Preconditioned descent algorithms for p-Laplacian. J. Sci. Comput. 32, 343–371 (2007)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google Scholar 

  27. 27.

    Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (2007)

    Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999)

    Google Scholar 

  30. 30.

    Lieb, E.H., Loss, M.: Analysis. AMS, Providence (2001)

    Google Scholar 

  31. 31.

    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Google 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)

    MathSciNet  Google Scholar 

  33. 33.

    Nocedal, J.: Theory of algorithms for unconstrained optimization. Acta Numer. 1, 199–242 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Quarteroni, A., Tuveri, M., Veneziani, A.: Computational vascular fluid dynamics: problems, models and methods. Comput. Vis. Sci. 2, 163–197 (2000)

    Article  MATH  Google 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)

    Article  Google 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)

    Article  Google 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. 205–227

  39. 39.

    Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Berlin (2008)

    Google Scholar 

  40. 40.

    Sun, W., Yuan, Y.-X.: Optimization Theory and Methods. Nonlinear Programming. Springer, New York (2006)

    Google Scholar 

  41. 41.

    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, vol. GDR. North Holland Publishing Company, Amsterdam (1978)

    Google Scholar 

  42. 42.

    Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa 27, 265–308 (1973)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13, 805–841 (2003)

    MathSciNet  Article  MATH  Google Scholar 

Download references

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”.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sergio González-Andrade.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

González-Andrade, S. A preconditioned descent algorithm for variational inequalities of the second kind involving the p-Laplacian operator. Comput Optim Appl 66, 123–162 (2017). https://doi.org/10.1007/s10589-016-9861-x

Download citation

Keywords

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

Mathematics Subject Classification

  • 47J20
  • 65K10
  • 65K15
  • 65N30
  • 76A05