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


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.

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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”.

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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

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

Mathematics Subject Classification

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