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Preconditioned Descent Algorithms for p-Laplacian

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In this paper, we examine some computational issues on finite element discretization of the p-Laplacian. We introduced a class of descent methods with multi-grid finite element preconditioners, and carried out convergence analysis. We showed that their convergence rate is mesh-independent. We studied the behavior of the algorithms with large p. Our numerical tests show that these algorithms are able to solve large scale p-Laplacian with very large p. The algorithms are then used to solve a variational inequality.

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References

  1. Ainsworth M., Kay D. (1999) The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs. Numer. Math. 83: 351–388

    Article  Google Scholar 

  2. Baranger J., Amri H.El. (1991) Estimateurs a posteriori d’erreur pour le calcul adaptatif d’ecoulements quasi-Newtoniens. R.A.I.R.O. M2AN 25: 31–48

    Google Scholar 

  3. Barrett J.W., Prigozhin L. (2000). Bean’s critical-state model as p→∞ limit of an evolutionary p-Laplacian equation. J. Nonlinear Anal. 42: 977–993

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Barrett J.W., Liu W.B. (1994) Finite element approximation of some degenerate quasi-linear problems. Lecture Notes Math. 303: 1–16

    Google Scholar 

  6. Barrett J.W., Liu W.B. (1994) Quasi-norm error bounds for finite element approximation of quasi-Newtonian flows. Numer. Math. 68: 437–456

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Bouchitte G., Buttazzo G., De Pascale L. (2003) A p-Laplacian approximation for some mass optimization problems. JOTA 118: 1–25

    Article  MATH  Google Scholar 

  9. Carstensen C., and Klose, R. (submitted). Guaranteed a posteriori finite element error control for p-Laplacian.

  10. Chow S.S. (1988) Finite element error estimates for non-linear elliptic equations of monitone type. Numer. Math. 54: 373–393

    Article  MATH  Google Scholar 

  11. Ciarlet P.G. (1978) The Finite Element Method for Elliptic Problems. North Holland, Amsterdam

    MATH  Google Scholar 

  12. Farhloul M. (1998) A mixed finite element method for a nonlinear Direchlet problem. IMA J. Numer. Anal. 18: 121–132

    Article  MATH  Google Scholar 

  13. Glowinski, R., and Lions, J. L., and Tremolieres, R. (1976). Numerical Analysis of Variational Inequalities, North-Holland, Netherlands.

  14. Glowinski R., Marrocco A. (1975) Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problémes de Dirichlet non linéaires (French). RAIRO Anal. Numer. 9(R-2): 41–76

    Google Scholar 

  15. Huang Y.Q., Shu S., Yu X.-J. (2006) Preconditioning higher order finite element systems by algebraic multigrid method of linear element. J. Comput. Math. 24(5): 657–664

    MATH  Google Scholar 

  16. Liu W.B., Barrett J.W. (1993) Higher order regularity for the solutions of some nonlinear degenerate elliptic equations. SIAM J. Math. Anal. 24(6): 1522–1536

    MATH  Google Scholar 

  17. Liu W.B., Barrett J.W. (1993) A remark on the regularity of the solutions of p-Laplacian and its applications to their finite element approximation. J. Math. Anal. Appl. 178: 470–488

    Article  MATH  Google Scholar 

  18. Liu W.B., Barrett J.W. (1993) A further remark on the regularity of the solutions of the p-Laplacian and its applications to their finite element approximation. J. Nonlinear. Anal. 21: 379–387

    Article  MATH  Google Scholar 

  19. Liu W.B. (2000) Degenerate quasilinear elliptic equations arising from bimaterial problems in elastic-plastic mechanics II. Numer. Math. 86: 491–506

    Article  MATH  Google Scholar 

  20. Liu W.B., Barrett J.W. (1996) Finite element approximation of some degenerate monotone quasi-linear elliptic systems. SIAM J. Numer. Anal. 33: 88–106

    Article  MATH  Google Scholar 

  21. Liu W.B., Yan N. (2001) Quasi-norm local error estimates for the finite element approximation of p-Laplacian. SIAM J. Numer. Anal. 39: 100–127

    Article  MATH  Google Scholar 

  22. Liu, W. B., and Yan, N. (2003). On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian. SIAM J. Numer. Anal.

  23. Manouzi H., Farhloul M. (2001) Mixed finite element analysis of a non-linear three-fields Stokes model. IMA J. Numer. Anal. 21: 143–164

    Article  MATH  Google Scholar 

  24. Padra C. (1997) A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows. SIAM J. Numer. Anal. 34: 1600–1615

    Article  MATH  Google Scholar 

  25. Simms, G. (1995). Finite element approximation of some nonlinear elliptic and parabolic problems, thesis. Imperial College, University of London.

  26. Verfürth R. (1994) A posteriori error estimates for non-linear problems. Math. Comp. 62: 445–475

    Article  MATH  Google Scholar 

  27. Tai X.C., Espedal M. (1998) Rate of convergence of some space decomposition method for linear and nonlinear elliptic problems. SIAM J. Numer. Anal. 38: 1558–1570

    Article  Google Scholar 

  28. Tai X.C., Xu J. (2002) Global convergence of subspace correction methods for convex optimization problems. Math. Comp. 74: 105–124

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China

    Y. Q. Huang

  2. KBS, and Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, England

    Ruo Li & Wenbin Liu

Authors
  1. Y. Q. Huang
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  2. Ruo Li
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  3. Wenbin Liu
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Correspondence to Y. Q. Huang.

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Huang, Y.Q., Li, R. & Liu, W. Preconditioned Descent Algorithms for p-Laplacian. J Sci Comput 32, 343–371 (2007). https://doi.org/10.1007/s10915-007-9134-z

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  • DOI: https://doi.org/10.1007/s10915-007-9134-z

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