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Quasi-Norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems

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Summary.

In this work, new interpolation error estimates have been derived for some well-known interpolators in the quasi-norms. The estimates are found to be essential to obtain the optimal a priori error bounds under the weakened regularity conditions for the piecewise linear finite element approximation of a class of degenerate equations. In particular, by using these estimates, we can close the existing gap between the regularity required for deriving the optimal error bounds and the regularity achievable for the smooth data for the 2-d and 3-d p-Laplacian.

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References

  1. Ainsworth, M., Day, K.: The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs. Numer. Math. 83, 354–388 (1999)

    Google Scholar 

  2. Ainsworth, M., Day, K.: Approximation theory for the hp-version finite element method and application to the non-linear Laplacian. Appl. Numer. Math. 34, 329–344 (2000)

    Google Scholar 

  3. Atkinson, C., Champion, C.R.: Some boundary-value problems for the equation ∇ .(| ∇ ϕ |N ∇ ϕ)=0. Quart. J. Mech. Appl. Math. 37, 401–419 (1984)

    Google Scholar 

  4. Baranger, J., Najib, K.: Numerical analysis of quasi-Newtonian flow obeying the power law or the Carreau law. Numer. Math. 58, 35–49 (1990)

    Google Scholar 

  5. Barrett, J.W., Bermejo, R.: An improved error bound for the discretization of the parabolic p-Laplacian and related degenerate quasilinear equations and variational inequalities. To appear

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

    Google Scholar 

  7. Barrett, J.W., Liu, WB.: Finite element approximation of some degenerate quasi-linear problems. Lecture Notes in Mathematics 303, Pitman, 1994, pp. 1–16

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

    Google Scholar 

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

    Google Scholar 

  10. Carstensen, C., Klose, R.: Guaranteed a posteriori finite element error control for the p-Laplace problem. Submitted

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

    Google Scholar 

  12. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978

  13. Dobrowolski, M., Rannacher, R.: Finite element methods for nonlinear elliptic systems of second order. Math. Nachr. 94, 155–172 (1980)

    Google Scholar 

  14. Ebmeyer, C., Liu, WB., Steinhauer, M.: Global regularity in fractional order Sobolev spaces for the p–Laplace equation on polyhedral domains. To be published in Zeit. Anal. Anwend., 2004

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

    Google Scholar 

  16. Frehse, J., Rannacher, R.: Asymptotic L∞-error estimates for linear finite element approximations of quasilinear boundary problems. SIAM J. Numer. Anal. 15, 418–431 (1978)

    Google Scholar 

  17. Glowinski, R., Marrocco, A.: Sur l’approximation par elements finis d’order un, et la resolution, par penalisation-dualite, d’une class de Dirichlet non lineaires. RAIRO Anal. Numer. 2, 41–76 (1975)

    Google Scholar 

  18. Kufner, A., John, O., Fučik, S.: Function Spaces. Prague: Academia 1977

  19. Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlin. Anal. 12, 1203–1219 (1988)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  22. Liu, WB., Barrett, J.W.: A remark on the regularity of the solutions of the p-Laplacian and its application to their finite element approximation. J. Math. Anal. Appl. 178, 470–487 (1993)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  25. Liu, WB., Yan, N.N.: Quasi-norm local error estimates for p-Laplacian. SIAM J. Numer. Anal. 39, 100–127 (2001)

    Google Scholar 

  26. Liu, WB., Yan, N.N.: Quasi-norm a posteriori error estimates for non-conforming FEM of p-Laplacian. Numer. Math. 89, 341–378 (2001)

    Google Scholar 

  27. Málek, J., Nečas, J., Rokyta, M., Rů žička, M.: Existence and regularity of solutions to evolutional partial differential equations. Applied Mathematics and Mathematical Computation, Vol. 13, Chapman and Hall, London, 1996

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

    Google Scholar 

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

    Google Scholar 

  30. Sandri, D.: A posteriori estimators for mixed finite element approximation of a fluid obeying the power law. Comput. Methods Appl. Mech. Eng. 166, 329–340 (1998)

    Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Seminar, Universität Bonn, Nussallee 15, D-53115, Bonn, Germany

    Carsten Ebmeyer

  2. KBS & IMS, University of Kent, Canterbury, CT2 7NF, England

    WB. Liu

Authors
  1. Carsten Ebmeyer
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  2. WB. Liu
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Correspondence to WB. Liu.

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Mathematics Subject Classification (1991): 65N30

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Ebmeyer, C., Liu, W. Quasi-Norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems. Numer. Math. 100, 233–258 (2005). https://doi.org/10.1007/s00211-005-0594-5

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  • DOI: https://doi.org/10.1007/s00211-005-0594-5

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