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Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic \(p\)-Laplacian

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Abstract

We generalize the a posteriori techniques for the linear heat equation in Verfürth (Calcolo 40(3):195–212, 2003) to the case of the nonlinear parabolic \(p\)-Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of the error and the residual, which is the key property for proving the error bounds.

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References

  1. Belenki, L., Berselli, L., Diening, L., R\(\breve{\rm u}\)žička, M.: On the finite element approximation of \(p\)-stokes systems. SIAM J. Numer. Anal. 50(2), 373–397 (2012)

  2. Belenki, L., Diening, L., Kreuzer, C.: Optimality of an adaptive finite element method for the \(p\)-laplacian equation. IMA J. Numer. Anal. 32(2), 484–510 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barrett, J.W., Liu, W.B.: Finite element approximation of the \(p\)-Laplacian. Math. Comput. 61(204), 523–537 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Barrett, J.W., Liu, W.B.: Finite element error analysis of a quasi-newtonian flow obeying the Carreau or power law. Numer. Math. 64(4), 433–453 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barrett, J.W., Liu, W.B.: Finite element approximation of the parabolic \(p\)-Laplacian. SIAM J. Numer. Anal. 31(2), 413–428 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Z., Feng, J.: An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comp. 73, 1167–1042 (2006)

    Article  MathSciNet  Google Scholar 

  7. Chow, S.-S.: Finite element error estimates for nonlinear elliptic equations of monotone type. Numer. Math. 54(4), 373–393 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ciarlet, P.G.: The finite element method for elliptic problems, Studies in mathematics and its applications, vol. 4. North-Holland Publishing Co., Amsterdam (1978)

    Google Scholar 

  9. Carstensen, C., Klose, R.: A posteriori finite element error control for the \(p\)-Laplace problem. SIAM J. Sci. Comput. 25(3), 792–814 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carstensen, C., Liu, W., Yan, N.: A posteriori FE error control for \(p\)-Laplacian by gradient recovery in quasi-norm. Math. Comput. 75(256), 1599–1616 (2006) (electroni)

    Google Scholar 

  11. Carstensen, C., Liu, W., Yan, N.: A posteriori error estimates for finite element approximation of parabolic \(p\)-Laplacian. SIAM J. Numer. Anal. 43(6), 2294–2319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Diening, L., Ettwein, F.: Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math. 3, 523–556 (2002)

    MathSciNet  Google Scholar 

  13. Diening, L., Ebmeyer, C., R\(\breve{\rm u}\)žička, M.: Optimal convergence for the implicit space-time discretization of parabolic systems with \(p\)-structure. SIAM J. Numer. Anal. 45(2), 457–472 (2007)

  14. Diening, L., Kreuzer, C.: Convergence of an adaptive finite element method for the \(p\)-Laplacian equation. SIAM J. Numer. Anal. 46(2), 614–638 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Diening, L., R\(\breve{\rm u}\)žička, M.: Interpolation operators in Orlicz-Sobolev spaces. Numer. Math. 107(1), 107–129 (2007)

  16. Diening, L., R\(\breve{\rm u}\)žička, M.: Non-Newtonian fluids and function spaces. Nonlinear Anal. Funct. Spaces Appl. 8, 95–143 (2007)

  17. Ebmeyer, C., Liu, W.: Quasi-norm interpolation error estimates for the piecewise linear finite element approximation of \(p\)-Laplacian problems. Numer. Math. 100(2), 233–258 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Mathematische Lehrbücher und Monographien. II. Abteilung. Band 38. Akademie-Verlag, Berlin. IX, 281 S., German (1974)

  19. Glowinski, R., Marrocco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires. Rev. Française Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numérique 9(R-2), 41–76 (1975)

  20. Kokilashvili, V., Krbec, M.: Weighted inequalities in Lorentz and Orlicz spaces. World Scientific Publishing Co. Pte. Ltd., Singapore (1991)

    Book  MATH  Google Scholar 

  21. Kreuzer, C., Möller, C.A., Schmidt, A., Siebert, K.G.: Design and convergence analysis for an adaptive discretization of the heat equation Preprint SM-DU-724 Universität Duisburg-Essen. IMA J. Numer. Anal, In (December 2010). (2011, to appear)

  22. Krasnosel’skij, M.A., Rutitskij, Y.B.: Convex functions and Orlicz spaces. P. Noordhoff Ltd., Groningen (1961)

    MATH  Google Scholar 

  23. Kreuzer, C.: A convergent adaptive Uzawa finite element method for the nonlinear Stokes problem, Ph.D. thesis, Mathematisches Institut, Universität Augsburg (2008).

  24. Kreuzer, C.: Analysis of an adaptive uzawa finite element method for the nonlinear Stokes problem. Math. Comput. 81, 21–55 (2012)

    Google Scholar 

  25. Liu, W., Yan, N.: Quasi-norm local error estimators for \(p\)-Laplacian. SIAM J. Numer. Anal. 39(1), 100–127 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, W., Yan, N.: On quasi-norm interpolation error estimation and a posteriori error estimates for \(p\)-Laplacian. SIAM J. Numer. Anal. 40(5), 1870–1895 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Musielak, J.: Orlicz spaces and modular spaces. Lecture notes in mathematics, vol. 1034. Springer-Verlag, Berlin (1983)

  28. Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. In: Pure and applied mathematics, vol. 146. Marcel Dekker, Inc., New York (1991)

  29. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Verfürth, R.: A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40(3), 195–212 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Verfürth, R.: Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J. Numer. Anal 43(4), 1783–1802 (2005). (electronic)

    Google Scholar 

  32. Zeidler, E.: Nonlinear functional analysis and its applications. II/A: Linear monotone operators. Transl. from the German by the author and by Leo F. Boron. Springer-Verlag, New York (1990)

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Acknowledgments

Part of this work was carried out during a stay at the Mathematical Institute of the University of Oxford. This stay was financed by the German Research Foundation DFG within the research grant Kr 3984/1-1. Last but not least I would like to thank Prof. Endre Süli for his great hospitality and many fruitful discussions.

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  1. Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44801 , Bochum, Germany

    Christian Kreuzer

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  1. Christian Kreuzer
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Kreuzer, C. Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic \(p\)-Laplacian. Calcolo 50, 79–110 (2013). https://doi.org/10.1007/s10092-012-0059-z

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