Linear convergence of an adaptive finite element method for the p-laplacian equation

SIAM Journal on Numerical Analysis

Volumn 46, Issue 2, 2008, Pages 614-638

  Diening, Lars ^a   Kreuzer, Christian ^b  

^a UNIVERSITY OF FREIBURG   (Germany)

^b UNIVERSITY OF AUGSBURG   (Germany)

Author keywords

A posteriori error estimator; Convergence; Error reduction; Oscillation; P Laplacian; Quasi norms

Indexed keywords

CONTROL THEORY; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; LAPLACE TRANSFORMS; LINEAR EQUATIONS; PIECEWISE LINEAR TECHNIQUES;

A POSTERIORI ERROR ESTIMATOR; CONVERGENCE; ERROR REDUCTION; OSCILLATION; P-LAPLACIAN; QUASI NORMS;

FINITE ELEMENT METHOD;

EID: 55349128827     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070681508     Document Type: Article

References (20)

1. J. W. BARRETT AND W. B. LIU, Finite element approximation of the p-Laplacian, Math. Comp., 61 (1993), pp. 523-537.
2. J. W. BARRETT AND W. B. LIU, Finite element approximation of degenerate quasilinear elliptic and parabolic problems, in Numerical Analysis 1993 (Dundee, 1993), Pitman Res. Notes Math. Ser. 303, Longman Scientific and Technical, Harlow, UK, 1994, pp. 1-16.
3. J. W. BARRETT AND W. B. LIU, Finite element approximation of the parabolic p-Laplacian, SIAM J. Numer. Anal., 31 (1994), pp. 413-428.
4. C. Carstensen and R. Klose, A posteriori finite element error control for the p-Laplace problem, SIAM J. Sci. Comput., 25 (2003), pp. 792-814.
5. C. CARSTENSEN, W. B. LIU, AND N. YAN, A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm, Math. Comp., 75 (1994), pp. 1599-1616.
6. L. DIENING AND F. ETTWEIN, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 20 (2008), pp. 523-556.
7. W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106-1124.
8. L. DIENING AND M. R̊ǔzǐcka, Interpolation operators in Orlicz-Sobolev spaces, Numer. Math., 107 (2007), pp. 107-129.
9. C. EBMEYER, Gobal regularity in Sobolev spaces for elliptic problems with p-structure on bounded domains, in Trends in Partial Differential Equations of Mathematical Physics, Progr. Nonlinear Differential Equations Appl. 61, Birkhäuser, Basel, 2005, pp. 81-89.
10. C. EBMEYER AND W. B. LIU, Quasi-norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems, Numer. Math., 100 (2005), pp. 233-258.
11. C. Ebmeyer, W. B. Liu, and M. Steinhauer, Global regularity in fractional order Sobolev spaces for the p-Laplace equation on polyhedral domains, Z. Anal. Anwendungen, 24 (2005), pp. 353-374.
12. E. GIUSTI, Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003.
13. R. Glowinski and A. Marocco, Sur l'approximation par ́eĺements finis d'ordre un, et la ŕesolution, par ṕenalisation-dualit́e, d'une classe de problèmes de Dirichlet non lineaires, Rev. Fraņcaise Automat. Informat. Recherche Oṕerationelle Ser. Rouge Anal. Nuḿer., 9 (1975), pp. 41-76.
14. W. LIU AND N. YAN, Quasi-norm local error estimators for p-Laplacian, SIAM J. Numer. Anal., 39 (2001), pp. 100-127.
15. W. Liu and N. Yan, On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian, SIAM J. Numer. Anal., 40 (2002), pp. 1870-1895.
16. K. Mekchay and R. H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal., 43 (2005), pp. 1803-1827.
17. P. MORIN, R. H. NOCHETTO, AND K. G. SIEBERT, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466-488.
18. M. M. RAO AND Z. D. REN, Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math. 146, Marcel Dekker, New York, 1991.
19. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
20. A. VEESER, Convergent adaptive finite elements for the nonlinear Laplacian, Numer. Math., 92 (2002), pp. 743-770.