Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm


Authors: Carsten Carstensen, W. Liu and N. Yan
Journal: Math. Comp. 75 (2006), 1599-1616
MSC (2000): Primary 65N30, 49J40
DOI: https://doi.org/10.1090/S0025-5718-06-01819-9
Published electronically: June 7, 2006
MathSciNet review: 2240626
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A posteriori error estimators based on quasi-norm gradient recovery are established for the finite element approximation of the p-Laplacian on unstructured meshes. The new a posteriori error estimators provide both upper and lower bounds in the quasi-norm for the discretization error. The main tools for the proofs of reliability are approximation error estimates for a local approximation operator in the quasi-norm.


References [Enhancements On Off] (What's this?)

  • [A] Adams, R.A. (1975). Sobolev Spaces. Academic Press, New York, London. MR 0450957 (56:9247)
  • [AO] Ainsworth, M. and Oden, J.T. (2000). A posteriori error estimation in finite element analysis. Wiley-Interscience [John Wiley & Sons], New York. xx+240. MR 1885308 (2003b:65001)
  • [BA] Baranger, J. and El Amri, H. (1991). Estimateurs a posteriori d'erreur pour le calcul adaptatif d'écoulements quasi-newtoniens. RAIRO Modél. Math. Anal. Numér., 25, 1, 31-47. MR 1086839 (91m:76070)
  • [BDR] Babuška, I., Durán, R. and Rodríguez, R. (1992). Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal., 29, 4, 947-964.MR 1173179 (93d:65096)
  • [BL1] Barrett, J.W. and Liu, W. B. (1993). Finite element approximation of the $ p$-Laplacian. Math. Comp., 61, 204, 523-537.MR 1192966 (94c:65129)
  • [BL2] Barrett, J.W. and Liu, W.B. (1994). Finite element approximation of some degenerate quasi-linear problems. Lecture Notes in Mathematics 303, Pitman, 1-16. MR 1267752 (95c:65151)
  • [BL3] Barrett, J.W. and Liu, W.B. (1994). Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math., 68, 4, 437-456. MR 1301740 (95h:65078)
  • [BS] Brenner, S.C. and Scott, L.R. (1994). The mathematical theory of finite element methods. Springer Verlag, New York, 15, xii+294. MR 1278258 (95f:65001)
  • [Ca] Carstensen, C. (1999). Quasi-interpolation and a posteriori error analysis in finite element method. M2AN Math. Model. Numer. Anal., 33, 6, 1187-1202. MR 1736895 (2001a:65135)
  • [CB] Carstensen, C. and Bartels, S. (2004). Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math. 99 225-249. MR 2107431
  • [CF1] Carstensen, C. and Funken, S.A. (2000)a. Constants in Clement-interpolation error and residual based a posteriori estimates in finite element methods. East-West J. Numer. Math., 8, 3, 153-175. MR 1807259 (2002a:65173)
  • [CF2] C. Carstensen and S. A. Funken. Averaging technique for FE-a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg., 190 (2001) 2483-2498. Part II: $ \lambda$-independent estimates. Comput. Methods Appl. Mech. Engrg., 190 (2001) 4663--4675. Part III: Locking-free conconforming FEM. Comput. Methods Appl. Mech. Engrg., 191 (2001), no. 8-10, 861-877. MR 1815651 (2002a:74114), MR 1840795 (2002d:65140), MR 1870519 (2002j:65106)
  • [CK] Carstensen, C. and Klose, R. (2003). Guaranteed a posteriori finite element error control for the p-Laplace problem. SIAM J. Scientific Computing 25 792-814. MR 2046112 (2005d:65203)
  • [Ch] Chow, S.-S. (1989). Finite element error estimates for nonlinear elliptic equations of monotone type. Numer. Math., 54, 4, 373-393. MR 0972416 (90a:65235)
  • [Ci] Ciarlet, P.G. (1978). The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam, xix+530. MR 0520174 (58:25001)
  • [EL] Ebmeyer, C. and Liu, W.B. Quasi-norm interpolation error estimates for finite element approximations of problems with p-structure. (to appear).
  • [GM] Glowinski, R. and Marrocco, A. (1975). 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. Francaise Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9, R-2, 41-76. MR 0388811 (52:9645)
  • [KJF] Kufner, A., John, O. and Fucík, S. (1977). Function spaces. Noordhoff International Publishing, Leyden, xv+454. MR 0482102 (58:2189)
  • [L] Lieberman, G.M. (1988). Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal., 12, 11, 1203-1219. MR 0969499 (90a:35098)
  • [LB5] Liu, W. B. and Barrett, J.W. (1996). Finite element approximation of some degenerate monotone quasilinear elliptic systems. SIAM J. Numer. Anal., 33, 1, 88-106.MR 1377245 (97j:65180)
  • [LY1] Liu, W. and Yan, N. (2001). Quasi-norm local error estimators for $ p$-Laplacian. SIAM J. Numer. Anal., 39, 1, 100-127. MR 1860718 (2002i:65131)
  • [LY2] Liu, W.B. and Yan, N. (2001). Quasi-norm a posteriori error estimates for non-conforming FEM of p-Laplacian. Numer. Math., 89, 341-378. MR 1855829 (2002g:65136)
  • [LY3] Liu, W. and Yan, N. (2001). Some a posteriori error estimators for $ p$-Laplacian based on residual estimation or gradient recovery. J. Sci. Comput., 16, 4, 435-477 (2002).MR 1881854 (2002k:65191)
  • [ODSD] Oden, J. T., Demkowicz, L., Strouboulis, T. and Devloo, P. (1986). Adaptive methods for problems in solid and fluid mechanics. (Accuracy estimates and adaptive refinements in finite element computations (Lisbon, 1984)), Wiley Ser. Numer. Methods Engrg., Wiley, Chichester, 249-280.MR 0879450 (88d:73010)
  • [P] Padra, C. (1997). A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows. SIAM J. Numer. Anal., 34, 4, 1600-1615.MR 1461798 (98h:65050)
  • [V1] Verfürth, R. (1994). A posteriori error estimates for nonlinear problems. (Finite element discretizations of elliptic equations), Math. Comp., 62, 206, 445-475.MR 1213837 (94j:65136)
  • [V2] Verfürth, R. (1996). A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner. MR 1643051

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 49J40

Retrieve articles in all journals with MSC (2000): 65N30, 49J40


Additional Information

Carsten Carstensen
Affiliation: Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
Email: cc@math.hu-berlin.de

W. Liu
Affiliation: CBS & Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, England
Email: W.B.Liu@ukc.ac.uk

N. Yan
Affiliation: Institute of System Sciences, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, People’s Republic of China
Email: yan@amss.ac.cn

DOI: https://doi.org/10.1090/S0025-5718-06-01819-9
Keywords: Finite element approximation, p-Laplacian, a posteriori error estimators, gradient recovery, quasi-norm error bounds
Received by editor(s): April 16, 2003
Received by editor(s) in revised form: May 3, 2005
Published electronically: June 7, 2006
Additional Notes: Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society