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

Carstensen, Carsten; Liu, W.; Yan, N.

doi:10.1090/S0025-5718-06-01819-9

A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm
HTML articles powered by AMS MathViewer

by Carsten Carstensen, W. Liu and N. Yan;

Math. Comp. 75 (2006), 1599-1616

DOI: https://doi.org/10.1090/S0025-5718-06-01819-9

Published electronically: June 7, 2006

PDF | Request permission

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

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 450957
Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
Jacques Baranger and Hassan El Amri, Estimateurs a posteriori d’erreur pour le calcul adaptatif d’écoulements quasi-newtoniens, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 31–47 (French, with English summary). MR 1086839, DOI 10.1051/m2an/1991250100311
Ivo Babuška, Ricardo Durán, and Rodolfo Rodríguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements, SIAM J. Numer. Anal. 29 (1992), no. 4, 947–964. MR 1173179, DOI 10.1137/0729058
John W. Barrett and W. B. Liu, Finite element approximation of the $p$-Laplacian, Math. Comp. 61 (1993), no. 204, 523–537. MR 1192966, DOI 10.1090/S0025-5718-1993-1192966-4
J. W. Barrett and Wen Bin Liu, Finite element approximation of degenerate quasilinear elliptic and parabolic problems, Numerical analysis 1993 (Dundee, 1993) Pitman Res. Notes Math. Ser., vol. 303, Longman Sci. Tech., Harlow, 1994, pp. 1–16. MR 1267752
John W. Barrett and W. B. Liu, Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math. 68 (1994), no. 4, 437–456. MR 1301740, DOI 10.1007/s002110050071
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Carsten Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187–1202. MR 1736895, DOI 10.1051/m2an:1999140
S. Bartels and C. Carstensen, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math. 99 (2004), no. 2, 225–249. MR 2107431, DOI 10.1007/s00211-004-0553-6
C. Carstensen and S. A. Funken, Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods, East-West J. Numer. Math. 8 (2000), no. 3, 153–175. MR 1807259
Carsten Carstensen and Stefan A. Funken, Averaging technique for FE—a posteriori error control in elasticity. I. Conforming FEM, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 18-19, 2483–2498. MR 1815651, DOI 10.1016/S0045-7825(00)00248-6
Carsten Carstensen and R. Klose, A posteriori finite element error control for the $p$-Laplace problem, SIAM J. Sci. Comput. 25 (2003), no. 3, 792–814. MR 2046112, DOI 10.1137/S1064827502416617
S.-S. Chow, Finite element error estimates for nonlinear elliptic equations of monotone type, Numer. Math. 54 (1989), no. 4, 373–393. MR 972416, DOI 10.1007/BF01396320
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Ebmeyer, C. and Liu, W.B. Quasi-norm interpolation error estimates for finite element approximations of problems with p-structure. (to appear).
R. Glowinski and A. Marrocco, 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 Sér. Rouge Anal. Numér. 9 (1975), no. R-2, 41–76 (French, with English summary). MR 388811, DOI 10.1051/m2an/197509R200411
Alois Kufner, Oldřich John, and Svatopluk Fučík, Function spaces, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leiden; Academia, Prague, 1977. MR 482102
Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. MR 969499, DOI 10.1016/0362-546X(88)90053-3
W. B. Liu and John W. Barrett, Finite element approximation of some degenerate monotone quasilinear elliptic systems, SIAM J. Numer. Anal. 33 (1996), no. 1, 88–106. MR 1377245, DOI 10.1137/0733006
Wenbin Liu and Ningning Yan, Quasi-norm local error estimators for $p$-Laplacian, SIAM J. Numer. Anal. 39 (2001), no. 1, 100–127. MR 1860718, DOI 10.1137/S0036142999351613
Wenbin Liu and Ningning Yan, Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of $p$-Laplacian, Numer. Math. 89 (2001), no. 2, 341–378. MR 1855829, DOI 10.1007/PL00005470
Wenbin Liu and Ningning Yan, Some a posteriori error estimators for $p$-Laplacian based on residual estimation or gradient recovery, J. Sci. Comput. 16 (2001), no. 4, 435–477 (2002). MR 1881854, DOI 10.1023/A:1013246424707
J. T. Oden, L. Demkowicz, T. Strouboulis, and P. Devloo, 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, 1986, pp. 249–280. MR 879450
Claudio Padra, A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows, SIAM J. Numer. Anal. 34 (1997), no. 4, 1600–1615. MR 1461798, DOI 10.1137/S0036142994278322
R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62 (1994), no. 206, 445–475. MR 1213837, DOI 10.1090/S0025-5718-1994-1213837-1
R. Verfürth, A review of a posteriori error estimation techniques for elasticity problems, Advances in adaptive computational methods in mechanics (Cachan, 1997) Stud. Appl. Mech., vol. 47, Elsevier Sci. B. V., Amsterdam, 1998, pp. 257–274. MR 1643051, DOI 10.1016/S0922-5382(98)80014-7