Asymptotically exact functional error estimators based on superconvergent gradient recovery

Numerische Mathematik

Volumn 102, Issue 3, 2006, Pages 543-558

  Ovall, Jeffrey S ^a  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29244458657     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-005-0655-9     Document Type: Article

References (28)

1. Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York, 2000
2. Babuška, I., Ihlenburg, F., Strouboulis, T., Gangaraj, S.K.: A posteriori error estimation for finite element solutions of Helmholtz' equation. II. Estimation of the pollution error. Internat. J. Numer. Methods Engrg. 40(21), 3883-3900 (1997)
3. Babuška, I., Miller, A.: The post-processing approach in the finite element method. Part 1. Calculation of displacements, stresses and other higher derivatives of the displacements. Internat. J. Numer. Methods Engrg. 20, 1085-1109 (1984)
4. Babuška, I., Miller, A.: The post-processing approach in the finite element method. Part 2. The calculation of stress intensity factors. Internat. J. Numer. Methods Engrg. 20, 1111-1129 (1984)
5. Babuška, I., Miller, A.: The post-processing approach in the finite element method. Part 3. A posteriori error estimates and adaptive mesh selection. Internat. J. Numer. Methods Engrg. 20, 2311-2324 (1984)
6. Bank, R.E. Pltmg: A software package for solving elliptic partial differential equations, users' guide 9.0. Technical report, University of California, San Diego, 2004
7. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators. I. Grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294-2312 (2003) (electronic)
8. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators. II. General unstructured grids. SIAM J. Numer. Anal. 41(6), 2313-2332 (2003) (electronic)
9. Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4(4), 237-264 (1996)
10. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1-102 (2001)
11. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, New York, 1994
12. Brezzi, F., Hughes, T.J.R., Marini, L.D., Russo, A., Süli, E.: A priori error analysis of residual-free bubbles for advection-diffusion problems. SIAM J. Numer. Anal. 36(6), 1933-1948 (1999) (electronic)
13. Du, L., Yan, N.: Gradient recovery type a posteriori error estimate for finite element approximation on non-uniform meshes. Adv. Comput. Math. 14(2), 175-193 (2001)
14. Durán, R., Muschietti, M.A., Rodríguez, R.: On the asymptotic exactness of error estimators for linear triangular finite elements. Numer. Math. 59, 107-127 (1991)
15. Estep, D.J., Holst, M.J., Larson, M.: Generalized green's functions and the effective domain of influence. SIAM J. Sci. Comput. 26(4), 1314-1339 (2005)
16. Giles, M., Süli, E.: Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality. In: Acta Numerica, vol. 11, Cambridge University Press, 2002, pp. 145-236
17. Heuveline, V., Rannacher, R.: Duality-based adaptivity in the hp-finite element method. J. Numer. Math. 11(2), 95-113 (2003)
18. Li, B., Zhang, Z.: Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements. Numer. Meth. Partial Differential Equations 15(2), 151-167 (1999)
19. Louis, A.: Acceleration of convergence for finite element solutions of the poisson equation. Numer. Math. 33, 43-53 (1979)
20. Ovall, J.S.: Duality-Based Adaptive Refinement for Elliptic PDEs. PhD Thesis, Department of Mathematics. University of California at San Diego, 2004
21. Pierce, N.A., Giles, M.B.: Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Rev. 42(2), 247-264 (2000) (electronic)
22. Prudhomme, S., Oden, J.T.: On goal-oriented error estimation for elliptic problems: application to control of pointwise errors. Comput. Methods Appl. Mech. Engrg. 1-4, 313-331 (1999)
23. Russo, A.: A posteriori error estimators via bubble functions. Math. Models Methods Appl. Sci. 6(1), 33-41 (1996)
24. Schatz, A.H., Wahlbin, L.B.: Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. II. The piecewise linear case. Math. Comp. 73(246), 517-523 (2004) (electronic)
25. Wahlbin, L.B.: Local behavior in finite element methods. In: Handbook of numerical analysis, Vol. II, Handb. Numer. Anal. II, North-Holland, Amsterdam, 1991, pp. 353-522
26. Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp. 73(247), 1139-1152 (2004) (electronic)
27. Yan, N., Zhou, A.: Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes. Comput. Methods Appl. Mech. Engrg. 190(32-33), 4289-4299 (2001)
28. Zhu, J.Z., Zienkiewicz, O.C.: Superconvergence recovery technique and a posteriori error estimators. Internat. J. Numer. Methods Engrg. 30(7), 1321-1339 (1990)