Some a Posteriori Error Estimators for p-Laplacian Based on Residual Estimation or Gradient Recovery

Authors:
Wenbin Liu;Ningning Yan
Affiliations:
CBS & Institute of Mathematics and Statistics, University of Kent, Canterbury CT2 7NF, England. W.B.Liu@ukc.ac.uk;Institute of System Sciences, Chinese Academy of Sciences, Beijing, China and CBS & Institute of Mathematics and Statistics, University of Kent, Canterbury CT2 7NF, England
Venue:
Journal of Scientific Computing
Year:
2002

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Abstract

In this paper, we first derive a posteriori error estimators of residual type for the finite element approximation of the p-Laplacian, and show that they are reliable, and efficient up to higher order terms. We then construct some a posteriori error estimators based on gradient recovery. We further compare the two types of a posteriori error estimators. It is found that there exist some relationships between the two types of estimators, which are similar to those held in the case of the Laplacian. It is shown that the a posteriori error estimators based on gradient recovery are equivalent to the discretization error in a quasi-norm provided the solution is sufficiently smooth and mesh is uniform. Under stronger conditions, superconvergnece properties have been established for the used gradient recovery operator, and then some of the gradient recovery based estimates are further shown to be asymptotically exact to the discretization error in a quasi-norm. Numerical results demonstrating these a posteriori estimates are also presented.