Quasi Norm a Posteriori Error Estimates Based on Gradient Recovery or Residual Estimation

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.