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Averaging techniques are popular tools in adaptive nite element methods for the numerical treatment of second order partial dierential equa- tions since they provide ecient a posteriori error estimates by a simple post- processing. In this paper, their reliablility is shown for conforming, noncon- forming, and mixed low order nite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possi- bly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

mathematics of computation

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM