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

Abstract

Averaging techniques are popular toolsin adaptive finite element methods for the numerical treatment ofsecond-order partial differential equations since they provide efficient aposteriori error estimates by simple post-processing. In this paper, theirreliability is shown for conforming, nonconforming, and mixed low-orderfinite element methods in a model situation: the Laplace equationwith mixed boundary conditions. Emphasis is on possibly unstructuredgrids, nonsmoothness of exact solutions, and a wide class of averagingtechniques. Theoretical and numerical evidence supports that thereliability depends on the smoothness of given right-hand sides.