RELIABLE AND EFFICIENT AVERAGING TECHNIQUES AS UNIVERSAL TOOL FOR A POSTERIORI FINITE ELEMENT ERROR CONTROL ON UNSTRUCTURED GRIDS

The striking simplicity of averaging techniques in a posteriori er- ror control of flnite element methods as well as their amazing accuracy in many numerical examples over the last decade have made them an extremely popular tool in scientiflc computing. Given a discrete stress or ∞ux ph and a post-processed approximation A(ph), the a posteriori error estimator reads ·A := kph i A(ph)k. There is not even a need for an equation to compute the estimator ·A and hence averaging techniques are employed everywhere. The most prominent example is occasionally named after Zienkiewicz and Zhu, and also called gradient recovery but preferably called averaging technique in the literature. The flrst mathematical justiflcation of the error estimator ·A as a computable approximation of the (unknown) error kp i phk involved the concept of super- convergence points. For highly structured meshes and a very smooth exact solution p, the error kpiA(ph)k of the post-processed approximation Aph may be (much) smaller than kp i phk of the given ph. Under the assumption that kpiA(ph)k = h.o.t. is in relative terms su-ciently small, the triangle inequal- ity immediately verifles reliability, i.e., kp i phkCrel ·A + h.o.t., and e-ciency, i.e., ·ACefi kp i phk + h.o.t., of the averaging error estimator ·A. However, the required assumptions on the symmetry of the mesh and the smoothness of the solution essentially contradict the use of adaptive grid reflning when p is singular and the proper treatment of boundary conditions remains unclear. This paper aims at an actual overview on the reliability and e-ciency of averaging a posteriori error control for unstructured grids. New aspects are new proofs of the e-ciency of all averaging techniques and for all problems.