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CARSTEN CARSTENSEN
Given a flux or stress approximation ph from a low-order finite element simulation of an elliptic PDE, for instance, averaging techniques aim to compute an improved approximation Aph by a (simple) post-processing of ph . One example, occasionally named after Zienkiewicz & Zhu, computes Aph (z) as the integral mean over the patch of the node z (this is the union of all finite elements which share the vertex z in a regular triangulation) and then linearly interpolates Aph on each element. Motivated by heuristic assumptions this estimator appeared to work very well in practice — engineers seemed to be extremely happy with this tool. The beginning of a mathematical justification of the error estimator ηA := kph − Aph k as a computable approximation of the (unknown) error kp − ph k involved the concept of super-convergence points. For highly structured meshes and a very smooth exact solution p, the error kp − Aph k of the post-processed approximation Aph may be (much) smaller than kp − ph k of the given ph . Under the assumption that kp − Aph k = h.o.t. is relatively sufficiently small, the triangle inequality immediately verifies reliability, i.e., kp − ph k ≤ Crel ηA + h.o.t., and efficiency, i.e., ηA ≤ Cef f kp − ph k + h.o.t., of the averaging error estimator ηA . However, the underlying assumptions essentially contradict the notion of adaptive grid refining for optimal experimental convergence rates when p is singular. Moreover, the proper treatment of boundary conditions lacks a serious inside. The presentation reports on old and new arguments for reliability and efficiency in the above sense with multiplicative constants Crel and Cef f and higher order terms h.o.t. Hi-lighted are the general class of meshes, averaging techniques, or finite element methods (conforming, nonconforming, and mixed elements) for elliptic PDEs. Numerical examples illustrate the amazing accuracy of ηA . The presentation closes with a discussion on current developments and the limitations as well as the perspectives of averaging techniques. Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Wiedner Hauptstraße 8-10/115, A-1040 Vienna, Austria E-mail address: carsten.carstensen@tuwien.ac.at
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