The striking simplicity of averaging techniques in a
posteriori error control of finite element methods as well as their
amazing accuracy in many numerical examples over the last decade have
made them an extremely popular tool in scientific computing. Given a
discrete stress or flux Ph and a post-processed approximation A(p(h)),
the a posteriori error estimator reads eta(A) := ||p(h) - A(p(h))||. There is not even a need
for an equation to compute the estimator eta(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 first mathematical justification of the error estimator eta(A) as a
computable approximation of the (unknown) error ||p - p(h)|| involved the concept of
superconvergence points. For highly structured meshes and a very smooth
exact solution p, the error ||p -
A(p(h))|| of the post-processed approximation
Ap(h) may be (much) smaller than ||p -
p(h)|| of the given p(h). Under the assumption
that ||p - A(p(h))||=
h.o.t. is in relative terms sufficiently small, the triangle inequality
immediately verifies reliability, i.e.,
|| p-p(h) || <= C-rel
eta(A) + h.o.t.,
and efficiency, i.e.,
eta(A) <= C-eff || p-p(h) || + h.o.t.,
of the averaging error estimator eta(A). However, the required
assumptions on the symmetry of the mesh and the smoothness of the
solution essentially contradict the use of adaptive grid refining when p
is singular and the proper treatment of boundary conditions remains unclear.
This paper aims at an actual overview on the reliability and efficiency
of averaging a posteriori error control for unstructured grids. New
aspects are new proofs of the efficiency of all averaging techniques and
for all problems.
