Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 173

There are five popular approaches to building a posteriori estimators.
1. Residual estimators.
We bound the error on an element T in terms of the size of the residual R T
and the jumps R e on the edges e ⊂ ∂T . These estimators are due to Babuska and
Rheinboldt [1978a].
2. Estimators based on local Neumann problems
On every triangle T we solve a local variational problem which is a discrete
analog of

− z = R T in T,

( 8 . 5 )

∂z
∂n = R e

on e ⊂ ∂T.

We choose the approximating space to contain polynomials whose degrees are
higher than those in the underlying finite element space. These estimators are
obtained using the energy norm

z 1 ,T , and are due to Bank and Weiser [1985].
See also the comment before Theorem 9.5.
3. Estimators based on a local Dirichlet problem.
For every element T , we solve a variational problem on the set ω T :

− z = f in ω T ,

( 8 . 6 )

z = u h on ∂ω T .

Again, we expand the approximating space to include polynomials of higher degree
than in the actual finite element space. Following Babuska and Rheinboldt [1978b],
the norm of the difference

z − u h 1 ,ω T

provides an estimator.

4. Estimators based on averaging.
We construct a continuous approximation σ h of ∇ u h by a two-step process.

At every node of the triangulation, let σ h be a weighted average of the gradients
∇ u h on the neighboring triangles, where the weight is proportional to the areas
of the triangles. We then extend σ h to the whole element by linear interpolation.
Then following Zienkiewicz and Zhu [1987], we use the difference between

∇ u h
and σ h as an estimator. An analysis without restrictive assumptions was done by
Rodriguez [1994] and by Carstensen and Bartels [2002].
5. Hierarchical estimators.
In principle the difference from a finite element approximation on an ex-
panded space is estimated. The difference can be estimated by using a strengthened
Cauchy inequality; see Deuflhard, Leinen, and Yserentant [1989]. The procedure
fits into Carl Runge's old and general concept (Runge's rule) . The error of a nu-
merical result is estimated by comparing it with the result of a more accurate
formula.