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S. Ferraz-Leite, C. Ortner, D. Praetorius:
"Convergence of simple adaptive Galerkin schemes based on h-h/2 error estimators";
Talk: RMMM 2009 - International Workshop on Reliable Methods of Mathematical Modeling, Berlin (invited); 06-24-2009 - 06-26-2009.

The (h-h/2)-strategy is one very basic and well-known technique for
the aposteriori error estimation. Let H be the energy space, which
is assumed to be a Hilbert space. Let u denote the exact solution.
One then considers

eta = || u_{h} - u_{h/2} ||

to estimate the error ||u-u_{h}||, where u_{h} is a Galerkin solution
with respect to a mesh T_{h} and u_{h/2} is a Galerkin solution for
a mesh T_{h/2} obtained from a uniform refinement of T_{h}. We stress
that eta is always efficient --- even with known efficiency constant
1. Reliability of eta is equivalent to the saturation assumption.

Based on the (h-h/2)-error estimation strategy, we introduce an
adaptive algorithm which is proven to be convergent under the
saturation assumption. The developed adaptive scheme is simple in
the sense that there is almost no implementational overhead for the
realization of the refinement indicators.

Besides finite element methods, our framework applies to boundary
integral formulations and yields a first convergence proof for
adaptive boundary element schemes. For a finite element model
problem, we extend the proposed adaptive scheme and prove convergence
even if the saturation assumption fails to hold in general.