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C. Carstensen, D. Praetorius:
"Convergence of adaptive boundary element methods";
Journal of Integral Equations and Applications, 24 (2012), 1; 1 - 23.

In many applications, adaptive mesh-refinement is observed to be an
efficient tool for the numerical solution of partial differential equations
and integral equations. Convergence of adaptive schemes to the correct
solution, however, is so far only understood for certain kind of differential
equations. In general, it cannot be excluded that the adaptive algorithm
computes a convergent sequence of discrete approximations with a limit
which is not the correct solution. This work proposes a feedback loop
which guarantees the convergence of the computed discrete approximations
to the correct solution. Although stated for Symm's integral equation of
the first kind, the main part of this work is written for a general audience
in the context of weak forms as Riesz representations in Hilbert spaces.
Numerical examples illustrate the adaptive strategies.

http://dx.doi.org/10.1216/JIE-2012-24-1-1