  In this paper, we are concerned with the numerical treatment of boundary
integral equations by means of the adaptive wavelet boundary element method
(BEM). In particular, we consider the second kind Fredholm integral equation
for the double layer potential operator on patchwise smooth manifolds contained
in $\mathbb{R}^3$. The corresponding operator equations are treated by means of
adaptive implementations that are in complete accordance with the underlying
theory. The numerical experiments demonstrate that adaptive methods really pay
off in this setting. The observed convergence rates fit together very well with
the theoretical predictions that can be made on the basis of a systematic
investigation of the Besov regularity of the exact solution.
  Keywords: Besov spaces, weighted Sobolev spaces, adaptive wavelet BEM,
non-linear approximation, integral equations, double layer potential operator,
regularity, manifolds.
