  We study the approximation properties of a harmonic function $u \in
H\sp{1-k}(\Omega)$, $k > 0$, on relatively compact sub-domain $A$ of $\Omega$,
using the Generalized Finite Element Method. For smooth, bounded domains
$\Omega$, we obtain that the GFEM--approximation $u_S$ satisfies $\|u -
u_S\|_{H\sp{1}(A)} \le C h^{\gamma}\|u\|_{H\sp{1-k}(\Omega)}$, where $h$ is the
typical size of the ``elements'' defining the GFEM--space $S$ and $\gamma \ge 0
$ is such that the local approximation spaces contain all polynomials of degree
$k + \gamma + 1$. The main technical result is an extension of the classical
super-approximation results of Nitsche and Schatz \cite{NitscheSchatz72} and,
especially, \cite{NitscheSchatz74}. It turns out that, in addition to the usual
``energy'' Sobolev spaces $H^1$, one must use also the negative order Sobolev
spaces $H\sp{-l}$, $l \ge 0$, which are defined by duality and contain the
distributional boundary data.
