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Abstract. We study the approximation properties of a harmonic function u ∈ H1−k(Ω), k> 0, on a relatively compact subset A of Ω, using the Gen-eralized Finite Element Method (GFEM). If Ω = O, for a smooth, bounded domain O, we obtain that the GFEM–approximation uS ∈ S of u satisfies ‖u−uS‖H1(A) ≤ Chγ‖u‖H1−k(O), where h is the typical size of the “elements” defining the GFEM–space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredi-ent is an extension of the classical super-approximation results of Nitsche and Schatz [20, 21]. In addition to the usual “energy ” Sobolev spaces H1(O), w