Interior numerical approximation of boundary value problems with a distributional data

We study the approximation properties of a harmonic function u ∈H ^1-k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = script O sign, for a smooth, bounded domain script O sign, we obtain that the GFEM-approximation u_s ∈ S of u satisfies ∥u - u _s∥H¹(A) ≤ Ch^γ∥u∥ _H1-k,(script O sign), 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 ingredient is an extension of the classical super-approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161-168; Math Comput 28 (1974), 937-958). In addition to the usual "energy" Sobolev spaces H¹(script O sign), we need also the duals of the Sobolev spaces H^m(script O sign), m ∈ ℤ₊.