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 = ??, for a smooth, bounded domain ??, we obtain that the GFEM‐approximation u_S ∈ S of u satisfies ‖u ? u_S‖ ≤ Ch^γ‖u‖, 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(??), we need also the duals of the Sobolev spaces H^m(??), m ∈ ?₊. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006