Interior Numerical Approximation of Boundary Value Problems with a Distributional Data

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 Generalized 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) ≤ Ch‖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 ingredient 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), we need also the duals of the Sobolev spaces Hm(O), m ∈ Z+.