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 relatively compact sub-domain A of Ω, using the Generalized Finite Element Method. For smooth, bounded domains Ω, we obtain that the GFEM–approximation uS satisfies ‖u−uS‖H1(A) ≤ Ch‖u‖H1−k(Ω), 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 + 1. The main technical result is an extension of the classical super-approximation results of Nitsche and Schatz [14] and, especially, [15]. It turns out that, in addition to the usual “energy” Sobolev spaces H1, one must use also the negative order Sobolev spaces H−l, l ≥ 0, which are defined by duality and contain the distributional boundary data.