Interior numerical approximation of boundary value problems with a distributional data

We study the approximation properties of a harmonic function u is an element of H1-k(Omega), k > 0, on a relatively compact subset A of Omega, using the generalized finite element method (GFEM). If Omega = O, for a smooth, bounded domain O, we obtain that the GFEM-approximation u(s) is an element of = S of u satisfies parallel to u - u(S)parallel to(1)(H)((A)) <= Ch gamma parallel to u parallel to(1-k)(H)((O)), where h is the typical size of the "elements" defining the GFEM-space S and gamma >= 0 is such that the local approximation spaces contain all polynomials of degree k + y. 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 H-1(O), we need also the duals of the Sobolev spaces H-m(C), m is an element of Z(+). (c) 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 22: 79-113, 2006.