Interior numerical approximation of boundary value problems with a distributional data

Abstract

We study the approximation properties of a harmonic function $u \in H\sp{1-k}(Ω)$, $k > 0$, 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 $u_S$ satisfies $\|u - u_S\|_{H\sp{1}(A)} \le C h^γ\|u\|_{H\sp{1-k}(Ω)}$, where $h$ is the typical size of the ``elements'' defining the GFEM--space $S$ and $γ\ge 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 \cite{NitscheSchatz72} and, especially, \cite{NitscheSchatz74}. It turns out that, in addition to the usual ``energy'' Sobolev spaces $H^1$, one must use also the negative order Sobolev spaces $H\sp{-l}$, $l \ge 0$, which are defined by duality and contain the distributional boundary data.