Some new error estimates for Ritz–Galerkin methods with minimal regularity assumptions

Abstract:

New uniform error estimates are established for finite element approximations $u_h$ of solutions $u$ of second-order elliptic equations $\mathcal {L} u = f$ using only the regularity assumption $\|u\|_1 \leq c\|f\|_{-1}$. Using an Aubin–Nitsche type duality argument we show for example that, for arbitrary (fixed) $\varepsilon$ sufficiently small, there exists an $h_0$ such that for $0 < h < h_0$ \[ \|u-u_h\|_0 \leq \varepsilon \|u-u_h\|_1. \] Here, $\|\cdot \|_s$ denotes the norm on the Sobolev space $H^s$. Other related results are established.