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