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

New uniform error estimates are established for finite element approximations u h of solutions u of second-order elliptic equations Lu = f using only the regularity assumption ∥u∥ 1 ≤ c∥f∥ −1 . Using an Aubin-Nitsche type duality argument we show for example that, for arbitrary (fixed) e sufficiently small, there exists an h 0 such that for 0<h<h 0 ∥u − u h ∥0 ≤ e∥u − u h ∥ 1 . Here, ∥.∥ s denotes the norm on the Sobolev space H s . Other related results are established.