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

New uniform error estimates are established for finite element approximations uh 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) ε sufficiently small, there exists an h0 such that for 0 < h < h0 ‖u− uh‖0 ≤ ε‖u− uh‖1. Here, ‖ · ‖s denotes the norm on the Sobolev space Hs. Other related results are established.