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New uniform error estimates are established for nite element ap- proximations uh of solutions u of second-order elliptic equationsLu =f using only the regularity assumptionkuk1 ckfk 1. Using an Aubin{Nitsche type duality argument we show for example that, for arbitrary (xed) " suciently small, there exists an h0 such that for 0 <h<h 0 k u uhk 0 "k u uhk 1 : Here,kks denotes the norm on the Sobolev space Hs. Other related results are established.

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