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

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 L u = f using only the regularity assumption ‖ u ‖ 1 ≤ c ‖ f ‖ − 1 \|u\|_1 ≤ c\|f\|₋₁ . Using an Aubin–Nitsche type duality argument we show for example that, for arbitrary (fixed) ε ε sufficiently small, there exists an h 0 h_0 such that for 0 > h > h 0 0 > h > h_0 \[ ‖ u − u h ‖ 0 ≤ ε ‖ u − u h ‖ 1 . \|u-u_h\|_0 ≤ ε \|u-u_h\|_1. \] Here, ‖ ⋅