On the div-curl lemma in a Galerkin setting

Given a sequence of Galerkin spaces Xh of square-integrable vector fields, we state necessary and sufficient conditions on Xh under which it is true that for any two sequences of vector fields uh,uh′∈Xh converging weakly in L2 and such that uh is discrete divergence free and curl uh′ is precompact in H−1, the scalar product uh⋅u′h converges in the sense of distributions to the right limit. The conditions are related to super-approximation and discrete compactness results for mixed finite elements, and are satisfied for Nédélec’s edge elements. We also provide examples of sequences of discrete divergence free edge element vector fields converging weakly to 0 in L2 but whose divergence is not precompact in Hloc−1.