On the div-curl lemma in a Galerkin setting

Abstract

Given a sequence of Galerkin spaces X h of square-integrable vector fields, we state necessary and sufficient conditions on X h under which it is true that for any two sequences of vector fields u h ,u h ′∈X h converging weakly in L2 and such that u h is discrete divergence free and curl u h ′ is precompact in H−1, the scalar product u h ⋅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 Nedelec’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 H −1 .