A Div-Curl Lemma for Edge Elements

Abstract

A variant of Murat and Tartar's div-curl lemma is stated and proved for Nedelec's edge elements. Given two sequences of vector fields of this type, converging weakly in L2 as the mesh width tends to 0, we prove that their scalar product converges in the sense of distributions when one of the sequences consists of so-called discrete divergence-free fields whereas the other has relatively compact $\curl$ in H-1. The proof uses a uniform norm equivalence related to discrete compactness properties of vector finite element spaces and a super-approximation property of scalar finite element spaces.