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We prove global and local versions of the so-called div-curl-lemma, a crucialresult in the homogenization theory of partial differential equations, formixed boundary conditions on bounded weak Lipschitz domains in 3D with weakLipschitz interfaces. We will generalize our results using an abstract Hilbertspace setting, which shows corresponding results to hold in arbitrarydimensions as well as for various differential operators. The crucial tools andthe core of our arguments are Hilbert complexes and related compact embeddings.

A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz  Domains and a Corresponding Generalized  $\mathrm{A}_{0}^{*}$-$\mathrm{A}_{1}$-Lemma in Hilbert Spaces