On discrete Sobolev inequalities for nonconforming finite elements under a semi-regular mesh condition

We derive a discrete $ L^q-L^p$ Sobolev inequality tailored for the Crouzeix--Raviart and discontinuous Crouzeix--Raviart finite element spaces on anisotropic meshes in both two and three dimensions. Subject to a semi-regular mesh condition, this discrete Sobolev inequality is applicable to all pairs $(q,p)$ that align with the local Sobolev embedding, including scenarios where $q \leq p$. Importantly, the constant is influenced solely by the domain and the semi-regular parameter, ensuring robustness against variations in aspect ratios and interior angles of the mesh. The proof employs an anisotropy-sensitive trace inequality that leverages the element height, a two-step affine/Piola mapping approach, the stability of the Raviart--Thomas interpolation, and a discrete integration-by-parts identity augmented with weighted jump/trace terms on faces. This Sobolev inequality serves as a mesh-robust foundation for the stability and error analysis of nonconforming and discontinuous Galerkin methods on highly anisotropic meshes.