On discrete Sobolev inequalities for nonconforming finite elements under a semiregular mesh condition

Abstract We derive a discrete $L^{q}-L^{p}$ Sobolev inequality tailored for the Crouzeix–Raviart (CR) and discontinuous CR finite-element spaces on anisotropic meshes in both two and three dimensions. Subject to a semiregular 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 semiregular 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.