arXiv:2611.13164v1 Announce Type: new Abstract: We prove Sobolev inequalities for the discrete de Rham (DDR) method, an arbitrary-order discretisation of the de Rham complex on polyhedral meshes. We show that, for any polyhedral domain $\Omega$, the DDR counterparts of the spaces $H({\bf grad},\Omega)$, $\boldsymbol{H}({\bf curl},\Omega)$ and $\boldsymbol{H}(\mathrm{div},\Omega)$ satisfy the same Sobolev embeddings as their continuous counterparts. For the curl and divergence spaces, this establishes on quasi-uniform meshes an $L^r$-bound (with $r>2$ in general, and $r=6$ if $\Omega$ is convex) on discrete vector fields that are orthogonal to discrete gradients (resp.~curls), in terms of the $L^2$-norms of the vector fields themselves and their discrete curl (resp.~divergence). These Sobolev embeddings are the first such results for a polytopal complex. Their proofs hinge on recently introduced quasi-interpolators and conforming liftings. As an application, we use these estimates to prove the convergence of a DDR scheme for the Navier--Stokes equations in the curl--curl formulation. The convergence proof uses a Maxwell compactness theorem for DDR and requires no smallness assumptions on the data.