Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations

Two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier-Stokes equations. Two discrete convective trilinear forms are proposed, a nonconservative one relying on Temam's device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure.