Discrete Compactness for p-Version of Tetrahedral Edge Elements

We consider the first family of $\text{Hcurl}$-conforming Ned'el'ec finite elements on tetrahedral meshes. Spectral approximation ($p$-version) is achieved by keeping the mesh fixed and raising the polynomial degree $p$ uniformly in all mesh cells. We prove that the associated subspaces of discretely weakly divergence free piecewise polynomial vector fields enjoy a long conjectured discrete compactness property as $p\to \mathrm{\infty }$. This permits us to conclude asymptotic spectral correctness of spectral Galerkin finite element approximations of Maxwell eigenvalue problems.