Interpolation of nullspaces for polynomial approximation of divergence-free functions in a cube

The aim of this paper is the interpolation between nullspaces of a fixed partial differential operator in different Sobolev spaces, for example the subspaces of divergence-free functions. We present several methods of proofs, which allow for handling various operators in different geometries, with or without boundary conditions. Our application is the optimal approximation of divergence-free functions in a cube by high degree polynomials, which is useful for the numerical analysis of the spectral discretization of the Stokes problem.