Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div)

Given an arbitrary function in $\boldsymbol{H}({\operatorname{div}})$, we show that the error attained by the global-best approximation by $\boldsymbol{H}({\operatorname{div}})$-conforming piecewise polynomial Raviart-Thomas-Nédélec elements under additional constraints on the divergence and normal flux on the boundary is, up to a generic constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The generic constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in $\boldsymbol{H}({\operatorname{div}})$, delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size $h$ and the polynomial degree $p$, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori$hp$-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.