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Given an arbitrary function in H(div), we show that the error attained by theglobal-best approximation by H(div)-conforming piecewise polynomialRaviart-Thomas-N\'ed\'elec elements under additional constraints on thedivergence and normal flux on the boundary, is, up to a generic constant,equivalent to the sum of independent local-best approximation errors overindividual mesh elements, without constraints on the divergence or normalfluxes. The generic constant only depends on the shape-regularity of theunderlying simplicial mesh, the space dimension, and the polynomial degree ofthe approximations. The analysis also gives rise to a stable, local, commutingprojector in H(div), delivering an approximation error that is equivalent tothe local-best approximation. We next present a variant of the equivalenceresult, where robustness of the constant with respect to the polynomial degreeis attained for unbalanced approximations. These two results together furtherenable us to derive rates of convergence of global-best approximations that arefully optimal in both the mesh size h and the polynomial degree p, for vectorfields that only feature elementwise the minimal necessary Sobolev regularity.We finally show how to apply our findings to derive optimal a priori hp-errorestimates for mixed and least-squares finite element methods applied to a modeldiffusion problem.

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