𝑝-robust equivalence of global continuous constrained and local discontinuous unconstrained approximation, a 𝑝-stable local commuting projector, and optimal elementwise ℎ𝑝 approximation estimates in 𝐻(^cp)

Let an open bounded Lipschitz polygon or polyhedron Ω \Omega , a function v \boldsymbol {v} in the Sobolev space H ( d i v , Ω ) \boldsymbol {H}(\mathrm {div},\Omega ) , and a simplicial mesh of Ω \Omega be given. We prove the equivalence of two piecewise (Raviart–Thomas) polynomial best approximations of v \boldsymbol {v} in the L 2 \boldsymbol {L}^2 -norm: (1) globally on the whole computational domain Ω \Omega , with the normal trace continuity requirement and a divergence constraint; (2) locally on each mesh element, without any interelement continuity requirement and without any constraint on the divergence. The former (global-best continuous constrained piecewise polynomial approximation) arises in numerical methods for partial differential equations related to the H ( d i v , Ω ) \boldsymbol {H}(\mathrm {div},\Omega ) space, whereas the latter (local-best discontinuous unconstrained piecewise polynomial approximation) is a key quantity in approximation theory. Crucially, we establish p p -robustness in that the equivalence constant only depends on the mesh shape regularity and the spatial dimension. This improves the recent result of Ern, Gudi, Smears, and Vohralík [IMA J. Numer. Anal. 42 (2022), pp. 1023–1049], where the equivalence constant was possibly dependent on the underlying polynomial degree. Consequently, we obtain fully h h - and p p - (mesh-size- and polynomial-degree-) optimal approximation estimates under the minimal Sobolev regularity only requested separately on each mesh element. These two results immediately follow by our construction of an operator from the infinite-dimensional Sobolev space H ( d i v , Ω