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## Section: New Results

### Local- and global-best equivalence, simple projector, and optimal $hp$ approximation in $𝐇\left(\mathrm{div}\right)$

Participants : Alexandre Ern, Thirupathi Gudi, Iain Smears, Martin Vohralík.

Figure 3. Equivalence between global-best and local-best approximation for any $𝐇\left(\mathrm{div}\right)$ function $𝐯$ with zero normal flux over part of the boundary and piecewise polynomial divergence

In [53], we prove that a global-best approximation in $𝐇\left(\mathrm{div}\right)$, with constraints on normal component continuity and divergence, is equivalent to the sum of independent local-best approximations, without any constraints, as illustrated in Figure 3. This may seem surprising on a first sight since the right term in Figure 3 is seemingly much smaller (since the minimization set is unconstrained and thus much bigger). This result leads to optimal a priori $hp$-error estimates for mixed and least-squares finite element methods, which were missing in the literature until 2019. Additionally, the construction we devise gives rise to a simple stable local commuting projector in $𝐇\left(\mathrm{div}\right)$, which delivers approximation error equivalent to the local-best approximation and applies under the minimal necessary Sobolev $𝐇\left(\mathrm{div}\right)$ regularity, which is another result that has been sought for a very long time.