Section: New Results
Local and globalbest equivalence, simple projector, and optimal $hp$ approximation in $\mathbf{H}\left(\mathrm{div}\right)$
Participants : Alexandre Ern, Thirupathi Gudi, Iain Smears, Martin Vohralík.

In [53], we prove that a globalbest approximation in $\mathbf{H}\left(\mathrm{div}\right)$, with constraints on normal component continuity and divergence, is equivalent to the sum of independent localbest 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 leastsquares 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 $\mathbf{H}\left(\mathrm{div}\right)$, which delivers approximation error equivalent to the localbest approximation and applies under the minimal necessary Sobolev $\mathbf{H}\left(\mathrm{div}\right)$ regularity, which is another result that has been sought for a very long time.