Local- and global-best equivalence, simple projector, and optimal hp approximation in H(div)


For any H(div) function v with zero normal flux over part of the boundary and piecewise polynomial divergence

Main results:

  • global-best approximation (constraints on normal component continuity and divergence) is equivalent to the sum of independent local-best approximations (no constraints);
  • gives rise to a simple stable local commuting projector in H(div);
  • this projector delivers approximation error equivalent to the local-best approximation;
  • leads to optimal hp approximation estimates in H(div);
  • applies under the minimal necessary Sobolev regularity;
  • gives optimal a priori hp-error estimates for mixed and least-squares finite element methods.


  • paper (preprint) with Alexandre Ern, Thirupathi Gudi, and Iain Smears, see also presentation;
  • local- and global-best equivalence in H(curl): paper (preprint) with Théophile Chaumont-Frelet.

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