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We design an operator from the infinite-dimensional Sobolev space${\boldsymbol H}(\mathrm{curl})$ to its finite-dimensional subspace formed bythe N\'ed\'elec piecewise polynomials on a tetrahedral mesh that has thefollowing properties: 1) it is defined over the entire ${\boldsymbolH}(\mathrm{curl})$, including boundary conditions imposed on a part of theboundary; 2) it is defined locally in a neighborhood of each mesh element; 3)it is based on simple piecewise polynomial projections; 4) it is stable in the${\boldsymbol L}^2$-norm, up to data oscillation; 5) it has optimal(local-best) approximation properties; 6) it satisfies the commuting propertywith its sibling operator on ${\boldsymbol H}(\mathrm{div})$; 7) it is aprojector, i.e., it leaves intact objects that are already in the N\'ed\'elecpiecewise polynomial space. This operator can be used in various parts ofnumerical analysis related to the ${\boldsymbol H}(\mathrm{curl})$ space. We inparticular employ it here to establish the two following results: i)equivalence of global-best, tangential-trace-and curl-constrained, andlocal-best, unconstrained approximations in ${\boldsymbol H}(\mathrm{curl})$including data oscillation terms; and ii) fully $h$- and $p$- (mesh-size- andpolynomial-degree-) optimal approximation bounds valid under the minimalSobolev regularity only requested elementwise. As a result of independentinterest, we also prove a $p$-robust equivalence of curl-constrained andunconstrained best-approximations on a single tetrahedron in the ${\boldsymbolH}(\mathrm{curl})$-setting, including $hp$ data oscillation terms.

A stable local commuting projector and optimal $hp$ approximation estimates in ${\boldsymbol H}(\mathrm{curl})$