Equivalence of local-best and global-best approximations in H(curl)

We derive results on equivalence of piecewise polynomial approximations of a given function in the Sobolev space \({\varvec{H}}{\mathrm{(curl)}}\). We namely show that the global-best approximation of a given \({\varvec{H}}{\mathrm{(curl)}}\) function in a \({\varvec{H}}{\mathrm{(curl)}}\)-conforming piecewise polynomial space imposing the continuity of the tangential trace can be bounded above and below by the Hilbertian sum of the respective local approximations from the elementwise spaces without any inter-element continuity requirement. In other words, the approximation of a \({\varvec{H}}{\mathrm{(curl)}}\) function by tangential-trace-continuous and discontinuous piecewise polynomials has comparable precision. We consider approximations of the curl of the target function in the \({\varvec{L}} ^2\)-norm, as well as approximations of the target function in the \({\varvec{L}} ^2\)-norm with a constraint on the curl; in the latter case, the constraint is removed in the local approximations. These best-approximation localizations hold under the minimal \({\varvec{H}}{\mathrm{(curl)}}\) regularity, on arbitrary shape-regular tetrahedral meshes, and include imposition of conditions on a part of the boundary. They extend to the \({\varvec{H}}{\mathrm{(curl)}}\) context some recent results from the \(H^1\) and \({\varvec{H}}{\mathrm{(div)}}\) spaces and have direct applications to a priori and a posteriori error analysis of numerical discretizations related to the \({\varvec{H}}{\mathrm{(curl)}}\) space, namely Maxwell’s equations.