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

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