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We analyze constrained and unconstrained minimization problems on patches oftetrahedra sharing a common vertex with discontinuous piecewise polynomial dataof degree p. We show that the discrete minimizers in the spaces of piecewisepolynomials of degree p conforming in the H1, H(curl), or H(div) spaces are asgood as the minimizers in these entire (infinite-dimensional) Sobolev spaces,up to a constant that is independent of p. These results are useful in theanalysis and design of finite element methods, namely for devising stable localcommuting projectors and establishing local-best/global-best equivalences in apriori analysis and in the context of a posteriori error estimation.Unconstrained minimization in H1 and constrained minimization in H(div) havebeen previously treated in the literature. Along with improvement of theresults in the H1 and H(div) cases, our key contribution is the treatment ofthe H(curl) framework. This enables us to cover the whole De Rham diagram inthree space dimensions in a single setting.

Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the de Rham complex