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