Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex

Abstract

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 \(H^1\), \({\varvec{H}}(\textbf{curl})\), or \({\varvec{H}}({\text {div}})\) 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 \(H^1\) and constrained minimization in \({\varvec{H}}({\text {div}})\) have been previously treated in the literature. Along with improvement of the results in the \(H^1\) and \({\varvec{H}}({\text {div}})\) cases, our key contribution is the treatment of the \({\varvec{H}}(\textbf{curl})\) framework. This enables us to cover the whole de Rham diagram in three space dimensions in a single setting.