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

TL;DR

This paper investigates the stability of local minimization problems in finite element methods across H1, H(curl), and H(div) spaces, providing p-independent bounds that enhance the design of stable projectors and error estimators.

Contribution

It extends the analysis of local minimizations to the H(curl) space, completing the coverage of the De Rham complex in 3D with p-independent stability results.

Findings

Discrete minimizers are as good as Sobolev space minimizers, up to a p-independent constant.

Provides stable local projectors for finite element methods.

Enables local and global error analysis in finite element discretizations.

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 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.