On the constants in inverse trace inequalities for polynomials orthogonal to lower-order subspaces

TL;DR

This paper establishes precise constants in inverse trace inequalities for orthogonal polynomials on simplices, aiding the analysis of advanced hybrid Galerkin numerical methods.

Contribution

It provides explicit, sharp constants in inverse trace inequalities for polynomial spaces orthogonal to lower degrees, enhancing $hp$-analysis of hybrid Galerkin methods.

Findings

Derived explicit constants for inverse trace inequalities.

Analyzed eigenvalues of face mass matrices for sharp bounds.

Applicable to $hp$-analysis of hybrid numerical methods.

Abstract

We derive sharp, explicit constants in inverse trace inequalities for polynomial functions belonging to $\mathbb{P}_p(T)$ (polynomial space with total degree $p$) that are orthogonal to the lower-order subspace $\mathbb{P}_n(T)$, $n\leq p$, where $T$ denotes a $d$-dimensional simplex. The proofs rely on orthogonal polynomial expansions on reference simplices and on a careful analysis of the eigenvalues of the relevant blocks of the face mass matrices, following the arguments developed in [9]. These results are very useful in the $hp$-analysis of the hybrid Galerkin methods, e.g. hybridizable discontinuous Galerkin methods, hybrid high-order methods, etc.