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

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.

💡 Research Summary

The paper addresses a gap in the theory of hp‑finite element methods concerning inverse trace inequalities for polynomial functions that are orthogonal to lower‑order subspaces. While the classical result of Warburton and Hesthaven (2003) provides sharp, h‑ and p‑explicit inverse trace bounds for all polynomials in the space (P_p(T)) on a d‑dimensional simplex (T), those bounds become overly pessimistic when the polynomial is known to be orthogonal to the subspace (P_n(T)) with (n\le p). The orthogonality eliminates the contribution of the low‑order modes, and consequently a tighter dependence on the polynomial degree should be possible.

The authors extend the methodology of