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

Zhaonan Dong, Tanvi Wadhawan

TL;DR

The paper derives sharp, explicit inverse-trace constants for polynomials in $\mathbb{P}_p(T)$ that are orthogonal to lower-order subspaces $\mathbb{P}_n(T)$ on $d$-dimensional simplices. By combining orthogonal polynomial expansions on reference elements with eigenvalue analysis of face-mass matrices, it extends Warburton–Hesthaven results to the orthogonal-to-lower-order setting and yields a unified bound $||\theta||^2_{L^2(F)} \le \frac{(p-n)(p+n+1+d)}{d} \frac{|F|}{|T|} ||\theta||^2_{L^2(T)}$ for $\theta = \xi - \Pi_T^n\xi$. The main contributions provide explicit constants in 1D, 2D, 3D, and general $d$-dimensional cases, with the $n=-1$ scenario recovering classical results. These results have direct implications for the hp-analysis of hybrid finite element methods (e.g., HDG, HHO), improving stability estimates by leveraging orthogonality to lower-order spaces.

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.

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

TL;DR

The paper derives sharp, explicit inverse-trace constants for polynomials in that are orthogonal to lower-order subspaces on -dimensional simplices. By combining orthogonal polynomial expansions on reference elements with eigenvalue analysis of face-mass matrices, it extends Warburton–Hesthaven results to the orthogonal-to-lower-order setting and yields a unified bound for . The main contributions provide explicit constants in 1D, 2D, 3D, and general -dimensional cases, with the scenario recovering classical results. These results have direct implications for the hp-analysis of hybrid finite element methods (e.g., HDG, HHO), improving stability estimates by leveraging orthogonality to lower-order spaces.

Abstract

We derive sharp, explicit constants in inverse trace inequalities for polynomial functions belonging to (polynomial space with total degree ) that are orthogonal to the lower-order subspace , , where denotes a -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 -analysis of the hybrid Galerkin methods, e.g. hybridizable discontinuous Galerkin methods, hybrid high-order methods, etc.

Paper Structure

This paper contains 6 sections, 5 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Contribution of this work.
  3. Preliminaries and notation
  4. Inverse inequality on a 1D interval
  5. Inverse inequality on a 2D triangle
  6. Inverse inequality on a tetrahedron and $d$-simplex

Key Result

Theorem 1.1

For a planar $d$-dimensional simplical domain $T$ the following estimate holds for all $\xi \in \mathbb{P}_p({T})$ with $n \leq p$, such that: where ${F}$ denotes a $(d-1)$-dimensional face of $T$ and $\Pi^{n}_{T}$ denotes the $L^2$-orthogonal projection onto the $\mathbb{P}_n({T})$. For $n=-1$, we set $\Pi^{-1}_T \xi := 0$.

Theorems & Definitions (9)

  • Theorem 1.1: Inverse trace inequality for polynomials orthogonal to lower-order subspaces
  • Theorem 3.1: Inverse trace inequality on a finite interval
  • proof
  • Theorem 4.1: Inverse trace inequality for the planar triangle
  • proof
  • Theorem 5.1: Inverse trace inequality on a tetrahedron
  • proof
  • Corollary 5.2: Inverse trace inequality on a $d$-simplex
  • proof