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

Zhaonan Dong, Tanvi Wadhawan

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Zhaonan Dong, Tanvi Wadhawan. On the constants in inverse trace inequalities for polynomials orthogonal to lower-order subspaces. 2025. hal-05419621

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Distributed under a Creative Commons Attribution 4.0 International License
On the constants in inverse trace inequalities for
polynomials orthogonal to lower-order subspaces

Zhaonan Dong* Tanvi Wadhawan*

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 $\mathbb{P}_n(T)$, $n \le p$, 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.

**Keywords.** *hp*-analysis, inverse estimate, discrete trace inequality

MSC. 65N30

# 1 Introduction

Inverse inequalities are a cornerstone of finite element theory and play an essential role in the analysis of modern numerical schemes. Classical norm-equivalence arguments, dating back to the early work of Ciarlet [2], provide general inverse estimates for simplicial finite element spaces. Over the past several decades, there has been substantial progress in deriving sharp and explicit constants in inverse inequalities. We refer, in particular, to the monograph by Schwab [8, Section 4.6], where several inverse inequalities are established with *hp*-explicit bounds up to a generic constant that is independent of *h* and *p*.

For inverse trace inequalities, a significant step forward was taken by Warburton and Hesthaven [9], who established constant-free, *h*- and *p*-explicit bounds for inverse inequalities for polynomials in the space $\mathbb{P}_p(T)$, consisting of polynomials of total degree most *p* on simplices *T*. In particular, the authors also identified extremal polynomials for which the inverse inequality becomes an identity. These inequalities play a central role in the stability analysis of discontinuous Galerkin methods and other nonconforming methods; see, e.g., [3, 1].

On the other hand, the results of Warburton and Hesthaven are sharp with respect to the mesh size *h*, the polynomial degree *p*. However, their results concern the trace of a polynomial in the space $\mathbb{P}_p(T)$. If one applies their results to the polynomial defined in $\mathbb{P}_p(T)$ but orthogonal to lower-order subspaces $\mathbb{P}_n(T)$, the bound will be over pessimistic in terms of *p*. This is due to the orthogonality with respect to the lower-order subspaces removes the contribution of the lower polynomial basis, which results in a sharp bound in *p*. Such estimates typically arise in hybrid finite element formulations such that the difference between a cell unknown and its local $L^2$-projection onto a lower-order space due to the hybrid variables defined on faces is defined in a lower-order polynomial spaces, such as the HDG methods, HHO schemes. As an application, we refer to [4].

<footnote n="1">¹Inria, 48 rue Barrault, 75647 Paris, France and CERMICS, ENPC, Institut Polytechnique de Paris, 6 & 8 avenue B. Pascal, 77455 Marne-la-Vallée, France</footnote>
**Contribution of this work.** In this work, we extend the results from [9] to the setting of inverse trace inequalities for the polynomials orthogonal to the lower-order subspace. The main result is the following lemma:

**Theorem 1.1** (Inverse trace inequality for polynomials orthogonal to lower-order subspaces). For a planar d-dimensional simplicial domain T the following estimate holds for all $\xi \in \mathbb{P}_p(T)$ with $n \le p$, such that:

$$ \|\xi - \Pi_T^n \xi\|_{L^2(F)}^2 \le \frac{(p-n)(p+n+1+d)}{d} \frac{|F|}{|T|} \|\xi - \Pi_T^n \xi\|_{L^2(T)}^2, \quad (1.1) $$

where F denotes a $(d-1)$-dimensional face of T and $\Pi_T^n$ denotes the $L^2$-orthogonal projection onto the $\mathbb{P}_n(T)$. For $n = -1$, we set $\Pi_T^{-1}\xi := 0$.

On the other hand, if we applying classical inverse trace inequality from [9] directly to $\theta := \xi - \Pi_T^n$, with $\theta = \xi - \Pi_T^n \xi \in \mathbb{P}_p(T)$, we infer

$$ \|\theta\|_{L^2(F)}^2 \le \frac{(p+1)(p+d)}{d} \frac{|F|}{|T|} \|\theta\|_{L^2(T)}^2. $$

However, recognizing that $\theta$ is orthogonal to $\mathbb{P}_n(T)$ which removes the contribution of the low-order polynomial modes. Consequently, the analysis provides the sharper bound

$$ \|\theta\|_{L^2(F)}^2 \le \frac{(p-n)(p+n+1+d)}{d} \frac{|F|}{|T|} \|\theta\|_{L^2(T)}^2. $$

It is easy to see that the new inverse trace inequality bound is always strictly than the bound using [9] for $0 \le n \le p$. For the case $n = -1$, the new bound will be equal to the other bound.

The remainder of the paper is organized as follows. Section 2 introduces notation and preliminary concepts. Section 3 presents one-dimensional proof, while Sections 4 extend the analysis to triangles. In Section 5, we generalize the result to tetrahedron and also d-dimensional simplex.

## 2 Preliminaries and notation

Let us start by introducing the following notations used throughout the paper:

* Let $T$ denote a $d$-dimensional simplicial domain with boundary $\partial T$, and let $F$ to be the $(d-1)$-dimensional simplicial face such that $F \subset \partial T$.

* For $p \ge 0$, $\mathbb{P}_p(T)$ is the space of polynomials of total degree at most $p$ on $T$.

* Any $u \in \mathbb{P}_p(T)$ can be expanded in an $L^2(T)$-orthonormal basis $\{\psi_n\}_{n=1}^{\dim \mathbb{P}_p(T)}$:

$$ u(x) = \sum_{n=1}^{\dim \mathbb{P}_p(T)} \hat{u}_n \psi_n(x), \quad \hat{u}_n = \int_T u(x) \psi_n(x) dx, $$

where $x := (x_1, x_2, \dots, x_d)^\top$.

* The $L^2$ norms on elements and faces are denoted by $\| \cdot \|_{L^2(T)}$ and $\| \cdot \|_{L^2(F)}$, respectively.

* For an integer $0 \le n \le p$, let $\Pi_T^n$ denote the $L^2$-projection onto $\mathbb{P}_n(T)$. For $n = -1$, we set $\Pi_T^{-1}\xi := 0$.
### 3 Inverse inequality on a 1D interval

In this section we prove the inverse trace inequality in one dimensional, which forms the basis for the higher-dimensional estimates.

**Theorem 3.1 (Inverse trace inequality on a finite interval).** Let $T = [a, b]$ be an interval, and denote by $\Pi_T^n$ the $L^2$-projection onto the space of polynomials of degree at most $n$ on $T$. The following bound holds for any $\xi \in \mathbb{P}_p(T)$, with $\theta := \xi - \Pi_T^n \xi$, such that

$$|\theta(a)|^2 \le \frac{(p-n)(p+n+2)}{|b-a|} \|\theta\|_{L^2(T)}^2. \quad (3.1)$$

*Proof.* Let us consider the reference interval $\hat{T} = [-1, 1]$ and $\{\psi_k\}_{k=0}^p$ denote the $L^2$-orthonormal Legendre polynomials. Then

$$\xi(\hat{x}) = \sum_{k=0}^{p} \xi_k \psi_k(\hat{x}), \quad \xi_k = \int_{\hat{T}} \xi(\hat{x}) \psi_k(\hat{x}) d\hat{x},$$

and the polynomial $\theta := \xi - \Pi_T^n \xi$ satisfies

$$\theta(\hat{x}) = \sum_{k=n+1}^{p} \xi_k \psi_k(\hat{x}), \quad |\theta(-1)|^2 = \Theta^\top L \Theta,$$

where $\Theta = (\xi_{n+1}, \dots, \xi_p)^\top$ and $L_{ij} = \psi_i(-1)\psi_j(-1) = (-1)^{i+j} \sqrt{\frac{(2i+1)(2j+1)}{4}}$, $i, j = n+1, \dots, p$. Since $L$ is rank-one matrix, its spectral radius is

$$\rho(L) = \sum_{k=n+1}^{p} \frac{2k+1}{2} = \frac{(p-n)(p+n+2)}{2}.$$

Hence, using Parseval identity,

$$|\theta(-1)|^2 \le \rho(L) \|\theta\|_{L^2(\hat{T})}^2 = \frac{(p-n)(p+n+2)}{2} \|\theta\|_{L^2(\hat{T})}^2.$$

Finally using a scaling argument yields the desired estimate (3.1). □

### 4 Inverse inequality on a 2D triangle

In the following we shall utilize this same approach to deduce the explicit form of the constant in Theorem 1.1.

**Theorem 4.1 (Inverse trace inequality for the planar triangle).** Let $T$ be a triangle with a face $F \subset \partial T$, and let $\Pi_T^n$ denote the $L^2$-projection onto $\mathbb{P}_n(T)$. The following bound holds for any $\xi \in \mathbb{P}_p(T)$, with $\theta := \xi - \Pi_T^n \xi$, such that

$$\|\theta\|_{L^2(F)}^2 \le \frac{(p-n)(p+n+3)}{2} \frac{|F|}{|T|} \|\theta\|_{L^2(T)}^2. \quad (4.1)$$

*Proof.* Consider the reference triangle $\hat{T} := \{(r, s) | -1 \le r, s \le 1, r + s \le 0\}$. To parametrize $\hat{T}$, we introduce the Duffy mapping with coordinates $a, b \in [-1, 1]$ via

$$r = \frac{(1+a)}{2}(1-b) - 1, \quad s = b, \quad -1 \le a, b \le 1,$$
which maps the unit square [-1, 1]² onto $\hat{T}$. We employ an orthonormal polynomial basis on $\hat{T}$, which traces its origins to Proriol [7] and was later refined and extended by Koornwinder [6], and Dubiner [5], indexed by integer pairs $(i, j)$ with $i + j \le p$:

$$ \psi_{ij}(r, s) = \left( \frac{P_i^{(0,0)}(a)}{\sqrt{\frac{2}{2i+1}}} \right) \left( \frac{\left(\frac{1-b}{2}\right)^i P_j^{(2i+1,0)}(b)}{\sqrt{\frac{1}{i+j+1}}} \right), $$

where $P_n^{(\alpha, \beta)}(x)$ denotes the n-th order Jacobi polynomial on [-1, 1]. Any polynomial $\xi \in \mathbb{P}_p(\hat{T})$ expands as

$$ \xi(r, s) = \sum_{i+j \le p} \xi_{(i,j)} \psi_{ij}(r, s). $$

The polynomial $\theta = \xi - \Pi_T^n \xi$ satisfies

$$ \theta(r, s) = \sum_{i+j \ge n+1}^{p} \xi_{(i,j)} \psi_{ij}(r, s). $$

Focusing on the face $\hat{F}$ where $s = -1$ (equivalently $b = -1$), we compute

$$ \int_{\hat{F}} \theta^2(r, -1) dr = \int_{-1}^{1} \theta^2(a, -1) da = \Theta^\top L \Theta, \qquad (4.2) $$

where $\Theta$ is the vector of coefficients $\xi_{(i,j)}$ with $i + j \in [n+1, p]$, and $L$ is the associated face mass matrix with entries

$$ \begin{aligned} L_{(ij)(kl)} &= \int_{-1}^{1} \psi_{ij}(r, -1) \psi_{kl}(r, -1) dr \\ &= \delta_{ik} (-1)^{j+l} \sqrt{i+j+1} \sqrt{k+l+1}, \end{aligned} $$

where $\delta_{ik}$ follows from the $L^2$-orthogonality of the Legendre polynomials $P_i^{(0,0)}(\hat{x})$.

The matrix $L$ is block-diagonal with blocks indexed by $i = 0, \dots, p$. To compute the spectral radius of $L$, we can consider each block separately. For the $i$-th block, $j$ ranges from $n+1-i$ to $p-i$, and each block is a rank-one matrix $Z^{(i)}$:

$$ Z^{(i)} = \mathbf{v}^{(i)}(\mathbf{v}^{(i)})^{\top}, \quad \mathbf{v}_{j}^{(i)} = (-1)^{j} \sqrt{i+j+1}. $$

The spectral radius $\rho(L)$ of $L$ equals to the maximum of the eigenvalues across all blocks. For any fixed $i$, we have $n+1-i \le j \le p-i$, so the eigenvalue of the $i$-th block $Z^{(i)}$ is

$$ \lambda^{(i)} = \sum_{j=n+1-i}^{p-i} (i + j + 1) = \frac{(p-n)(p+n+3)}{2}, $$

which is independent of $i$. Consequently, the spectral radius $\rho(L)$ equals to $\lambda^{(i)}$, since every block has the same eigenvalue. Thus,

$$ \rho(L) = \frac{(p-n)(p+n+3)}{2}. $$

Using (4.2), we obtain

$$ \|\theta\|_{L^2(\hat{F})}^2 \le \rho(L) \|\theta\|_{L^2(\hat{T})}^2 = \frac{(p-n)(p+n+3)}{2} \|\theta\|_{L^2(\hat{T})}^2. $$

Finally, a standard scaling argument yields the desired estimate (4.1). □
# 5 Inverse inequality on a tetrahedron and d-simplex

In this section we establish a sharp inverse trace inequality on tetrahedral domain T.

**Theorem 5.1 (Inverse trace inequality on a tetrahedron).** Let T be a tetrahedron with a face F ⊂ ∂T, and let Π_T^n denote the L^2-projection onto P_n(T). The following bound holds for any ξ ∈ P_p(T), with θ := ξ − Π_T^n ξ, such that

$$ \|\theta\|_{L^2(F)}^2 \le \frac{(p-n)(p+n+4)}{3} \frac{|F|}{|T|} \|\theta\|_{L^2(T)}^2, \quad (5.1) $$

*Proof.* Consider a reference tetrahedron $\hat{T} := \{(r, s, t) | -1 \le r, s, t \le 1; r + s + t \le -1\}$. Next, to parametrize $\hat{T}$, we introduce the Duffy mapping with coordinates $a, b, c \in [-1, 1]$ such that

$$ r = \frac{(1+a)}{2} (1-b) (1-c) - 1, \quad s = \frac{(1+b)}{2} (1-c) - 1, \quad t = c - 1 \le a, b, c \le 1 $$

and the mapping maps a unit cube into $\hat{T}$. Next, we utilize the orthonormal polynomial basis for $\hat{T}$ indexed by integer triplets $(i, j, k)$ introduced in [5] as

$$ \psi_{ijk}(r, s, t) = \left( \frac{P_i^{(0,0)}(a)}{\sqrt{\frac{2}{2i+1}}} \right) \left( \frac{\left(\frac{1-b}{2}\right)^i P_j^{(2i+1,0)}(b)}{\sqrt{\frac{2}{2(i+j)+2}}} \right) \left( \frac{\left(\frac{1-c}{2}\right)^{i+j} P_k^{(2(i+j)+2,0)}(c)}{\sqrt{\frac{2}{2(i+j+k)+3}}} \right), $$

where, $P_n^{(\alpha, \beta)}(x)$ denotes the n-th order Jacobi polynomial defined on $[-1, 1]$. Using these basis, any polynomial $\xi \in P_p(\hat{T})$ can be expanded as $\xi(r, s, t) = \sum_{ijk} \xi_{(i,j,k)} \psi_{ijk}(r, s, t)$. Thus the polynomial $\theta = \xi - \Pi_T^n \xi$ satisfies

$$ \theta(r, s, t) = \sum_{i+j+k \ge n+1}^{p} \xi_{(i,j,k)} \psi_{ijk}(r, s, t). $$

Next, we focus on the face $\hat{F}$ where $c = -1$, and compute the face integral as

$$ \int_{\hat{F}} \theta^2(r, s, -1) dr ds = \int_{-1}^{1} \int_{-1}^{1} \theta^2(a, b, -1) da db = \Theta^{\top} L \Theta, \quad (5.2) $$

where $\Theta$ is the vector of coefficients $\hat{\xi}_{ijk}$ corresponding to $i + j + k \in [n + 1, p]$, and L is the associated face matrix. The entries of face matrix L are given by

$$ \begin{aligned} L_{(ijk)(lmn)} &= \int_{-1}^{1} \int_{-1}^{1} \psi_{ijk}(a, b, -1) \psi_{lmn}(a, b, -1) da db \\ &= \delta_{il} \delta_{jm} (-1)^{(k+n)} \sqrt{\frac{2(i+j+k)+3}{2}} \sqrt{\frac{2(\ell+m+n)+3}{2}}. \end{aligned} $$

Thus, L is block-diagonal with blocks indexed by $\alpha = i + j$ ranging from 0 to $p$. Within each block, $k$ ranges from $n + 1 - \alpha \le k \le p - \alpha$. Thus each block is rank one matrix and can be written as

$$ Z_{(\alpha)} = \nu^{(\alpha)}(\nu^{(\alpha)})^{\top}, \text{ with } \nu_k^{(\alpha)} = (-1)^k \sqrt{\frac{2(\alpha + k) + 3}{2}}. $$

The eigenvalue of each rank-one block equals the squared norm of $\nu^{(\alpha)}$:

$$ \lambda^{(\alpha)} = \sum_{k=n+1-|\alpha|}^{p-|\alpha|} \frac{2(\alpha + k) + 3}{2} = \frac{(p-n)(p+n+4)}{2}, $$
which is independent of $\alpha$. Therefore, the spectral radius $\rho(L)$ of the full face matrix $L$ is the maximum of these eigenvalues which implies

$$ \rho(L) = \frac{(p-n)(p+n+4)}{2}. $$

Using the relation (5.2), we obtain

$$ \|\theta\|_{L^2(\bar{F})}^2 \le \rho(L)\|\theta\|_{L^2(\hat{T})}^2 = \frac{(p-n)(p+n+4)}{2} \|\theta\|_{L^2(\hat{T})}^2. $$

Finally, a standard scaling argument yields the desired estimate (5.1). □

Next, we will discuss about the extension to $d$-dimensional simplex.

**Corollary 5.2** (Inverse trace inequality on a $d$-simplex). Let $T$ be a $d$-dimensional simplex with a face $F \subset \partial T$, and let $\Pi_T^n$ denote the $L^2$-projection onto $\mathbb{P}_n(T)$. The following bound holds for any $\xi \in \mathbb{P}_p(T)$, with $\theta := \xi - \Pi_T^n\xi$, such that

$$ \|\theta\|_{L^2(F)}^2 \le \frac{(p-n)(p+n+1+d) |F|}{d |T|} \|\theta\|_{L^2(T)}^2, \quad (5.3) $$

*Proof.* The result follows by generalizing the proof of Lemma 5.1 (3D case) to a $d$-dimensional simplex. Detailed proof is therefore omitted for brevity. □

## References

[1] A. CANGIANI, Z. DONG, E. H. GEORGOULIS, AND P. HOUSTON, hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes, SpringerBriefs in Mathematics, Springer, Cham, 2017.

[2] P. G. CIARLET, The finite element method for elliptic problems, Society for Industrial and Applied Mathematics, 2002.

[3] D. DI PIETRO AND A. ERN, Mathematical aspects of discontinuous Galerkin methods, vol. 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer, Heidelberg, 2012.

[4] Z. DONG, A. ERN, AND T. WADHAWAN, hp-a posteriori error estimates of hybrid high-order methods for biharmonic problems. In preparation, 2025.

[5] M. DUBINER, Spectral methods on triangles and other domains, J. Sci. Comput., 6 (1991), pp. 345–390.

[6] T. KOORNWINDER, Two-variable analogues of the classical orthogonal polynomials, in Theory and Application of Special Functions, R. Askey, ed., Academic Press, 1975, pp. 435–495.

[7] J. PRORIOL, Sur une famille de polynomes à deux variables orthogonaux dans un triangle, C. R. Acad. Sci. Paris, 257 (1957), pp. 2459–2461.

[8] C. SCHWAB, p- and hp-Finite element methods: Theory and applications in solid and fluid mechanics, Oxford University Press: Numerical mathematics and scientific computation, 1998.

[9] T. WARBURTON AND J. S. HESTHAVEN, On the constants in hp-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 2765–2773.