#  $hp$-optimal convergence of the original DG method for linear hyperbolic
problems on special simplicial meshes

Z. Dong, L. Mascotto Inria, 2 rue Simone Iff, 75589 Paris, France and CERMICS,
Ecole des Ponts, 77455 Marne-la-Vallée 2, France,
<EMAIL_ADDRESS>di Matematica e Applicazioni, Università di
Milano Bicocca, 20125 Milan, Italy<EMAIL_ADDRESS>IMATI-CNR,
27100, Pavia, Italy; Fakultät für Mathematik, Universität Wien, 1090 Vienna,
Austria<EMAIL_ADDRESS>
###### Abstract

We prove $hp$-optimal error estimates for the original DG method when
approximating solutions to first-order hyperbolic problems with constant
convection fields in the $L^{2}$ and DG norms. The main theoretical tools used
in the analysis are novel $hp$-optimal approximation properties of the special
projector introduced in [Cockburn, Dong, Guzman, SINUM, 2008]. We assess the
theoretical findings on some test cases.

AMS subject classification: 65N12; 65N30; 65N50.

Keywords: discontinuous Galerkin; optimal convergence; linear hyperbolic
problems; a priori error estimation; $p$-version.

##  1 Introduction

The discontinuous Galerkin method was introduced in [19] for the approximation
of solutions to the neutron transport equation, i.e., a first-order linear
hyperbolic problem. This method was first analysed in [16]; advances on the
convergence of the scheme were given in [13]. The convergence rates in the
$L^{2}$ norm of the $h$-version of the method proven therein are suboptimal by
half an order; this was also apparent from the numerical experiments in [18].
On special classes of meshes, optimal convergence for the $h$-version was
established in [7].

Fewer results are available for the $p$\- and $hp$-versions of the method. In
[11], $hp$ error estimates were derived for a streamline diffusion version of
the discontinuous Galerkin method on fairly general quadrilateral meshes.
Shortly after [12], for the original method and linear polynomial convection
fields, $p$ convergence was shown suboptimal by half a order for the
convection-diffusion case, but optimal for the linear hyperbolic case; for
more general vector fields, suboptimality by one order and a half was
discussed as well. The main theoretical tools in this reference are $hp$
optimal approximation properties of the $L^{2}$ projector on the boundary of
tensor product elements. More recently [9] and again for tensor product
elements, such a suboptimality was reduced to half an order only for a special
class of convective fields.

This paper aims at extending even more the knowledge on the convergence
analysis of the $hp$-DG for first order hyperbolic problems in the following
aspects: for constant convection fields,

  * •

we prove $hp$-optimal convergence of the method in the $L^{2}$ norm, thus
generalizing the results in [7] to the $p$-version of the method;

  * •

we prove $hp$-optimal convergence of the method in the DG norm also on special
simplicial meshes as in [7], thus generalizing the results in [11] to the case
of simplicial meshes.

To the aim, we analyse the $hp$ approximation properties of the CDG projector
introduced in [8]. In particular, we generalise the one dimensional results in
[20, Lemmas $3.5$ and $3.6$] to simplices in two and three dimensions.

A possible reason why the simplicial mesh case was not contemplated in [11,
12] is that $hp$-optimal convergence estimates for the trace of the polynomial
$L^{2}$-projection operator on simplices crucial in the analysis of the CDG
projector properties were derived later [6, 17].

#### Notation.

Let $D$ be a Lipschitz domain in $\mathbb{R}^{d}$, $d=1$, $2$, and $3$, with
boundary $\partial D$. The space of Lebesgue measurable and square integrable
functions over $D$ is $L^{2}(D)$. The Sobolev space of positive integer order
$s$ is $H^{s}(D)$. We also write $L^{2}(D)=H^{0}(D)$. We endow $H^{s}(D)$ with
the inner product, seminorm, and norm

$(\cdot,\cdot)_{s,D},\qquad\qquad\qquad{\left|{\cdot}\right|}_{s,D},\qquad\qquad\qquad{\left\|{\cdot}\right\|}_{s,D}.$

Interpolation theory is used to construct Sobolev spaces of positive
noninteger order; duality is used to define negative order Sobolev spaces.

The trace theorem is valid for $H^{s}(D)$, $1/2<s<3/2$. In particular, given
$s$ in the above range and $g$ in $H^{s-\frac{1}{2}}(\Gamma_{g})$, being
$\Gamma_{g}$ any subset of $\partial D$ with nonzero measure in $\partial D$,
we are allowed to define the space

$H^{s}_{g}(D,\Gamma_{g}):=\\{v\in H^{s}(D,\Gamma_{D})\mid
v_{|\Gamma_{D}}=g\\}.$

The space of polynomials of nonnegative degree $p$ over $D$ is
$\mathbb{P}_{p}(D)$.

#### The continuous problem.

Given $\Omega$ a Lipschitz domain in $\mathbb{R}^{d}$, $d=1,2,3$, with
boundary $\Gamma$, consider $\boldsymbol{\beta}$ in $\mathbb{R}^{d}$ and $c$
in $L^{\infty}(\Omega)$. We introduce the inflow part $\Gamma_{-}$ of the
boundary of $\Omega$ as follows: given $\mathbf{n}_{\Gamma}(\mathbf{x})$ the
outward normal to $\Gamma$ at $\mathbf{x}$,

$\Gamma_{-}:=\\{\mathbf{x}\in\partial\Omega\mid\boldsymbol{\beta}\cdot\mathbf{n}_{\Gamma}(\mathbf{x})<0\\}.$

The outflow part $\Gamma_{+}$ of $\Gamma$ is given by
$\Gamma\setminus\Gamma_{-}$. We are interested in the approximation of
solutions to convection-reaction problems: given $g$ in $L^{2}(\Gamma_{-})$,

$\begin{cases}\text{find }u\text{ such that }\\\ \boldsymbol{\beta}\cdot\nabla
u+c\ u=f&\text{in }\Omega\\\ u=g&\text{on }\Gamma_{-}.\end{cases}$ (1)

Introduce the graph space

$V:=\\{v\in L^{2}(\Omega)\mid\boldsymbol{\beta}\cdot\nabla v\in
L^{2}(\Omega)\\},$

which we endow with the graph norm

${\left\|{v}\right\|}_{V}^{2}:={\left\|{v}\right\|}_{0,\Omega}^{2}+{\left\|{\boldsymbol{\beta}\cdot\nabla
v}\right\|}_{0,\Omega}^{2}.$ (2)

We can define a trace operator from the graph space onto the space

$L^{2}(|\boldsymbol{\beta}\cdot\mathbf{n}_{\Gamma}|,\Gamma):=\\{v\text{
measurable on
}\Gamma\mid(|\boldsymbol{\beta}\cdot\mathbf{n}_{\Gamma}|v,v)_{0,\Gamma}<\infty\\}.$

We denote the space of functions in the graph space with trace $g$ in
$L^{2}(|\boldsymbol{\beta}\cdot\mathbf{n}_{\Gamma}|,\Gamma)$ by $V_{g}$. We
further introduce the bilinear form on $V_{g}\times L^{2}(\Omega)$ as

$b(u,v)=(\boldsymbol{\beta}\cdot\nabla u,v)_{0,\Omega}+(c\ u,v)_{0,\Omega}.$

The weak formulation of problem (1) reads

$\begin{cases}\text{find }u\in V_{g}\text{ such that}\\\
b(u,v)=(f,v)_{0,\Omega}\qquad\forall v\in L^{2}(\Omega).\end{cases}$ (3)

Henceforth, we assume that there exists a positive constant $\overline{c}_{0}$
such that

$\overline{c}:=c-\frac{1}{2}\nabla\cdot\boldsymbol{\beta}\geq\overline{c}_{0}.$
(4)

Problem (3) is well posed with respect to the graph norm in (2); see, e.g.,
[3].

#### Structure of the paper.

In Section 2, we introduce admissible simplicial meshes in the sense of [7]
and recall the original DG method from [19]. The main technical result of the
paper, i.e., $hp$ approximation properties of the CDG projector are derived in
Section 3 in one, two, and three dimensions. Such estimates are used to derive
$hp$-optimal convergence of the method in the $L^{2}$ norm in Section 4;
there, we also show $hp$-optimal convergence of the method in a DG norm. We
assess the theoretical results with several numerical experiments in Section 5
and draw some conclusions in Section 6.

##  2 Admissible meshes and the method

We introduce admissible simplicial meshes for the forthcoming analysis, and
define associated Sobolev and polynomial broken spaces in Section 2.1, and
recall the original DG method for problem (3) in Section 2.2.

###  2.1 Admissible simplicial meshes

We follow [7, Section 1] and introduce admissible simplicial meshes that are
instrumental in deriving the main result of the paper; see Section 3 below.

Let $\mathcal{T}_{h}$ be a shape-regular (with parameter $\sigma$) simplicial
mesh of the domain $\Omega$ and $\mathcal{F}_{h}$ be its set of
($d-1$)-dimensional facets. We distinguish the facets in $\mathcal{F}_{h}$
into internal and boundary facets; the former are those $F$ not contained in
$\partial\Omega$. The union of the interior facets of $\mathcal{T}_{h}$ is
$\Gamma_{h}^{\operatorname{in}}$.

With each element $T$ of $\mathcal{T}_{h}$ we associate its diameter $h_{T}$
and its outward unit vector $\mathbf{n}_{T}$, which is defined almost
everywhere on $\partial T$. The set of ($d-1$)-dimensional facets of $T$ is
$\mathcal{F}^{T}$. With each facet $F$ in $\mathcal{F}_{h}$ we associate its
diameter $h_{F}$ and a unit normal vector $\mathbf{n}_{F}$, which is pointing
outward $\Omega$ if $F$ is a boundary facet.

We distinguish the facets in $\mathcal{F}^{T}$ into outflow, inflow, and
characteristics (with respect to $\boldsymbol{\beta}$) facets, depending on
whether they satisfy either of the two following properties:

$\mathbf{n}_{T}{}_{|F}\cdot\boldsymbol{\beta}>0,\qquad\qquad\qquad\mathbf{n}_{T}{}_{|F}\cdot\boldsymbol{\beta}<0,\qquad\qquad\qquad\mathbf{n}_{T}{}_{|F}\cdot\boldsymbol{\beta}=0.$

The union of the outflow and inflow facets is
$\Gamma^{T}_{\operatorname{out}}$ and $\Gamma^{T}_{\operatorname{in}}$,
respectively.

We require that each element $T$ of $\mathcal{T}_{h}$ satisfy the two
following properties:

  * (A1)

$T$ has only one outflow facet $F_{T}^{+}$;

  * (A2)

each interior facet $F$ that is an inflow facet for $T$ is included in the
outflow facet for another simplex in the mesh.

The second assumption above implies that $\mathcal{T}_{h}$ can be
nonconforming, i.e., hanging facets are allowed. Meshes satisfying the above
properties can be always constructed; see, e.g., [7, Appendix].

The focus of the paper is on $p$-optimal estimates; therefore, we pick quasi-
uniform meshes and denote the mesh size of $\mathcal{T}_{h}$ by $h$.

Given $p$ in $\mathbb{N}$ and $s$ in $\mathbb{R}^{+}$, we associate with a
mesh $\mathcal{T}_{h}$ as in Section 2.1 the spaces

$\mathbb{P}_{p}(\mathcal{T}_{h}):=\\{q_{p}\in L^{2}(\Omega)\mid
q_{p}{}_{|T}\in\mathbb{P}_{p}(T)\quad\forall T\in\mathcal{T}_{h}\\}$

and

$H^{s}(\mathcal{T}_{h}):=\\{v\in L^{2}(\Omega)\mid v_{|T}\in
H^{s}(T)\quad\forall T\in\mathcal{T}_{h}\\}.$

Differential operators defined piecewise over $\mathcal{T}_{h}$ are denoted
with the same symbol of the original operator with an extra subcript $h$; for
instance, the broken gradient is $\nabla_{h}$.

Given an interior facet $F$, we denote by $T^{+}$ and $T^{-}$ the two elements
of $\mathcal{T}_{h}$ such that $F$ is an outflow and inflow facet for $T^{+}$
and $T^{-}$, respectively. Given $v_{h}$ in $\mathbb{P}_{p}(\mathcal{T}_{h})$
and $\mathbf{x}$ in the face $F$, we write

$v_{h}^{\pm}(\mathbf{x})=\lim_{\delta\downarrow
0}v_{h}(\mathbf{x}\pm\delta\boldsymbol{\beta}).$

We introduce the jump operator
$\left[\\!\left[\cdot\right]\\!\right]:H^{1}(\mathcal{T}_{h})\to
L^{2}(\Gamma_{h}^{\operatorname{in}})$ given by
$\left[\\!\left[v_{h}\right]\\!\right]:=v_{h}^{+}-v_{h}^{-}$.

Henceforth, given two positive quantities $a$ and $b$, we write $a\lesssim b$
is there exists a positive constant $c$ possibly depending on the shape-
regularity parameter $\sigma$ only, such that $a\leq c\ b$. If $a\lesssim b$
and $b\lesssim a$ at once, we write $a\approx b$.

###  2.2 The original DG method

Consider the space $V_{h}$ given by $\mathbb{P}_{p}(\mathcal{T}_{h})$. The
original [19] DG method for (3) reads

$\begin{cases}\text{find }u_{h}\in V_{h}\text{ such that}\\\
\mathcal{B}(u_{h},v_{h})=(f,v_{h})_{0,\Omega}-(g,\mathbf{n}_{\Gamma}\cdot\boldsymbol{\beta}\
v_{h})_{0,\Gamma_{-}}\qquad\forall v_{h}\in V_{h}.\end{cases}$ (5)

where

$\mathcal{B}(u_{h},v_{h}):=(\boldsymbol{\beta}\cdot\nabla_{h}u_{h}+c\
u_{h},v_{h})_{0,\Omega}-\sum_{T\in\mathcal{T}_{h}}(\left[\\!\left[u_{h}\right]\\!\right],\mathbf{n}_{T}\cdot\boldsymbol{\beta}\
v_{h}^{+})_{0,\Gamma^{T}_{\operatorname{in}}}-(u_{h},\mathbf{n}_{\Gamma}\cdot\boldsymbol{\beta}\
v_{h})_{0,\Gamma_{-}}.$

The solution $u_{h}$ can be computed starting at the inflow boundary
$\Gamma_{-}$ as a time-marching scheme. In fact, the communication between
elements takes place as an upwind scheme.

An equivalent alternative formulation used, e.g., in [7] is derived using an
integration by parts and the fact that $\boldsymbol{\beta}$ is a constant
field. Notably, an integration by parts implies that the bilinear form
$\mathcal{B}(\cdot,\cdot)$ can be rewritten as

$\begin{split}\mathcal{B}(u_{h},v_{h})&=-(u_{h},\boldsymbol{\beta}\cdot\nabla_{h}v_{h})_{0,\Omega}\\!+\\!\sum_{T\in\mathcal{T}_{h}}(u_{h}^{-},\mathbf{n}_{T}\cdot\boldsymbol{\beta}\
\left[\\!\left[v_{h}\right]\\!\right])_{0,\Gamma^{T}_{\operatorname{in}}\setminus\Gamma_{-}}\\!+\\!(c\
u_{h},v_{h})_{0,\Omega}+(u_{h},\mathbf{n}_{\Gamma}\cdot\boldsymbol{\beta}\
v_{h})_{0,\Gamma_{+}}.\end{split}$ (6)

Method (5) is well posed. In fact, a priori estimates with respect to the data
are derived in [11, Lemma 2.4 and Section 5].

######  Proposition 2.1.

Let $\overline{c}$ be given in (4). The following bound holds true:

$\overline{c}{\left\|{u_{h}}\right\|}_{0,\Omega}^{2}+\sum_{T\in\mathcal{T}_{h}}{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}\left[\\!\left[u_{h}\right]\\!\right]}\right\|}_{0,\Gamma^{T}_{\operatorname{in}}}^{2}+{\left\|{|\mathbf{n}_{\Gamma}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}u_{h}}\right\|}_{0,\Gamma}^{2}\leq\overline{c}^{-1}{\left\|{f}\right\|}_{0,\Omega}+2{\left\|{g}\right\|}_{0,\Gamma_{-}}^{2}.$

All constants appearing in the stability estimates of Proposition 2.1 are
explicit, and independent of $h$ and $p$. This is essential to derive the
optimal error estimates in Section 4 below. Former stability results display
stability estimates that are suboptimal in the polynomial degree; see, e.g.,
[13, Theorem 2.1].

Method (5) is consistent. The following Galerkin orthogonality property
follows: given $u$ and $u_{h}$ the solutions to (3) and (5),

$\mathcal{B}(u-u_{h},v_{h})=0\qquad\qquad\qquad\forall v_{h}\in V_{h}.$ (7)

In particular, the bilinear form $\mathcal{B}(\cdot,\cdot)$ is also well
defined on $V_{g}\times V_{h}$.

##  3 The CDG projector and its approximation properties

We recall some notation and technical results from the theory of orthogonal
polynomials in Section 3.1; we define of the CDG projector and state the main
result in Section 3.2; we prove $p$-optimal approximation properties of the
CDG operator in one, two, and three dimensions in Sections 3.3, 3.4, and 3.5,
respectively; in Section 3.6 we prove trace-type estimates for the CDG
projectors.

###  3.1 Preliminary results

We recall $hp$ approximation properties of the $L^{2}$ projector on a simplex
$T$ in the $L^{2}$ norm on the boundary of $T$. This result traces back to [6,
Theorem 2.1] and [17, Theorem 1.1]. Let
$\Pi^{0}_{p}:L^{2}(T)\to\mathbb{P}_{p}(T)$ denote the $L^{2}$ projector
defined as

$(v-\Pi^{0}_{p}v,q_{p})_{0,T}=0\qquad\qquad\qquad\forall
q_{p}\in\mathbb{P}_{p}(T).$ (8)

######  Lemma 3.1.

Under the notation of Section 2.1, there exists a positive $C$ independent of
$\sigma$ and $d$ such that

${\left\|{u-\Pi^{0}_{p}u}\right\|}_{0,F_{T}^{+}}\leq
C\left(\frac{h_{T}}{p}\right)^{\frac{1}{2}}{\left|{u}\right|}_{1,T}\qquad\qquad\qquad\forall
u\in H^{1}(T).$

As a consequence, for a possibly larger constant $C$, we also have

${\left\|{u-\Pi^{0}_{p}u}\right\|}_{0,F_{T}^{+}}\leq
C\left(\frac{h_{T}}{p}\right)^{k+\frac{1}{2}}{\left|{u}\right|}_{k+1,T}\qquad\qquad\qquad\forall
u\in H^{k+1}(T).$

As we are interested in deriving estimates explicit in the polynomial degree
and the estimates explicit in the element size are already known from [8], we
shall prove approximation properties on reference elements.

Set $\widehat{I}:=[-1,1]$ the reference interval. We recall some properties of
orthogonal polynomials; see, e.g., [21]. Let $\\{L_{j}\\}_{j=0}^{+\infty}$ be
the $L^{2}$ orthogonal set of the Legendre polynomials over $\widehat{I}$
satisfying [21, Corollary 3.6, $\alpha=0$, $\beta=0$]

$(L_{i},L_{j})_{0,\widehat{I}}=\frac{2}{2j+1}\delta_{i,j}\qquad\qquad\forall
i,j\in\mathbb{N}_{0}.$ (9)

Given $\Gamma(\cdot)$ the Gamma function, further let
$\\{J_{j}^{\ell}\\}_{j=0}^{+\infty}$, $\ell>-1$, be the weighted-$L^{2}$
orthogonal set of Jacobi polynomials over $\widehat{I}$ satisfying [21,
Corollary 3.6, $\alpha=\ell$, $\beta=0$]

$((1-x)^{\ell}J_{i}^{\ell},J_{j}^{\ell})_{0,\widehat{I}}=\frac{2^{\ell+1}}{2j+\ell+1}\frac{\Gamma(j+\ell+1)\Gamma(\ell+1)}{\Gamma(\ell+1)\Gamma(j+\ell+1)}\delta_{j,\ell}=\frac{2^{\ell+1}}{2j+\ell+1}\delta_{j,\ell}\qquad\qquad\forall
i,j\in\mathbb{N}_{0}.$ (10)

For future convenience, we introduce the $H^{1}$-projector
$\Pi^{1}_{p}:H^{1}(\widehat{I})\to\mathbb{P}_{p}(\widehat{I})$ defined as

$\begin{cases}(\nabla(v-\Pi^{0}_{p}v),\nabla q_{p})_{0,T}=0\qquad\qquad\forall
q_{p}\in\mathbb{P}_{p}(T)\\\ (v-\Pi^{0}_{p}v,1)_{0,T}=0.\end{cases}$ (11)

###  3.2 The CDG projector and the main result

The CDG operator was introduced for the analysis of superconvergent DG methods
for second-order elliptic problems in [8]. It can be defined for simplices in
any dimensions satisfying assumptions (A1)–(A2). In particular, recall that
$F_{T}^{+}$ is the only outflow facet of $T$.

The CDG operator
$\mathcal{P}_{p}:H^{\frac{1}{2}+\varepsilon}(T)\to\mathbb{P}_{p}(T)$ is
defined as

$\displaystyle(v-\mathcal{P}_{p}v,q_{p-1}^{T})_{0,T}=0\qquad\qquad\qquad\forall
q_{p-1}^{T}\in\mathbb{P}_{p-1}(T),$ (12a)
$\displaystyle(v-\mathcal{P}_{p}v,q_{p}^{F_{T}^{+}})_{0,F_{T}^{+}}=0\qquad\qquad\qquad\forall
q_{p}^{F_{T}^{+}}\in\mathbb{P}_{p}(F_{T}^{+}).$ (12b)

In [8], the following error estimate was proven for sufficiently smooth
functions:

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,T}\leq C\
h^{p+1}{\left|{v}\right|}_{p+1,\Omega}.$

The constant $C$ above depends on $\sigma$ and the polynomial degree $p$. The
remainder of the section is devoted to carefully detail the dependence on $p$.
Namely, we shall prove the following result.

######  Theorem 3.2.

Let the assumptions (A1) and (A2) be valid, and $T$ be in $\mathcal{T}_{h}$.
Then, there exist positive constants $C_{1}$ and $C_{2}$ independent of $h$
and $p$, such that, for sufficiently smooth functions $v$, we have

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,T}\leq
C_{1}\Big{(}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,T}+\frac{h_{T}}{p}{\left|{v-\Pi^{1}_{p}v}\right|}_{1,T}\Big{)}\leq
C_{2}\frac{h^{\min(p,k)+1}}{p^{k+1}}{\left\|{v}\right\|}_{k+1,T}.$

Above, $\Pi^{0}_{p}$ and $\Pi^{1}_{p}$ are the operators in (8) and (11).

###  3.3 The 1D case

We prove Theorem 3.2 in the case of one dimensional simplices, i.e., on
intervals. This was already done in [20, Lemmas 3.5 and 3.6]. However, we deem
that reviewing the main steps from [20] is beneficial for the understanding of
the extension to the two and three dimensional cases in Sections 3.4 and 3.5
below, respectively.

Without loss of generality, we pick $T=\widehat{I}$ and assume $-1$ to be the
outflow facet. In 1D, the operator $\mathcal{P}_{p}$ is defined imposing
orthogonality up to order $p-1$ over $\widehat{I}$ and imposing that
$\mathcal{P}_{p}v(-1)=v(-1)$; this last condition replaces the orthogonality
on the outflow facet.

The proof of optimal $p$-convergence can be split in the following steps:

  1. 1.

we write $\mathcal{P}_{p}v$ as a combination of Legendre polynomials and
compute the corresponding coefficients; up to order $p-1$, the coefficients
are the same as those of the expansion of $v$; the coefficient of order $p$ is
given by a tail of the other Legendre coefficients;

  2. 2.

we relate such a tail to the coefficients of $v$ minus its $L^{2}$ projection
on the triangle restricted to the outflow facet;

  3. 3.

we deduce the assertion using trace error estimates as in Lemma 3.1.

#### Step 1: writing $\mathcal{P}_{p}v$ with respect to the Legendre basis and
with explicit coefficients.

Given $v$ in $L^{2}(\widehat{I})$, we consider its expansion with respect to
Legendre polynomials

$v(x)=\sum_{j=0}^{+\infty}\mathbf{v}_{j}\ L_{j}(x).$

Consider also the truncated expansion up to order $p$ given by

$\mathcal{P}_{p}v(x)=\sum_{j=0}^{p}\widehat{\mathbf{v}}_{j}\ L_{j}(x).$

The orthogonality condition (12a) with respect to polynomials of degree $p-1$
gives

$\mathcal{P}_{p}v(x)=\sum_{j=0}^{p-1}\mathbf{v}_{j}\
L_{j}(x)+\widehat{\mathbf{v}}_{p}\ L_{p}(x).$

The last coefficient is given imposing the condition
$v(-1)=\mathcal{P}_{p}v(-1)$. More precisely, since $L_{j}(-1)=(-1)^{j}$ for
all $j$ in $\mathbb{N}$, we write

$v(-1)=\sum_{j=0}^{+\infty}(-1)^{j}\mathbf{v}_{j},\qquad\qquad\mathcal{P}_{p}v(-1)=\sum_{j=0}^{p-1}(-1)^{j}\mathbf{v}_{j}+(-1)^{p}\widehat{\mathbf{v}}_{p}.$

We deduce

$(-1)^{p}\widehat{\mathbf{v}}_{p}=\sum_{j=p}^{+\infty}\mathbf{v}_{j}(-1)^{j},$

whence we get

$\mathcal{P}_{p}v(x)=\sum_{j=0}^{p-1}\mathbf{v}_{j}\
L_{j}(x)-\big{(}\sum_{j=p}^{+\infty}(-1)^{j-p}\mathbf{v}_{j}\big{)}\
L_{p}(x).$ (13)

Given $\Pi^{0}_{p}$ the $L^{2}$ projection as in (8), we arrive at

$\begin{split}(v-\mathcal{P}_{p}v)(x)&=\sum_{j=p}^{+\infty}\mathbf{v}_{j}\
L_{j}(x)-\big{(}\sum_{j=p}^{+\infty}(-1)^{j-p}\mathbf{v}_{j}\big{)}\
L_{p}(x)\\\ &=\sum_{j=p+1}^{+\infty}\mathbf{v}_{j}\
L_{j}(x)+\mathbf{v}_{p}L_{p}(x)-\mathbf{v}_{p}L_{p}(x)-\big{(}\sum_{j=p+1}^{+\infty}(-1)^{j-p}\mathbf{v}_{j}\big{)}\
L_{p}(x)\\\ &=\sum_{j=p+1}^{+\infty}\mathbf{v}_{j}\
L_{j}(x)-\big{(}\sum_{j=p+1}^{+\infty}(-1)^{j-p}\mathbf{v}_{j}\big{)}\
L_{p}(x)\\\
&=(v-\Pi^{0}_{p}v)(x)-(-1)^{-p}\big{(}\sum_{j=p+1}^{+\infty}(-1)^{j}\mathbf{v}_{j}\big{)}\
L_{p}(x).\end{split}$ (14)

#### Step 2: relate the tail of the Legendre coefficients to the trace of
$v-\Pi^{0}_{p}v$ on the outflow facet.

Observe that

$(v-\Pi^{0}_{p}v)(x)=\sum_{j=p+1}^{+\infty}\mathbf{v}_{j}L_{j}(x),\qquad(v-\Pi^{0}_{p}v)(-1)=\sum_{j=p+1}^{+\infty}\mathbf{v}_{j}L_{j}(-1)=\sum_{j=p+1}^{+\infty}(-1)^{j}\mathbf{v}_{j}.$
(15)

Combining (14) and (15) gives

$(v-\mathcal{P}_{p}v)(x)=(v-\Pi^{0}_{p}v)(x)-(-1)^{p}[(v-\Pi^{0}_{p}v)(-1)]L_{p}(x)$

Recall from (9) that

${\left\|{L_{p}}\right\|}^{2}_{0,\widehat{I}}=\frac{2}{2p+1}.$ (16)

We deduce

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,\widehat{I}}^{2}\leq{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,\widehat{I}}^{2}+|(v-\Pi^{0}_{p}v)(-1)|^{2}\frac{2}{2p+1}.$

#### Step 3: deduce the desired bound using Lemma 3.1.

The 1D version of Lemma 3.1 reads

$|(v-\Pi^{0}_{p}v)(-1)|^{2}\lesssim(p+1)^{-1}{\left|{v}\right|}_{1,\widehat{I}}^{2}.$

Combining the two equations above yields

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,\widehat{I}}\lesssim{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,\widehat{I}}+(p+1)^{-1}{\left|{v}\right|}_{1,\widehat{I}}.$

Given $\Pi^{1}_{p}:H^{1}(\widehat{I})\to\mathbb{P}_{p}(\widehat{I})$ the
$H^{1}$ projection as in (11), noting that $\mathcal{P}_{p}$ and $\Pi^{0}_{p}$
preserve polynomials of maximum degree $p$, we have

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,\widehat{I}}\lesssim{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,\widehat{I}}+(p+1)^{-1}{\left|{v-\Pi^{1}_{p}v}\right|}_{1,\widehat{I}}.$

The above bound and standard polynomial approximation results entail Theorem
3.2 in 1D.

###  3.4 The 2D case

We prove Theorem 3.2 in two dimensions. Compared to the 1D case in Section
3.3, the role of orthogonal basis functions is now played by tensor product
Jacobi polynomials (with different weights) collapsed on the triangle via the
Duffy transformation; in other words, we use Koornwinder polynomials following
the construction in [4, 6]; see also [10, 15].

First, we introduce a reference 2D simplex $T$:

$\widehat{T}:=\left\\{\left(z\frac{1-y}{2},y\right)\in\mathbb{R}^{2}\
\middle|\ (z,y)\in[-1,1]^{2}\right\\}.$ (17)

The reference triangle $\widehat{T}$ corresponds to the reference square
$\widehat{Q}:=[-1,1]^{2}$ collapsed under the Duffy transformation

$(z,y)\in\widehat{Q}\qquad\Longrightarrow\qquad\left(z\frac{1-y}{2},y\right).$

This particular choice of $\widehat{T}$ is convenient for our purposes, as we
shall analyse the approximation properties of $\mathcal{P}_{p}$ assuming that
$F^{+}_{\widehat{T}}$ is the segment $[-1,1]\times\\{-1\\}$.

We introduce the orthogonal polynomial basis (known as Koornwinder polynomial
basis [15]) over $\widehat{T}$ given by

$\Phi_{j,\ell}(x,y):=L_{j}\left(\frac{2x}{1-y}\right)J_{\ell}^{2j+1}(y)\left(\frac{1-y}{2}\right)^{j}\qquad\qquad\forall
j,\ \ell\in\mathbb{N}.$ (18)

This is an orthogonal basis over $\widehat{T}$. In fact, using the Duffy
transformation, we have

$\int_{\widehat{T}}\Phi_{i,j}(x,y)\Phi_{k,\ell}(x,y)dx\
dy=\left(\int_{-1}^{1}L_{i}(z)L_{k}(z)\
dz\right)\left(\int_{-1}^{1}J_{j}^{2i+1}(y)J_{\ell}^{2k+1}(y)\left(\frac{1-y}{2}\right)^{i+k+1}dy\right).$

If $i\neq k$, the above inner product is zero. Thus, without loss of
generality, we can assume $i=k$. Using (10), we write

$\begin{split}&\int_{\widehat{T}}\Phi_{i,j}(x,y)\Phi_{i,\ell}(x,y)dx\
dy=\left(\int_{-1}^{1}L_{i}(z)L_{i}(z)\
dz\right)\left(\int_{-1}^{1}J_{j}^{2i+1}(y)J_{\ell}^{2i+1}(y)\left(\frac{1-y}{2}\right)^{2i+1}dy\right)\\\
&=\frac{2}{2j+1}2^{-2j-1}\frac{2^{2j+2}}{2\ell+2j+2}\delta_{j,\ell}=\frac{2}{2j+1}2^{-2j-1}\frac{2^{2j+1}}{j+\ell+1}\delta_{j,\ell}=\frac{2}{2j+1}\frac{1}{j+\ell+1}\delta_{j,\ell}.\end{split}$
(19)

With this at hand, we extend the results in Section 3.3 to 2D simplices.
Henceforth, let $v$ denote a sufficiently regular function over $\widehat{T}$.

#### Step 1: writing $\mathcal{P}_{p}v$ with respect to the Koornwinder basis
and with explicit coefficients.

We expand $v$ with respect to the Koornwinder basis in (18):

$v(x,y)=\sum_{j+\ell=0}^{+\infty}\mathbf{v}_{j,\ell}\Phi_{j,\ell}(x,y).$

Since $\mathcal{P}_{p}v$ belongs to $\mathbb{P}_{p}(\widehat{T})$, we expand
it with respect to the Koornwinder basis:

$\mathcal{P}_{p}v(x,y)=\sum_{j+\ell=0}^{p}\widehat{\mathbf{v}}_{j,\ell}\Phi_{j,\ell}(x,y).$

By the orthogonality property (12a), we readily get

$\mathcal{P}_{p}v(x,y)=\sum_{j+\ell=0}^{p-1}\mathbf{v}_{j,\ell}\Phi_{j,\ell}(x,y)+\sum_{j+\ell=p}\widehat{\mathbf{v}}_{j,\ell}\Phi_{j,\ell}(x,y).$

The coefficients corresponding to the Koornwinder’s polynomials of order $p$
are found using the facet orthogonality condition (12b) on the facet
$F^{+}_{\widehat{T}}$. Notably, we impose the identity

$\int_{F^{+}_{\widehat{T}}}\Big{[}\sum_{j+\ell\geq
p+1}\mathbf{v}_{j,\ell}\Phi_{j,\ell}(x,-1)+\sum_{j+\ell=p}(\mathbf{v}_{j,\ell}-\widehat{\mathbf{v}}_{j,\ell})\Phi_{j,\ell}(x,-1)\Big{]}L_{k}(x)\
dx=0\qquad\forall k=0,\dots,p.$

Using (18) and the fact that $J_{\ell}^{2i+1}(-1)=(-1)^{\ell}$, this identity
can be rewritten as

$\begin{split}&\sum_{j+\ell=p}\int_{F^{+}_{\widehat{T}}}(\mathbf{v}_{j,\ell}-\widehat{\mathbf{v}}_{j,\ell})(-1)^{p-j}L_{j}(x)L_{k}(x)dx\\\
&\qquad\qquad+\sum_{\widetilde{p}=p+1}^{+\infty}\sum_{j+\ell=\widetilde{p}}\int_{F^{+}_{\widehat{T}}}\mathbf{v}_{j,\ell}(-1)^{\widetilde{p}-j}L_{j}(x)L_{k}(x)dx=0\quad\forall
k=0,\dots,p.\end{split}$

Further using (16) and the orthogonality property (9) of the Legendre
polynomials, we end up with

$(-1)^{p-k}\frac{2}{2k+1}[\mathbf{v}_{k,p-k}-\widehat{\mathbf{v}}_{k,p-k}]+\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-k}\frac{2}{2k+1}\mathbf{v}_{k,\widetilde{p}-k}=0\qquad\qquad\forall
k=0,\dots,p.$

In other words, the order $p$ coefficients of $\mathcal{P}_{p}v$ are given by

$(-1)^{p-k}\widehat{\mathbf{v}}_{k,p-k}=\sum_{\widetilde{p}=p}^{+\infty}(-1)^{\widetilde{p}-k}\mathbf{v}_{k,\widetilde{p}-k}\qquad\qquad\forall
k=0,\dots,p.$

This is the 2D counterpart of (13). For completeness, we collect the above
identities and arrive at

$\mathcal{P}_{p}v(x,y)=\sum_{j+\ell=0}^{p-1}\mathbf{v}_{j,\ell}\Phi_{j,\ell}(x,y)+\sum_{j+\ell=p}\Big{(}\sum_{\widetilde{p}=p}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}\Big{)}\Phi_{j,\ell}(x,y).$

As in the 1D case, we deduce

$\begin{split}(v-\mathcal{P}_{p}v)(x,y)&=\sum_{j+\ell=p}^{+\infty}\mathbf{v}_{j,p-j}\Phi_{j,\ell}(x,y)-\sum_{j+\ell=p}(\sum_{\widetilde{p}=p}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j})\Phi_{j,\ell}(x,y)\\\
&=\sum_{j+\ell=p+1}^{+\infty}\mathbf{v}_{j,p-j}\Phi_{j,\ell}(x,y)-\sum_{j+\ell=p}(\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j})\Phi_{j,\ell}(x,y)\\\
&=(v-\Pi^{0}_{p}v)(x,y)-\sum_{j+\ell=p}(\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j})\Phi_{j,\ell}(x,y).\end{split}$
(20)

#### Step 2: relate the tail of the Koornwinder coefficients to the trace of
$v-\Pi^{0}_{p}v$ on the outflow facet.

Using that $J_{\ell}^{2j+1}(-1)=(-1)^{\ell}$ for all $\ell$ and $j$ in
$\mathbb{N}$, we write explicitly the trace of $v-\Pi^{0}_{p}v$ on
$F^{+}_{\widehat{T}}$:

$(v-\Pi^{0}_{p}v)(x,y)=\sum_{j+\ell=p+1}^{+\infty}\mathbf{v}_{j,\ell}\Phi_{j,\ell}(x,y),\qquad(v-\Pi^{0}_{p}v)(x,-1)=\sum_{j+\ell=p+1}^{+\infty}(-1)^{\ell}\mathbf{v}_{j,\ell}L_{j}(x).$

The orthogonality property (9) of the Legendre polynomials over
$\widehat{I}:=[-1,1]$ and (16), we write

$\begin{split}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{\widehat{T}}}^{2}&=\sum_{j=0}^{+\infty}{\left\|{L_{j}}\right\|}_{0,\widehat{I}}^{2}\Big{(}\sum_{\ell=\max(p+1-j,0)}^{+\infty}(-1)^{\ell}\mathbf{v}_{j,\ell}\Big{)}^{2}\\\
&=\sum_{j=0}^{+\infty}\frac{2}{2j+1}\Big{(}\sum_{\ell=\max(p+1-j,0)}^{+\infty}(-1)^{\ell}\mathbf{v}_{j,\ell}\Big{)}^{2}.\end{split}$

Next, we focus on the second term on the right-hand side of (20). Using (19),
we can write

$\begin{split}&{\left\|{\sum_{j+\ell=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}\big{)}\Phi_{j,\ell}}\right\|}_{0,\widehat{T}}^{2}=\sum_{j+\ell=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}\big{)}^{2}{\left\|{\Phi_{j,\ell}}\right\|}_{0,\widehat{T}}^{2}\\\
&=\sum_{j+\ell=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}\big{)}^{2}\frac{2}{2j+1}\frac{1}{j+\ell+1}=\frac{1}{p+1}\sum_{j=0}^{p}\frac{2}{2j+1}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}\big{)}^{2}\\\
&\leq\frac{1}{p+1}\sum_{j=0}^{+\infty}\frac{2}{2j+1}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}\big{)}^{2}=\frac{1}{p+1}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{\widehat{T}}}^{2}.\end{split}$
(21)

The last identity follows from the fact that the quantity in the parenthesis
is squared and the only relevant fact is the alternating sign of the
$(-1)^{\widetilde{p}-p}$ term.

#### Step 3: deduce the desired bound using Lemma 3.1.

Combining the above inequality with (20) and using the 2D version of Lemma
3.1, we deduce

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,\widehat{T}}\leq{\left\|{v-\Pi^{0}_{p}}\right\|}_{0,\widehat{T}}+(p+1)^{-\frac{1}{2}}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{\widehat{T}}}\lesssim{\left\|{v-\Pi^{0}_{p}}\right\|}_{0,\widehat{T}}+(p+1)^{-1}{\left|{v}\right|}_{1,\widehat{T}}.$

Given $\Pi^{1}_{p}$ the $H^{1}$ projection as in (11), noting that
$\mathcal{P}_{p}$ and $\Pi^{0}_{p}$ preserve polynomials of maximum degree
$p$, we have

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,\widehat{T}}\lesssim{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,\widehat{T}}+(p+1)^{-1}{\left|{v-\Pi^{1}_{p}v}\right|}_{1,\widehat{T}}.$

The above bound and standard polynomial approximation results entail Theorem
3.2 in 2D.

###  3.5 The 3D case

We prove Theorem 3.2 in three dimensions. Compared to the 1D and 2D cases of
Sections 3.3 and 3.4, the role of orthogonal basis functions is now played by
tensor product Jacobi polynomials (of different orders) collapsed on the
tetrahedron using the 3D Duffy transformation; in other words, we use 3D
Koornwinder polynomials following the construction in [6, 22, 14].

Introduce the reference cube $\widehat{Q}=[-1,1]^{3}$ and the reference
tetrahedron $\widehat{T}$ of vertices

$(-1,-1,-1);\qquad\qquad(1,-1,-1);\qquad\qquad(0,1,-1);\qquad\qquad(0,0,1).$

We denote the lower face of $\widehat{T}$ by $F^{+}_{\widehat{T}}$; it
coincides with the reference triangle in Section 3.4 at $z=1$.

We consider the 3D Duffy transformation
$\mathcal{D}:\widehat{Q}\to\widehat{T}$ that maps each
$(z_{1},z_{2},z_{3})=\mathbf{z}$ in $\widehat{Q}$ into
$(x_{1},x_{2},x_{3})=\mathbf{x}$ in $\widehat{T}$ as follows:

$x_{1}=z_{1}\frac{1-z_{2}}{2}\frac{1-z_{3}}{2},\qquad\qquad
x_{2}=z_{2}\frac{1-z_{3}}{2},\qquad\qquad x_{3}=z_{3}.$

The Jacobian of the transformation is

$\det\begin{bmatrix}\frac{1-z_{2}}{2}\frac{1-z_{3}}{2}&0&0\\\
-\frac{1}{2}z_{1}\frac{1-z_{3}}{2}&\frac{1-z_{3}}{2}&0\\\
-\frac{1}{2}z_{1}\frac{1-z_{2}}{2}&-\frac{z_{2}}{2}&1\\\
\end{bmatrix}=\frac{1-z_{2}}{2}\left(\frac{1-z_{3}}{2}\right)^{2}.$

We also compute the formal inverse $\mathcal{D}^{-1}$ of the transformation
$\mathcal{D}$:

$z_{3}=x_{3},\qquad\qquad
z_{2}=x_{2}\frac{2}{1-z_{3}}=\frac{2x_{2}}{1-x_{3}},\qquad\qquad
z_{1}=x_{1}\frac{4}{(1-z_{2})(1-z_{3})}=\frac{4x_{1}}{1-2x_{2}-x_{3}}.$

We introduce the following $L^{2}$-orthogonal basis over $\widehat{T}$: given
a multi-index $\mathbf{j}=(j_{1},j_{2},j_{3})$ in $\mathbb{N}^{3}$,

$\Phi_{\mathbf{j}}(\mathbf{x})=L_{j_{1}}\left(\frac{4x_{1}}{1-2x_{2}-x_{3}}\right)J_{j_{2}}^{2j_{1}+1}\left(\frac{2x_{2}}{1-x_{3}}\right)\left(\frac{1-2x_{2}-x_{3}}{2(1-x_{3})}\right)^{j_{1}}J_{j_{3}}^{2j_{1}+2j_{2}+2}(x_{3})\left(\frac{1-x_{3}}{2}\right)^{j_{1}+j_{2}}.$

We only check the norm of the above Koornwinder polynomials. Transforming
$\widehat{T}$ into $\widehat{Q}$ by $\mathcal{D}^{-1}$ and recalling (10), we
deduce

$\begin{split}&\int_{\widehat{T}}\Phi_{\mathbf{j}}(\mathbf{x})\Phi_{\mathbf{j}}(\mathbf{x})d\mathbf{x}\\\
&=\int_{\widehat{Q}}L_{j_{1}}(z_{1})^{2}J_{j_{2}}^{2j_{1}+1}(z_{2})^{2}\left(\frac{1-z_{2}}{2}\right)^{2j_{2}+1}J_{j_{3}}^{2j_{1}+2j_{2}+2}(z_{3})^{2}\left(\frac{1-z_{3}}{2}\right)^{2j_{1}+2j_{2}+2}\\!\\!\\!d\mathbf{z}\\\
&=\int_{\widehat{I}}L_{j_{1}}(z_{1})^{2}\
dz_{1}\int_{\widehat{I}}J_{j_{2}}^{2j_{1}+1}(z_{2})^{2}\left(\frac{1-z_{2}}{2}\right)^{2j_{2}+1}\\!\\!\\!dz_{2}\int_{\widehat{I}}J_{j_{3}}^{2j_{1}+2j_{2}+2}(z_{3})^{2}\left(\frac{1-z_{3}}{2}\right)^{2j_{1}+2j_{2}+2}\\!\\!\\!dz_{3}\\\
&=\frac{2}{2j_{1}+1}\frac{2}{2j_{1}+2j_{2}+2}\frac{2}{2j_{1}+2j_{2}+2j_{3}+2}\qquad\forall\mathbf{j}=(j_{1},j_{2},j_{3})\in\mathbb{N}^{3}.\end{split}$
(22)

Next, we prove $p$-optimal approximation properties in the $L^{2}$ norm of the
operator $\mathcal{P}_{p}$ given as in (12) with the role of outflow face
played by the facet $F^{+}_{\widehat{T}}$.

#### Step 1: writing $\mathcal{P}_{p}v$ with respect to the 3D version
Koornwinder basis and with explicit coefficients.

Consider the following expansion of $v$:

$v(\mathbf{x})=\sum_{|\mathbf{j}|=0}^{+\infty}\mathbf{v}_{\mathbf{j}}\Phi_{\mathbf{j}}(\mathbf{x}).$

Using (12a), we realize that

$v(\mathbf{x})=\sum_{|\mathbf{j}|=0}^{p-1}\mathbf{v}_{\mathbf{j}}\Phi_{\mathbf{j}}(\mathbf{x})+\sum_{|\mathbf{j}|=p}\widehat{\mathbf{v}}_{\mathbf{j}}\Phi_{\mathbf{j}}(\mathbf{x}),$

where the coefficients $\widehat{\mathbf{v}}_{\mathbf{j}}$, $|\mathbf{j}|=p$
are to be determined using (12b).

First, we write the trace of any polynomial basis function on
$F^{+}_{\widehat{T}}$, i.e., we impose the passage through $x_{3}=-1$:

$\Phi_{\mathbf{j}}(\mathbf{x}){}_{|F^{+}_{\widehat{T}}}=L_{j_{1}}\left(\frac{2x_{1}}{1-x_{2}}\right)J_{j_{2}}^{2j_{1}+1}(x_{2})\left(\frac{1-x_{2}}{2}\right)^{j_{1}}(-1)^{j_{3}}.$

Let $\Psi_{\ell_{1},\ell_{2}}$ be the 2D Koornwinder basis in (18) on the
reference triangle $F^{+}_{\widehat{T}}$. We impose (12b):

$\begin{split}&\sum_{|\mathbf{j}|=p}\int_{F^{+}_{\widehat{T}}}[\mathbf{v}_{\mathbf{j}}-\widehat{\mathbf{v}}_{\mathbf{j}}](-1)^{j_{3}}L_{j_{1}}\left(\frac{2x_{1}}{1-x_{2}}\right)L_{j_{2}}(x_{2})\left(\frac{1-x_{2}}{2}\right)^{j_{1}}\Psi_{\ell_{1},\ell_{2}}(x_{1},x_{2})dx_{1}dx_{2}\\\
&+\sum_{\widetilde{p}=p+1}^{+\infty}\sum_{|\mathbf{j}|=\widetilde{p}}\int_{F^{+}_{\widehat{T}}}\mathbf{v}_{\mathbf{j}}(-1)^{j_{3}}L_{j_{1}}\left(\frac{2x_{1}}{1-x_{2}}\right)L_{j_{2}}(x_{2})\left(\frac{1-x_{2}}{2}\right)^{j_{1}}\Psi_{\ell_{1},\ell_{2}}(x_{1},x_{2})dx_{1}dx_{2}=0,\end{split}$

We rewrite the above relation as

$\begin{split}&\sum_{|\mathbf{j}|=p}\int_{F^{+}_{\widehat{T}}}[\mathbf{v}_{\mathbf{j}}-\widehat{\mathbf{v}}_{\mathbf{j}}](-1)^{j_{3}}\Psi_{j_{1},j_{2}}(x_{1},x_{2})\Psi_{\ell_{1},\ell_{2}}(x_{1},x_{2})dx_{1}dx_{2}\\\
&+\sum_{\widetilde{p}=p+1}^{+\infty}\sum_{|\mathbf{j}|=\widetilde{p}}\int_{F^{+}_{\widehat{T}}}\mathbf{v}_{\mathbf{j}}(-1)^{j_{3}}\Psi_{j_{1},j_{2}}(x_{1},x_{2})\Psi_{\ell_{1},\ell_{2}}(x_{1},x_{2})dx_{1}dx_{2}=0,\end{split}$

Using the orthogonality property (19) of the Koornwinder polynomials in 2D, we
deduce

$\begin{split}&(-1)^{p-\ell_{1}-\ell_{2}}\frac{2}{2\ell_{2}}\frac{1}{\ell_{1}+\ell_{2}+1}(\mathbf{v}_{\ell_{1},\ell_{2},p-\ell_{1}-\ell_{2}}-\widehat{\mathbf{v}}_{\ell_{1},\ell_{2},p-\ell_{1}-\ell_{2}})\\\
&+\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-(\ell_{1}+\ell_{2})}\frac{2}{2\ell_{2}+1}\frac{1}{\ell_{1}+\ell_{2}}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\qquad\qquad\forall\ell_{1}+\ell_{2}\leq
p.\end{split}$

In other words, we have

$\widehat{\mathbf{v}}_{\ell_{1},\ell_{2},p-\ell_{1}-\ell_{2}}=\sum_{\widetilde{p}=p}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\qquad\qquad\forall\ell_{1}+\ell_{2}\leq
p.$

This provides us with the representation

$\mathcal{P}_{p}v(\mathbf{x})=\sum_{|\mathbf{j}|=0}^{p-1}\mathbf{v}_{\mathbf{j}}\Phi_{\mathbf{j}}(\mathbf{x})+\sum_{|\mathbf{j}|=p}\big{(}\sum_{\widetilde{p}=p}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}\Phi_{\mathbf{j}}(\mathbf{x}).$

We get

$\begin{split}&(\mathbf{v}-\mathcal{P}_{p}\mathbf{v})(\mathbf{x})=\sum_{|\mathbf{j}|=p}^{+\infty}\mathbf{v}_{j_{1},j_{2},p-j_{1}-j_{2}}\Phi_{j}(\mathbf{x})-\sum_{|\mathbf{j}|=p}\big{(}\sum_{\widetilde{p}=p}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}\Phi_{\mathbf{j}}(\mathbf{x})\\\
&=\sum_{|\mathbf{j}|=p+1}^{+\infty}\mathbf{v}_{j_{1},j_{2},p-j_{1}-j_{2}}\Phi_{j}(\mathbf{x})-\sum_{|\mathbf{j}|=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}\Phi_{\mathbf{j}}(\mathbf{x})\\\
&=(\mathbf{v}-\Pi^{0}_{p}\mathbf{v})(\mathbf{x})-\sum_{|\mathbf{j}|=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}\Phi_{\mathbf{j}}(\mathbf{x}).\end{split}$
(23)

#### Step 2: relate the tail of the 3D version Koornwinder coefficients to the
trace of $v-\Pi^{0}_{p}v$ on the outflow facet.

The restriction of $\mathbf{v}-\Pi^{0}_{p}\mathbf{v}$ on the outflow facet
$F^{+}_{\widehat{T}}$ reads

$\begin{split}(\mathbf{v}-\mathcal{P}_{p}\mathbf{v})(\mathbf{x})_{|F^{+}_{\widehat{T}}}&=\sum_{|\mathbf{j}|=p+1}^{+\infty}\mathbf{v}_{\mathbf{j}}L_{j_{1}}\left(\frac{2x_{1}}{1-x_{2}}\right)J_{j_{2}}^{2j_{1}+1}(x_{2})\left(\frac{1-x_{2}}{2}\right)^{j_{1}}(-1)^{j_{3}}\\\
&=\sum_{|\mathbf{j}|=p+1}^{+\infty}\mathbf{v}_{\mathbf{j}}\Psi_{j_{1},j_{2}}(\mathbf{x})(-1)^{j_{3}}.\end{split}$

We take the $L^{2}$ norm on both sides over $F^{+}_{\widehat{T}}$, use the
orthogonality property (19) of Koornwinder polynomials over
$F^{+}_{\widehat{T}}$, and get, for $\widetilde{j}$ equal to
$\max(p+1-(j_{1}+j_{2}),0)$,

${\left\|{\mathbf{v}-\mathcal{P}_{p}\mathbf{v}}\right\|}_{0,F^{+}_{\widehat{T}}}=\sum_{j_{1}+j_{2}=0}^{+\infty}{\left\|{\Psi_{j_{1},j_{2}}}\right\|}_{0,F^{+}_{\widehat{T}}}^{2}\big{(}\sum_{j_{3}=\widetilde{j}}(-1)^{j_{3}}\mathbf{v}_{\mathbf{j}}\big{)}^{2}=\sum_{j_{1}+j_{2}=0}^{+\infty}\frac{2}{2j_{2}+1}\frac{1}{j_{1}+j_{2}+1}\big{(}\sum_{j_{3}=\widetilde{j}}(-1)^{j_{3}}\mathbf{v}_{\mathbf{j}}\big{)}^{2}.$

We show a bound on the $L^{2}$ norm over $\widehat{T}$ of the second term on
the right-hand side of (23) in terms of the $L^{2}$ norm of
$\mathbf{v}-\mathcal{P}_{p}\mathbf{v}$ on $F^{+}_{\widehat{T}}$. To this aim,
we use (22):

$\begin{split}&{\left\|{\sum_{|\mathbf{j}|=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}\Phi_{\mathbf{j}}(\mathbf{x})}\right\|}_{0,\widehat{T}}^{2}=\sum_{|\mathbf{j}|=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}^{2}{\left\|{\Phi_{\mathbf{j}}}\right\|}_{0,\widehat{T}}^{2}\\\
&=\sum_{|\mathbf{j}|=p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}^{2}\frac{2}{2j_{1}+1}\frac{1}{j_{1}+j_{2}+1}\frac{1}{j_{1}+j_{2}+j_{3}+1}\\\
&=\frac{1}{p+1}\sum_{j_{1}+j_{2}=0}^{p}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}^{2}\frac{2}{2j_{1}+1}\frac{1}{j_{1}+j_{2}+1}\\\
&\leq\frac{1}{p+1}\sum_{j_{1}+j_{2}=0}^{+\infty}\big{(}\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{\ell_{1},\ell_{2},\widetilde{p}-\ell_{1}-\ell_{2}}\big{)}^{2}\frac{2}{2j_{1}+1}\frac{1}{j_{1}+j_{2}+1}=\frac{1}{p+1}{\left\|{\mathbf{v}-\Pi^{0}_{p}\mathbf{v}}\right\|}_{0,F^{+}_{\widehat{T}}}^{2}.\end{split}$

#### Step 3: deduce the desired bound using optimal trace error estimates.

Combining the above inequality with (20) and using the 3D version of Lemma
3.1, we deduce

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,\widehat{T}}\leq{\left\|{v-\Pi^{0}_{p}}\right\|}_{0,\widehat{T}}+(p+1)^{-\frac{1}{2}}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{\widehat{T}}}\lesssim{\left\|{v-\Pi^{0}_{p}}\right\|}_{0,\widehat{T}}+(p+1)^{-1}{\left|{v}\right|}_{1,\widehat{T}}.$

Given $\Pi^{1}_{p}$ the $H^{1}$ projection as in (11), noting that
$\mathcal{P}_{p}$ and $\Pi^{0}_{p}$ preserve polynomials of maximum degree
$p$, we have

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,\widehat{T}}\lesssim{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,\widehat{T}}+(p+1)^{-1}{\left|{v-\Pi^{1}_{p}v}\right|}_{1,\widehat{T}}.$

The above bound and standard polynomial approximation results entail Theorem
3.2 in 3D.

###  3.6 Trace-type estimates for the CDG operator

The CDG operator satisfies $hp$-optimal approximation properties in the
$L^{2}$ norm; see Theorem 3.2. Here, we investigate $hp$-optimal approximation
properties in the $L^{2}$ norm on the boundary of a simplex $T$, i.e., the
counterpart of Lemma 3.1 for the projector $\mathcal{P}_{p}$.

To the aim, we show separate bounds on the facets. First, we consider the case
of the (unique) outflow facet $F^{+}_{T}$. By (12b) and Lemma 3.1, we get

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,F^{+}_{T}}\leq{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{T}}\lesssim\left(\frac{h_{T}}{p}\right)^{\frac{1}{2}}{\left|{v}\right|}_{1,T}\qquad\qquad\qquad\forall
v\in H^{1}(T).$

Approximation properties on the inflow facets are more elaborated and are
discussed in the next result.

######  Proposition 3.3.

Let the assumptions (A1) and (A2) be valid, $T$ be in $\mathcal{T}_{h}$, and
$F^{-}_{T}$ be any of the inflow facets of $T$. Then, there exist positive
constants $C_{1}$ and $C_{2}$ independent of $h$ and $p$ such that, for
sufficiently smooth functions, the following estimate holds true:

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,F^{-}_{T}}\leq
C_{1}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,\partial T}\leq
C_{2}\frac{h^{\min(p,k)+\frac{1}{2}}}{p^{k+\frac{1}{2}}}{\left|{v}\right|}_{k+1,T}.$
(24)

Above, $\Pi^{0}_{p}$ is the operator in (8).

###### Proof.

The proof of the 1D case is fairly simple. The two and three dimensional cases
are more elaborated; we provide details for the 2D case, as the 3D case can be
dealt with similarly. Throughout, we employ the notation of Sections 3.2–3.5.
We prove the assertion on the reference simplex $\widehat{T}$; the general
statement follows from a scaling argument.

#### Proof of (24) in 1D.

Here $\widehat{T}=[-1,1]$. From (14), (15), and $L_{j}(1)=1$ for all $j$ in
$\mathbb{N}$, we deduce

$|(v-\mathcal{P}_{p}v)(1)|\leq|(v-\Pi^{0}_{p}v)(-1)|+|(v-\Pi^{0}_{p}v)(1)|.$

The assertion follows from Lemma 3.1.

#### Proof of (24) in 2D.

Here, $\widehat{T}$ is as in (17). Introduce

$\operatorname{\mathcal{R}}:=\sum_{j+\ell=p}(\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j})\Phi_{j,\ell}(x,y).$

Introduce

$\widehat{C}:=\max_{j+\ell=p}{\left\|{\Phi_{j,\ell}(x,y)}\right\|}_{0,F^{-}_{\widehat{T}}}.$

We have

${\left\|{\operatorname{\mathcal{R}}}\right\|}_{0,F^{-}_{\widehat{T}}}^{2}\lesssim\sum_{j+\ell=p}|\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}|^{2}{\left\|{\Phi_{j,\ell}(x,y)}\right\|}_{0,F^{-}_{\widehat{T}}}^{2}\leq\widehat{C}^{2}\sum_{j+\ell=p}|\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}|^{2},$

whence we deduce

$\begin{split}{\left\|{\operatorname{\mathcal{R}}}\right\|}_{0,F^{-}_{\widehat{T}}}^{2}&\lesssim\widehat{C}^{2}\sum_{j+\ell=p}|\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}|^{2}\frac{2}{2j+1}\frac{2j+1}{2}\\\
&\leq\widehat{C}^{2}\frac{2p+1}{2}\sum_{j+\ell=p}|\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}|^{2}\frac{2}{2j+1}.\end{split}$

We proceed as in (21) and arrive at

${\left\|{\operatorname{\mathcal{R}}}\right\|}_{0,F^{-}_{\widehat{T}}}^{2}\lesssim\widehat{C}^{2}\frac{2p+1}{2}\sum_{j+\ell=p}|\sum_{\widetilde{p}=p+1}^{+\infty}(-1)^{\widetilde{p}-p}\mathbf{v}_{j,\widetilde{p}-j}|^{2}\frac{2}{2j+1}\leq\widehat{C}^{2}\frac{2p+1}{2}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{\widehat{T}}}^{2}.$

Combining (20) with the inequality above yields

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,F^{-}_{\widehat{T}}}\leq{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{-}_{\widehat{T}}}+{\left\|{\operatorname{\mathcal{R}}}\right\|}_{0,F^{-}_{\widehat{T}}}\lesssim{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{-}_{\widehat{T}}}+\widehat{C}p^{\frac{1}{2}}{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{\widehat{T}}}.$
(25)

If we were able to prove that $\widehat{C}\lesssim p^{-\frac{1}{2}}$, then the
assertion would follow from Lemma 3.1 and (25). In fact, we would write

${\left\|{v-\mathcal{P}_{p}v}\right\|}_{0,F^{-}_{\widehat{T}}}\leq{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{-}_{\widehat{T}}}+{\left\|{\operatorname{\mathcal{R}}}\right\|}_{0,F^{-}_{\widehat{T}}}\lesssim{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{-}_{\widehat{T}}}+{\left\|{v-\Pi^{0}_{p}v}\right\|}_{0,F^{+}_{\widehat{T}}}.$

We are left with showing $\widehat{C}\lesssim p^{-\frac{1}{2}}$. For the sake
of presentation, we fix $F^{-}_{\widehat{T}}$ to be the inflow facet of
$\widehat{T}$ with negative $x$ coordinates. If this is the case, then a
direct application of definition (18) gives

$\Phi_{j,\ell}{}_{|F^{-}_{\widehat{T}}}=(-1)^{j}J_{\ell}^{2j+1}(y)\left(\frac{1-y}{2}\right)^{j}.$

Taking into account the Jacobian of the transformation mapping the facet
$F^{-}_{\widehat{T}}$ into the “vertical” facet $\\{-1\\}\times[-1,1]$ and
recalling (10), we deduce

${\left\|{\Phi_{j,\ell}}\right\|}_{0,F^{-}_{\widehat{T}}}^{2}=\int_{-1}^{1}\left[J_{\ell}^{2j+1}(y)\right]^{2}\left(\frac{1-y}{2}\right)^{2j+1}=\frac{2^{2j+2}}{2j+1+2\ell+1}\frac{1}{2^{2j+1}}=\frac{1}{1+j+\ell}.$

Since we are considering couples $(j,\ell)$ such that $j+\ell=p$, we arrive at

${\left\|{\Phi_{j,\ell}}\right\|}_{0,F^{-}_{\widehat{T}}}^{2}=\frac{1}{p+1},$

i.e., $\widehat{C}\lesssim p^{-\frac{1}{2}}$. ∎

##  4 A priori analysis

We prove $hp$-optimal a priori error estimates for method (5). The proof is
essentially the $hp$-version of that of [7, Theorem $2.2$]. We report here the
details for three reasons: for the sake of completeness; since we are
interested in $hp$-optimal a priori bounds; because we have different
assumptions, namely we require (4) for well posedness of the method, but we
have no restrictions on the mesh size $h$ as for instance required in [13].

######  Theorem 4.1.

Let $\mathcal{T}_{h}$ be a shape-regular, quasi uniform mesh satisfying the
assumptions (A1)–(A2), and $u$ and $u_{h}$ be the solutions to (3) and (5). If
(4) holds true and $u$ belongs to $H^{k+1}(\Omega)$, then there exists a
positive constant $C$ independent of $h$ and $p$ but possibly depending on
$\sigma$ such that

${\left\|{u-u_{h}}\right\|}_{0,\Omega}\leq
C\frac{h^{\min(k,p)+1}}{p^{k+1}}{\left\|{u}\right\|}_{k+1,\Omega}.$

###### Proof.

Introduce

$e_{h}:=u_{h}-\mathcal{P}_{p}u.$

The rewriting of the bilinear form in (6) and the Galerkin orthogonality (7)
imply

$\mathcal{B}(e_{h},v_{h})=\mathcal{B}(u-\mathcal{P}_{p}u,v_{h})=\sum_{j=1}^{3}T_{j},$

where

$\begin{split}&T_{1}:=-(u-\mathcal{P}_{p}u,\boldsymbol{\beta}\cdot\nabla_{h}v_{h})_{0,\Omega},\\\
&T_{2}:=\sum_{T\in\mathcal{T}_{h}}((u-\mathcal{P}_{p}u)^{-},\mathbf{n}_{T}\cdot\boldsymbol{\beta}\
\left[\\!\left[v_{h}\right]\\!\right])_{0,\Gamma^{T}_{\operatorname{in}}}+\sum_{T\in\mathcal{T}_{h}}(u-\mathcal{P}_{p}u,\mathbf{n}_{T}\cdot\boldsymbol{\beta}\
v_{h})_{0,\partial T\cap\Gamma_{+}},\\\
&T_{3}:=(c(u-\mathcal{P}_{p}u),v_{h})_{0,\Omega}.\end{split}$

The term $T_{1}$ vanishes due to (12a).

As for the term $T_{2}$, some manipulations imply

$T_{2}=-\sum_{T\in\mathcal{T}_{h}}(u-\mathcal{P}_{p}u,\mathbf{n}_{T}\cdot\boldsymbol{\beta}\left[\\!\left[v_{h}\right]\\!\right])_{0,\Gamma^{T}_{\operatorname{out}}}+\sum_{T\in\mathcal{T}_{h}}(u-\mathcal{P}_{p}u,\mathbf{n}_{T}\cdot\boldsymbol{\beta}\
v_{h})_{0,\partial T\cap\Gamma_{+}}.$

The last identity is a consequence of the definition of outflow facets.
Recalling (12b), also $T_{2}$ vanishes. We end up with

$\mathcal{B}(e_{h},v_{h})=(c(u-\mathcal{P}_{p}u),v_{h})_{0,\Omega}.$

This means that $e_{h}$ solves method (5) with data $f$ and $g$ given by
$c(u-\mathcal{P}_{p}u)$ and $0$, respectively.

This identity and the stability estimate in Proposition 2.1 imply

${\left\|{e_{h}}\right\|}_{0,\Omega}\lesssim{\left\|{c(u-\mathcal{P}_{p}u)}\right\|}_{0,\Omega}.$

The assertion follows from the triangle inequality and the $hp$-optimal
approximation properties of the CDG operator displayed in Theorem 3.2. ∎

We also prove $hp$-optimal convergence for method (5) in the full DG norm

$|||v_{h}|||_{\operatorname{DG}}^{2}:=\\!\\!\sum_{T\in\mathcal{T}_{h}}\left({\left\|{c\
v_{h}}\right\|}_{0,T}^{2}+{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}v_{h}^{-}}\right\|}_{0,\Gamma^{T}_{\operatorname{in}}\cap\Gamma_{-}}^{2}\\!\\!\\!\\!+{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}v_{h}^{+}}\right\|}_{0,\Gamma^{T}_{\operatorname{out}}\cap\Gamma_{+}}^{2}\\!\\!\\!\\!+{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}\left[\\!\left[v_{h}\right]\\!\right]}\right\|}_{0,\Gamma^{T}_{\operatorname{in}}\setminus\Gamma_{-}}^{2}\right).$

######  Theorem 4.2.

Let $\mathcal{T}_{h}$ be a shape-regular, quasi uniform mesh satisfying the
assumptions (A1)–(A2), and $u$ and $u_{h}$ be the solutions to (3) and (5). If
(4) holds true and $u$ belongs to $H^{k+1}(\Omega)$, then there exists a
positive constant $C$ independent of $h$ and $p$ but possibly depending on
$\sigma$ such that

$|||u-u_{h}|||_{\operatorname{DG}}\leq
C\frac{h^{\min(k,p)+\frac{1}{2}}}{p^{k+\frac{1}{2}}}{\left\|{u}\right\|}_{k+1,\Omega}.$

###### Proof.

This is the simplicial version of [11, Theorem 3.7]. As in the proof of that
result, given $\eta:=u-\Pi^{0}_{p}u$, it is possible to show that

$\begin{split}&|||u-u_{h}|||_{\operatorname{DG}}\\\
&\lesssim(\sum_{T\in\mathcal{T}_{h}}{\left\|{\eta}\right\|}_{0,T}^{2})^{\frac{1}{2}}+\Big{(}\sum_{T\in\mathcal{T}_{h}}\Big{(}{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}\eta^{+}}\right\|}_{0,\Gamma^{T}_{\operatorname{out}}\cap\Gamma_{+}}^{2}+{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}\eta^{+}}\right\|}_{0,\Gamma^{T}_{\operatorname{in}}\cap\Gamma_{-}}^{2}\\\
&\qquad\qquad+{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}\eta^{-}}\right\|}_{0,\Gamma^{T}_{\operatorname{out}}\setminus\Gamma_{-}}^{2}+{\left\|{|\mathbf{n}_{T}\cdot\boldsymbol{\beta}|^{\frac{1}{2}}\eta^{+}}\right\|}_{0,\Gamma^{T}_{\operatorname{out}}\setminus\Gamma_{-}}^{2}\Big{)}\Big{)}^{\frac{1}{2}}\\\
&=:A+B.\end{split}$

The hidden constant above is independent of $h$ and $p$.

The term $A$ is dealt with using standard $hp$ approximation properties of the
$L^{2}$ projection; see, e.g., [1, Lemma 4.5]. The term $B$ is dealt with
using Lemma 3.1. ∎

Theorems 4.1 and 4.2 improve the current state of the art of the literature
along different directions:

  * •

[7, Theorem 2.2] shows $h$-optimal convergence in the $L^{2}$ norm; here, we
provide full explicit track of the $p$-convergence;

  * •

[11, Theorem 3.7] and [12, Theorem 5.1] display $hp$-optimal convergence in
the full DG norm on Cartesian-type meshes; here, we show optimal $hp$-optimal
convergence in the full DG norm on special simplicial meshes and also improved
convergence (on simplicial meshes) in the $L^{2}$ norm.

The theoretical limitations of Theorems 4.1 and 4.2 are the use

  * •

of the special meshes in Section 2.1 from [7];

  * •

of assumption (4).

In Section 5 below, we investigate whether the two limitations above can be
overcome in practice or are necessary condition for the $hp$-optimality.

##  5 Numerical experiments

We assess the convergence rates of method (5) in the $L^{2}$ and full
$\operatorname{DG}$ norms proven in Theorems 4.1 and 4.2. We focus on the
$p$-version of the method in two dimensions, fix two triangular meshes of $50$
and $32$ triangles partitioning the square domain $\Omega:=(-1,1)^{2}$, see
Figure 1, and use polynomial degrees $p$ from $1$ to $40$. Standard composite
Gaussian quadrature is employed throughout. We pick an $L^{2}$ orthonormal
basis over simplices as that constructed in [14].

We consider two test cases. The first one involves a constant convection field
such that assumption (4) holds true and the meshes satisfy the construction in
Section 2.1. Instead, a second test case is devoted to check the performance
of the scheme even if the convection field and the meshes do not satisfy the
assumptions needed in the analysis of Sections 3 and 4.

Figure 1:  _Left-panel_ : mesh consisting of 50 triangles. _Right-panel_ :
mesh consisting of 32 triangles.

#### Test case 1.

As in [5, Section 5.2], we consider the convection field
$\boldsymbol{\beta}=(1,1)$, the reaction coefficient $c=1$, and the exact
solution, for given positive $\alpha$,

$u(x,y)=\begin{cases}\cos(\pi y/2)&\text{ in }(-1,0)\times(-1,1)\\\ \cos(\pi
y/2)+x^{\alpha}&\text{ in }(-1,0)\times(-1,1).\\\ \end{cases}$ (26)

The right-hand side $f$ is computed accordingly. The two meshes in Figure 1
are such that each element has only one outflow facet for the given convection
field $\boldsymbol{\beta}$.

The solution $u$ belongs to $H^{1/2+\alpha-\epsilon}(\Omega)$ for any positive
and arbitrarily small $\epsilon$. The singularity is located in the interior
of a mesh cell for the mesh in Figure 1 (_left-panel_) and at the interface of
several cell elements for the mesh in Figure 1 (_right-panel_).

We report the results in Figure 2 and 3 for the $50$ and $32$ triangles cases,
respectively. We pick $\alpha=0.5$, $1.5$, and $2.5$.

Figure 2: $p$-version for the case of constant convection field
$\boldsymbol{\beta}$. We consider the exact solution $u$ in (26) and fix the
triangular mesh with 50 elements as in Figure 1 so as the mesh assumptions in
Section 2.1 are satisfied. The singularity lies in the interior of a cell
element. _Left-panel_ : $L^{2}$ norm error. _Right-panel_ : DG norm error.

Figure 3: $p$-version for the case of constant convection field
$\boldsymbol{\beta}$. We consider the exact solution $u$ in (26) and fix the
triangular mesh with 32 elements as in Figure 1 so as the mesh assumptions in
Section 2.1 are satisfied. The singularity lies at the interface of several
cell elements. _Left-panel_ : $L^{2}$ norm error. _Right-panel_ : DG norm
error.

The convergence rate of the DG scheme under $p$ refinement in the $L^{2}$-norm
error is of order $\mathcal{O}(p^{-(1/2+\alpha)})$, which is optimal. The
convergence rate in the DG norm is also of order
$\mathcal{O}(p^{-(1/2+\alpha)})$, i.e., we observe super-convergence.

For the case of 50 triangular meshes, the singularity lies at the interface of
several mesh cell elements. Therefore, the usual doubling of the convergence
takes place, see, e.g., [2], and the convergence rate is
$\mathcal{O}(p^{-2\alpha})$ in both norms.

#### Test case 2.

As in [12, Example 1], we consider the convection field
$\boldsymbol{\beta}=(2-y^{2},2-x)$, the reaction coefficient
$c=1+(1+x)(1+y^{2})$, and the exact solution given by $u$ in (26). The right-
hand side $f$ is computed accordingly. The two meshes in Figure 1 do not
satisfy the mesh assumptions in Section 2.1 for the given convection field
$\boldsymbol{\beta}$. The regularity of the exact solution $u$ is already
discussed in Test case 1.

We report the results in Figure 4 and 5 for the $50$ and $32$ triangles cases,
respectively. We pick $\alpha=0.5$, $1.5$, and $2.5$.

Figure 4: $p$-version for the case of variable convection field
$\boldsymbol{\beta}$. We consider the exact solution $u$ in (26) and fix the
triangular mesh with 50 elements as in Figure 1. The mesh assumptions in
Section 2.1 are not satisfied. The singularity lies in the interior of a cell
element. _Left-panel_ : $L^{2}$ norm error. _Right-panel_ : DG norm error.

Figure 5: $p$-version for the case of variable convection field
$\boldsymbol{\beta}$. We consider the exact solution $u$ in (26) and fix the
triangular mesh with 32 elements as in Figure 1. The mesh assumptions in
Section 2.1 are not satisfied. The singularity lies at the interface of
several cell elements. _Left-panel_ : $L^{2}$ norm error. _Right-panel_ : DG
norm error.

We observe convergence rates as those for the constant convection field case.
This is in accordance with the numerical observation in [12, Example 1].

In both test cases, the use of an orthonormal basis implies that the condition
number remains moderate, whence the convergence rates do not deteriorate for
high $p$.

##  6 Conclusions

We analyzed $p$-optimal approximation properties of the CDG operator in the
$L^{2}$ norm and the $L^{2}$ norm of the trace on simplices with only one
outflow facet in 1D, 2D, and 3D. These results are instrumental in deriving
$p$-optimal error estimates for the original DG method on special simplicial
meshes. Numerical experiments validate the predicted convergence rates
measured in the $L^{2}$ norm also for convection fields and meshes not
satisfying the assumptions used in our proofs; super-convergence is observed
for the error measured in the full DG norm.

## References

  * [1] I. Babuška and M. Suri.  The $hp$ version of the finite element method with quasiuniform meshes.  ESAIM Math. Model. Numer. Anal., 21(2):199–238, 1987.
  * [2] I. Babuška and M. Suri.  The optimal convergence rate of the $p$-version of the finite element method.  SIAM J. Numer. Anal., 24(4):750–776, 1987.
  * [3] C. Bardos.  Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport.  Ann. Sci. de l’Ecole Norm. Superieure, 3(2):185–233, 1970.
  * [4] S. Beuchler and J. Schöberl.  New shape functions for triangular $p$-FEM using integrated Jacobi polynomials.  Numer. Math., 103(3):339–366, 2006.
  * [5] P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau.  An a priori error analysis of the local discontinuous Galerkin method for elliptic problems.  SIAM J. Numer. Anal., 38(5):1676–1706, 2000.
  * [6] A. Chernov.  Optimal convergence estimates for the trace of the polynomial ${L}^{2}$-projection operator on a simplex.  Math. Comp., 81(278):765–787, 2012.
  * [7] B. Cockburn, B. Dong, and J. Guzmán.  Optimal convergence of the original DG method for the transport-reaction equation on special meshes.  SIAM J. Numer. Anal., 46(3):1250–1265, 2008.
  * [8] B. Cockburn, B. Dong, and J. Guzmán.  A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems.  Math. Comp., 77(264):1887–1916, 2008.
  * [9] Z. Dong and L. Mascotto.  On the suboptimality of the $p$-version discontinuous Galerkin methods for first order hyperbolic problems.  Numerical Methods and Algorithms in Science and Engineering, 700, 2021.
  * [10] M. Dubiner.  Spectral methods on triangles and other domains.  J. Sci. Comput., 6:345–390, 1991.
  * [11] P. Houston, Ch. Schwab, and E. Süli.  Stabilized $hp$-finite element methods for first-order hyperbolic problems.  SIAM J. Numer. Anal., 37(5):1618–1643, 2000.
  * [12] P. Houston, Ch. Schwab, and E. Süli.  Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems.  SIAM J. Numer. Anal., 39(6):2133–2163, 2002.
  * [13] C. Johnson and J. Pitkäranta.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation.  Math. Comp., 46(173):1–26, 1986.
  * [14] G. E. Karniadakis and S. Sherwin.  Spectral/$hp$ element methods for computational fluid dynamics.  Oxford University Press, 2005.
  * [15] T. Koornwinder.  Two-variable analogues of the classical orthogonal polynomials.  In Theory and application of special functions, pages 435–495. Elsevier, 1975.
  * [16] P. Lasaint and P.-A. Raviart.  On a finite element method for solving the neutron transport equation.  In Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), pages 89–123. Academic Press, New York-London, 1974.
  * [17] J. M. Melenk and T. Wurzer.  On the stability of the boundary trace of the polynomial ${L}^{2}$-projection on triangles and tetrahedra.  Comp. Math. Appl. Sc., 67(4):944–965, 2014.
  * [18] T. E. Peterson.  A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation.  SIAM J. Numer. Anal., 28(1):133–140, 1991.
  * [19] W. H. Reed and T. R. Hill.  Triangular mesh methods for the neutron transport equation.  Technical report, Los Alamos Scientific Lab., 1973.
  * [20] D. Schötzau and Ch. Schwab.  Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method.  SIAM J. Numer. Anal., 38(3):837–875, 2000.
  * [21] J. Shen, T. Tang, and L.-L. Wang.  Spectral methods: algorithms, analysis and applications, volume 41.  Springer Science & Business Media, 2011.
  * [22] T. Warburton and J. S. Hesthaven.  On the constants in $hp$-finite element trace inverse inequalities.  Comput. Methods Appl. Mech. Engrg., 192(25):2765–2773, 2003.


# Packing and coloring $r$-bounded axis-parallel rectangles

Marco Caoduro
Univ. Grenoble Alpes, Laboratoire G-SCOP, Grenoble-INP, Grenoble, France
<EMAIL_ADDRESS>
###### Abstract

Let $\mathcal{R}$ be a family of axis-parallel rectangles in the plane. The
_transversal number_ $\tau(\mathcal{R})$ is the minimum number of points
needed to pierce all the rectangles. The _independence number_ $\nu(R)$ is the
maximum number of pairwise disjoint rectangles. Given a positive real number
$r$, we say that $\mathcal{R}$ is an _r-bounded_ family if, for any rectangle
in $\mathcal{R}$, the aspect ratio of the longer side over the shorter side is
at most $r$.

Gyárfás and Lehel asked if it is possible to bound the transversal number
$\tau(\mathcal{R})$ with a linear function of the independence number
$\nu(\mathcal{R})$. Ahlswede and Karapetyan claimed a positive answer for the
particular case of $r$-bounded families, but without providing a proof.
Chudnovsky et al. confirmed the result proving the bound
$\tau\leq(14+2r^{2})\nu$.

This note aims at giving a simple proof of $\tau\leq 2(r+1)(\nu-1)+1$,
slightly improving the previous results. As a consequence of this new
approach, we also deduce a constant factor bound for the ratio
$\frac{\chi}{\omega}$ in the case of $r$-bounded family.

###### Acknowledgements

I would like to thank my thesis supervisor András Sebő, for providing guidance
and insightful feedback throughout the writing of this note.

##  1 Introduction

###  1.1 Basic definitions

Let $\mathcal{R}$ be a family of axis-parallel rectangles in the plane and $r$
a positive real number. We say that $\mathcal{R}$ is an _r-bounded_ family if,
for any rectangle in $\mathcal{R}$, the aspect ratio of the longer side over
the shorter side is at most $r$.

We define some general parameters: the _transversal number_
$\tau(\mathcal{R})$ is the minimum number of points needed to pierce all the
rectangles in $\mathcal{R}$; the _independence number_ $\nu(\mathcal{R})$ is
the maximum number of pairwise disjoint rectangles in $\mathcal{R}$; the
_chromatic number_ $\chi(\mathcal{R})$ is the minimum number of classes in a
partition of $\mathcal{R}$ into pairwise disjoint rectangles, and the _clique
number_ $\omega(\mathcal{R})$ is the maximum number of pairwise intersecting
rectangles in $\mathcal{R}$. It is straightforward to see that
$\tau(\mathcal{R})\geq\nu(\mathcal{R})$, since if we have a set of pairwise
disjoint rectangles we need at least one point for each, and
$\chi(\mathcal{R})\geq\omega(\mathcal{R})$, since if we have a set of pairwise
intersecting rectangles we need a different color for each.

The _intersection graph_ $G(\mathcal{R})$ is the graph having the set of axis-
parallel rectangles as its vertex set and an edge for each pair of
intersecting rectangles. It is easy to see the relations between
$\nu(\mathcal{R})$, $\chi(\mathcal{R})$, $\omega(\mathcal{R})$ and usual graph
parameters
$\alpha(G(\mathcal{R})),\chi(G(\mathcal{R})),\omega(G(\mathcal{R}))$ (see
[9]). Furthermore, due to the Helly property, $\tau(\mathcal{R})$ corresponds
to the minimum number of cliques covering all the vertices of
$G(\mathcal{R})$. A family $\mathcal{R}$ is said to be _$k$ -degenerate_, if
its intersection graph is $k$-degenerate, i.e., the minimum degree of any
subgraph of $G(\mathcal{R})$ is at most $k$.

###  1.2 Packing results

In 1965, Wegner [15] conjectured that, for a family $\mathcal{R}$ of axis-
parallel rectangles, we have $\tau(R)\leq 2\nu(R)-1$. Gyárfás and Lehel [10]
asked whether there exists a constant $c$ such that $\tau(R)\leq c\nu(R)$.

Almost sixty years later, both problems are still open. Jelínek [8]
constructed a family of rectangles which shows that $c\geq 2$. Moreover, Chen
and Dumitrescu [6] and Sebő [14] recently found examples which prove that, if
confirmed, Wegner’s conjecture is tight for $\nu\leq 4$.

Besides these, there are several partial results in the literature. Kim et al.
[11] give an answer to Gyárfás and Lehel’s question in general for any set of
homothetic images of a fixed convex body with a constant of 16. Moreover, Chan
and Har-Peled [5] proved the corresponding statement for cross-free families
of rectangles (i.e., families of rectangles in which the intersection of two
rectangles is either empty or contains at least a corner of one of them).
Complementing this, Asplund and Grünbaum [2] observed that the intersection
graph of a crossing family of rectangles $\mathcal{R}_{c}$ is perfect, so
$\nu(\mathcal{R}_{c})=\tau(\mathcal{R}_{c})$.

Ahlswede and Karapetyan [1] published a note with some results without
providing the proofs. In Statement 1, they claimed that for $r$-bounded
families $\tau\leq 2(r+1)\nu$ holds, offering a particular solution of Gyárfás
and Lehel’s question. Lately, this statement was confirmed in a paper of
Chudnovsky et al. [7] but with a larger factor $(14+2r^{2})$. In this note we
give a simple proof of $\tau\leq 2(r+1)(\nu-1)+1$, slightly improving the
previous results.

###  1.3 Coloring results

Another challenging problem in Geometric Combinatorics is relating the
chromatic number and the clique number of family of convex bodies. In 1948,
Bielecki [3] asked if it is possible to bound the chromatic number of a family
of axis-parallel rectangles using a function of the clique number. Asplund and
Grünbaum [2] solved the problem providing a quadratic bound which remained
asymptotically best for exactly sixty years. Only recently Chamerlsook and
Walczak [4] improved the result proving
$\chi(\mathcal{R})=\mathcal{O}\left(\omega(\mathcal{R})\log(\omega(\mathcal{R}))\right)$.
For a lower bound, Krawczyk and Walczak [12] constructed a family
$\mathcal{R}$ of axis-parallel rectangles with
$\chi(\mathcal{R})=3\omega(\mathcal{R})-2$ providing the best known lower
bound for the problem.

According to the current state of the art and our personal experience, it
seems reasonable to ask whether $\frac{\chi}{\omega}$ can be bounded by a
constant for families of axis-parallel rectangles. As an addition to our main
theorem, we observe that our proof technique implies a positive answer for
$r$-bounded families.

##  2 Results

Let $\mathcal{R}$ be a family of axis-parallel rectangles. Denote for brevity
$\tau=\tau(\mathcal{R})$ and $\nu=\nu(\mathcal{R})$.

######  Theorem 1.

Let $\mathcal{R}$ be a set of axis-parallel rectangles and $r$ be the maximum
aspect ratio of a rectangle in $\mathcal{R}$. Then, $\tau\leq
2(r+1)(\nu-1)+1.$

###### Proof.

We prove the result by induction on $\nu$. If $\nu=1$, then by the Helly
property $\tau=1$, so the inequality holds.
Assume now that $\nu\geq 2$ and the result is true for every family having
$\nu^{\prime}<\nu$. Let $\epsilon:=\min_{R\in\mathcal{R}}\\{a_{R}\\}$ where
$a_{R}$ is the length of the smaller side of $R$ and let $R_{\epsilon}$ be the
rectangle which realizes this minimum. We assume, without loss of generality,
that $a_{\mathcal{R}_{\epsilon}}=\epsilon$ is the height of $R_{\epsilon}$ and
$b_{\mathcal{R}_{\epsilon}}$ is its width. Define $P:=\\{p_{1},...\
,p_{r+1},q_{1},...\ ,q_{r+1}\\}$ as the set of $2(r+1)$ points which subdivide
the upper and the lower sides of $R_{\epsilon}$ into segments of length at
most $\frac{b_{\mathcal{R}_{\epsilon}}}{r}\leq\epsilon$, as showed in Figure
1.

$p_{1}$$p_{2}$…$p_{r+1}$$q_{1}$$q_{2}$…$q_{r+1}$$a_{R_{\epsilon}}$$\frac{b_{R_{\epsilon}}}{r}$
Figure 1: Definition of $P$

######  Claim 1.

If $R\in\mathcal{R}$ and $R\cap R_{\epsilon}\neq\emptyset$, then $R\cap
P\neq\emptyset$.

Indeed, since $R\cap R_{\epsilon}\neq\emptyset$, then $R$ intersects at least
one of the $r$ rectangles with corner points in $P$ partitioning
$\mathcal{R}_{\epsilon}$ (Figure 1). We denote this rectangle as $P(R)$:
$P(R)\cap R\neq\emptyset$. Both sides of $P(R)$ are of length at most
$\epsilon$. Since both sides of $R$ are of length at least $\epsilon$, $R$
contains a corner of $P(R)$. Hence, $R\cap P\neq\emptyset$ and the claim is
proved.

######  Claim 2.

Let $\mathcal{R}^{\prime}:=\\{R\in\mathcal{R},\ R\cap
R_{\epsilon}=\emptyset\\}$, then:

  1. a)

$\nu(\mathcal{R})\geq\nu(\mathcal{R}^{\prime})+1$;

  2. b)

$\tau(\mathcal{R})\leq\tau(\mathcal{R}^{\prime})+2(r+1)$.

Indeed, $a)$ follows by noting that adding $R_{\epsilon}$ to any independent
set of $\mathcal{R}^{\prime}$, we get an independent set again; $b)$ is an
immediate consequence of Claim 1, since $|P|=2r+1$.

Claim 2 finishes the proof of Theorem 1 by induction. ∎

######  Corollary 2.

Let $\mathcal{S}$ be a family of squares in the plane, then
$\tau(\mathcal{S})\leq 4\nu(\mathcal{S})-3.$

In the proof of Theorem 1, Claim 1 shows that every rectangle intersecting
$R_{\epsilon}$ contains a point of $P$. Since any point can be contained in at
most $\omega(\mathcal{R})$ elements of $\mathcal{R}$, $R_{\epsilon}$ has at
most $2(r+1)(\omega(\mathcal{R})-1)$ neighbors. We can conclude the following:

######  Lemma 3.

Let $G$ be the intersection graph of an $r$-bounded family of axis-parallel
rectangles. Then, $G$ is $2(r+1)(\omega(G)-1)$-degenerate.

######  Theorem 4.

Let $\mathcal{R}$ be a set of axis-parallel rectangles and $r$ be the maximum
aspect ratio of a rectangle in $\mathcal{R}$. Then, $\chi(\mathcal{R})\leq
2(r+1)(\omega(\mathcal{R})-1)+1.$

###### Proof.

It is enough to use Lemma 3 and notice that any $k$-degenerate graph has
chromatic number at most $k+1$. ∎

Theorem 4, together with the following result of Perepelitsa [13], directly
implies another result for squares.

######  Theorem 5.

(Theorem 7, Corollary 8 [13]) Let G be the triangle-free intersection graph of
a finite number of rectangles. If the intersection of any pair of rectangles
is either empty or contains at least a corner of one of them, then G is a
plane graph. Moreover, G is 3-colorable.

######  Corollary 6 (Statement 3 [1]).

Let $\mathcal{S}$ be a family of squares in the plane, then
$\chi(\mathcal{S})\leq 4\omega(\mathcal{S})-3\ $. Moreover, if
$G(\mathcal{S})$ is triangle-free, then $\chi(\mathcal{S})\leq 3$.

##  3 Conclusion

In this note we offer a simple proof and a slight improvement of a result
often cited in the literature but so far without proof. In addition, we state
some new results about the chromatic number of $r$-bounded families.

Finally, we would like to point out that a further improvement of the bound
proposed by Theorem 1, either replacing $r+1$ by $r$ or removing the
multiplicative factor 2, would lead to an affirmative answer of Wegner’s
conjecture for families of squares.

## References

  * [1] R. Ahlswede and I. Karapetyan  “Intersecting graphs of rectangles and segments”  In _LNCS_ 4123, 2006, pp. 1064–1065
  * [2] E. Asplund and B. Grünbaum  “On a coloring problem”  In _Math. Scand._ 8, 1960, pp. 181–188
  * [3] A. Bielecki  “Problem 56”  In _Colloq. Math_ 1, 1948, pp. 333
  * [4] P. Chalermsook and B. Walczak  “Coloring and Maximum Weight Independent Set of Rectangles”, 2020  arXiv:2007.07880
  * [5] T.. Chan and S. Har-Peled  “Approximation algorithms for maximum independent set of pseudo-disks”  In _Discrete Computational Geometry_ 48, 2012, pp. 373–392
  * [6] K. Chen and A. Dumitrescu  “On Wegner’s inequality for axis-parallel rectangles”  In _Discrete Mathematics_ 343.12, 2020
  * [7] M. Chudnovsky, S. Sprinkl and S. Zerbib  “Piercing axis-parallel boxes”  In _The Electronic Journal of Combinatorics_ 25(1), 2018, pp. P1.70
  * [8] J. Correa, L. Feuilloley, P. Pérez-Lantero and J.. Soto  “Independent and hitting sets of rectangles intersecting a diagonal line: algorithms and complexity”  In _Discrete Computational Geometry_ 53, 2015, pp. 344–365
  * [9] Reinhard Diestel  “Graph Theory (Graduate Texts in Mathematics)”  Springer, Hardcover, 2005
  * [10] A. Gyárfás and J. Lehel  “Covering and coloring problems for relatives of intervals”  In _Discrete Mathematics_ 55, 1985, pp. 161–166
  * [11] S.. Kim, K. Nakprasit, M.. Pelsmajer and J. Skokan  “Transversal numbers of translates of a convex body”  In _Discrete Mathematics_ 306.18, 2006, pp. 2166–2173
  * [12] Tomasz Krawczyk and Bartosz Walczak  “On-Line Approach to Off-Line Coloring Problems on Graphs with Geometric Representations”  In _Combinatorica_ 37.6, 2017, pp. 1139–1179
  * [13] I.G. Perepelitsa  “Bounds on the chromatic number of intersection graphs of sets in the plane”  In _Discrete Mathematics_ 262, 2003, pp. 221–227
  * [14] András Sebő, Personal Communication, 2020
  * [15] G. Wegner  “Über eine kombinatorisch geometrische Frage von Hadwiger and Debrunner”  In _Israel J. Math_ 3, 1965, pp. 187–198