Saturation estimates for hp-finite element methods

Abstract

In this paper we will prove saturation estimates for the adaptive $hp$-finite element method for linear, second order partial differential equations. More specifically we will consider a sequence of nested finite element discretizations where we allow for both, local mesh refinement and locally increasing the polynomial order. We will prove that the energy norm of the error on the finer level can be estimated by the sum of a contraction of the old error and data oscillations. We will derive estimates of the contraction factor which are explicit with respect to the local mesh width and the local polynomial degree. In order to cover $p$-refinement of finite element spaces new polynomial projection operators will be introduced and new polynomial inverse estimates will be derived.

Appendices

Appendix 1: Lower bound for the constant $c_{\pi }$: numerical experiments

In this appendix we will invest the dependence of the stability constant $c_{\pi }$ of the polynomial projection operator $\Pi _{z}$ [cf. (5.8)] on the polynomial degree $p$. We consider mainly two cases: pure $p$-refinement and $h$-refinement.

1.1 $p$-Refinement

First, we will rewrite the definition of $c_{\pi }$ as an algebraic eigenvalue problem which we will solve numerically. We have performed numerical experiments for the two-dimensi- onal setting on stars as described in this paper but also considered the one-dimensional case where $\omega _{z}$ consists of the two intervals which have $z$ as a common endpoint.

1.1.1 Equivalent formulation

The goal is to investigate the dependence of the constant

$$\begin{aligned} c_{\pi }:=\inf _{v\in \mathbb {P}_{p}\left( \omega _{z}\right) \backslash \left\{ 0\right\} }\frac{\left( v,\Pi _{z}^{p}v\right) _{L^{2}(\omega _{z})} }{\left\| \varPhi _{z}^{1/2}v\right\| _{L^{2}\left( \omega _{z}\right) }^{2}} \end{aligned}$$

(7.1)

on the polynomial degree $p$ numerically. Let $d$ denote the spatial dimension. Let $\omega _{z}$ consists of $q\ge d$ simplices $K_{i}$, $1\le i\le q$.

By employing a global affine map we can pull back the star $\omega _{z}$ to a reference configuration, where $K_{1}=\widehat{K}$ is the unit simplex, on the expense that $c_{\pi }$ in (7.1) depends additionally on the shape regularity of $K_{1}$. Let $\chi _{i}:\widehat{K}\rightarrow K_{i}$ denote affine bijection with the special choice $\chi _{1}=\mathrm{id}$. Then,

$$\begin{aligned} \left\| \varPhi _{z}^{1/2}v\right\| _{L^{2}\left( \omega _{z}\right) } ^{2}=\sum _{i=1}^{q}\frac{\left| K_{i}\right| }{\left| \widehat{K}\right| }\int _{\widehat{K}}\widehat{\varPhi _{K}}\widehat{v}_{i}^{2}, \end{aligned}$$

where $\widehat{v}_{i}=v\circ \chi _{i}\$and $\varPhi _{\widehat{K}}$ denotes the product of barycentric coordinates. Let $\left( P_{n}\right) _{n\in \iota _{p}}$ denote a basis of $\mathbb {P}_{p}\left( \omega _{z}\right)$ for a suitable index set $\iota _{p}$. We write

$$\begin{aligned} v=\sum _{n=0}^{p}v_{n}P_{n} \end{aligned}$$

(7.2)

and obtain

$$\begin{aligned} \left\| \varPhi _{z}^{1/2}v\right\| _{L^{2}\left( \omega _{z}\right) } ^{2}=\mathbf {v}^{\intercal }\mathbf {M}^{\left( p\right) }\mathbf {v}, \end{aligned}$$

where

$$\begin{aligned}&\left( \mathbf {M}_{i}^{\left( p\right) }\right) _{n,m}\!:=\!\frac{\left| K_{i}\right| }{\left| \widehat{K}\right| }\int _{\widehat{K} }\widehat{\varPhi _{K}}\left( P_{n}\circ \chi _{i}\right) \left( P_{m}\circ \chi _{i}\right) ,\quad \!n,m\in \iota _{p}\\&\quad \text {and}\quad \mathbf {M}^{\left( p\right) }:=\sum _{i=1}^{q}\mathbf {M}_{i}^{\left( p\right) }. \end{aligned}$$

For the special case that $\widehat{\varPhi _{\widehat{K}}}$ is the polynomial bubble function we can choose an orthogonal basis for $\mathbb {P}_{p}\left( \widehat{K}\right)$ (cf. [23, 31]) so that $\mathbf {M}_{1}^{\left( p\right) }$ is a diagonal matrix.

In order to invest (7.1) we introduce a matrix representation of $\Pi _{z}^{p}v$ with $v$ as in (7.2) via the ansatz

$$\begin{aligned} \Pi _{z}^{p}v|_{K_{i}}=\sum _{m\in \iota _{p}}w_{m,i}P_{m}\circ \chi _{i}^{-1} \end{aligned}$$

The coefficients $\mathbf {w}_{i}=(w_{m,i})_{m\in \iota _{p}}$ are determined via

$$\begin{aligned}&\mathbf {w}_{i}\!=\!\left( {\mathbf {M}_{1}^{\left( p-1\right) }}\right) ^{-1}\mathbf {W}_{i}\mathbf {v\quad }\text {with}\\&\left( \mathbf {W} _{i}\right) _{m,n}\!:=\!\frac{\left| K_{i}\right| }{\left| \widehat{K}\right| }\int _{\widehat{K}}\widehat{\varPhi _{K}}P_{n}\circ \chi _{i}P_{m}\quad \text {for}\quad m\in \iota _{p-1},n\in \iota _{p}. \end{aligned}$$

Hence,

$$\begin{aligned} \left( v,\Pi _{z}^{p}v\right) _{z}=\mathbf {v}^{\intercal }\mathbf {Bv\quad }\text {with}\quad \mathbf {B}:=\sum _{i=1}^{q}\mathbf {W}_{i}^{\intercal }\left( {\mathbf {M}_{1,1}^{\left( p-1\right) }}\right) ^{-1}\mathbf {W}_{i} \end{aligned}$$

so that the constant $c_{\pi }$ has the algebraic representation

$$\begin{aligned} c_{\pi }=\inf _{\mathbf {v}\in \mathbb {R}^{\iota _{p}}}\frac{\mathbf {v}^{\intercal }\mathbf {Bv}}{\mathbf {v}^{\intercal }\mathbf {M}^{\left( p\right) }\mathbf {v} }. \end{aligned}$$

Hence, $c_{\pi }$ is the smallest eigenvalue of

$$\begin{aligned} \left( {\mathbf {M}^{\left( p\right) }}\right) ^{-1/2}\mathbf {B}\left( {\mathbf {M}^{\left( p\right) }}\right) ^{-1/2}. \end{aligned}$$

1.1.2 The one-dimensional case

In this case we have $\widehat{K}=\left[ -1,1\right]$ and $P_{n}$ are the Jacobi polynomials $P_{n}^{\left( 1,1\right) }$ which are defined as follows

$$\begin{aligned} P_{n}^{\left( \alpha ,\beta \right) }(x)=\frac{(2)_{n}}{n!}\left. _{2} F_{1}\right. \!\left( \begin{matrix} {-n,n+\alpha +\beta +1}\\ \alpha +1 \end{matrix} ;\frac{1-x}{2}\right) , \end{aligned}$$

where $\left( \cdot \right) _{n}$ is Pochhammer’s symbol and $\left. _{2}F_{1}\right.$ is the terminating Gauss hypergeometric function

$$\begin{aligned} \left. _{2}F_{1}\right. \!\left( \genfrac{}{}{0.0pt}{}{-n,b}{c} ;z\right) =\sum _{k=0}^{n}\frac{(-n)_{k}(b)_{k}}{(c)_{k}k!}z^{k}. \end{aligned}$$

We consider $K_{1}=\widehat{K}$ and $K_{2}=\left[ 1,1+\delta \right]$ for some $\delta >0$. Note that $\mathbf {M}_{1}^{\left( p\right) }$ in this case is given by

$$\begin{aligned} \mathbf {M}_{1}^{\left( p\right) }=\mathrm{diag}\left[ 8\frac{\left( n+1\right) }{\left( 2n+3\right) \left( n+2\right) }:n\in \iota _{p}\right] . \end{aligned}$$

The mapping $\chi _{2}$ is defined by

$$\begin{aligned} \chi _{2}\left( \hat{x}\right) =\frac{1-\hat{x}}{2}+\frac{1+\hat{x}}{2}\left( 1+\delta \right) . \end{aligned}$$

To observe the behaviour of $c_{\pi }$ with respect to $p$ and $\delta$, we consider three different cases: $\delta =0.5,\,\delta =1,\,\delta =2,\,\delta =4$. The following observations can be obtained from Fig. 4:

• $c_{\pi }$ converges to a positive constant with respect to $p$,

• $c_{\pi }$ is properly bounded from below,

• $c_{\pi }$ is decreasing as $\delta$ goes to zero.

Fig. 4

Performance of $c_{\pi }$ versus $p$ for the one-dimensional case

1.1.3 The two-dimensional case

Now we consider Jacobi bivariate polynomials as our basis functions on the reference triangle, which are defined as follows:

$$\begin{aligned} P_{n,k}^{1,1,1}(x,y):=(1-x)^{k}P_{n-k}^{(1,3+2k)}(1-2x)P_{k}^{(1,1)}\left( 1-\frac{2y}{1-x}\right) , \end{aligned}$$

which is a polynomial of degree $n$ in $x$ and $y$.

We study different triangulations. Again we assume that $K_{1}$ is the unit simplex and the common point of all triangles in the patch is $(0,0)$. The meshes consist of the following nodes and are illustrated in Fig. 5:

$$\begin{aligned}&v_{1}=\{(0,0),(1,0),(0,1),(-1,1),(-1,0),(0,-1),(1,-1)\},\nonumber \\&v_{2} =\{(0,0),(1,0),(0,1),(-1,1),(-2,0),(-2,-1),(-1,-3),\nonumber \\&\qquad (0,-3),(1,-1)\},\nonumber \\&v_{3} =\{(0,0),(1,0),(0,1),(-1,1),(-3,0),(-4,-2),(-3,-3),\nonumber \\&\qquad (-1,-4),(0,-4),(1,-2)\},\\&v_{4}=\{(0,0),(1,0),(0,1),(-1,-1)\},\nonumber \\&v_{5}=\{(0,0),(1,0),(0,1),(-0.1,-0.2)\},\nonumber \\&v_{6}=\{(0,0),(1,0),(0,1),(-4,3),(-4,0),(-4,-4),(0,-4),\nonumber \\&\qquad (1,-0.1)\}.\nonumber \end{aligned}$$

(7.3)

Fig. 5

Illustration of the geometric configuations described in (7.3)

Fig. 6

Performance of $c_{\pi }$ versus $p$ for the two-dimensional cases

Figure 6 shows the behaviour of $c_{\pi }$ with respect to $p$ in each case and we summarize the main observations.

• (a) In the first three cases, i.e., the number of triangles (at least six) is varying while the shape regularity constant is always moderately bounded, the lower bound of $c_{\pi }$ is approximately $1$. It also shows that the constant $c_{\pi }$ is robust with respect to the elongation of the triangles which is in analogy to the one-dimensional observation ($\delta$ increases).

• (b) If we consider the minimal number (three) elements, again, with moderate shape regular constant, we still get a proper lower bound. Recall that the dimension of the image space $\mathbb {P}_{p-1}\left( \mathcal {T} _{z}\right)$ [in (5.1)] increases with the number of triangles so that we expect that the constant $c_{\pi }$ becomes larger with increasing number of triangles.

• (c) On the other hand, if we consider the minimal configuration with only three triangles and large shape regularity constant (the area of the triangles is highly varying) as described by $v_{5}$, then the constant $c_{\pi }$ becomes smaller as expected.

• (d) Configuration $v_{6}$ supports the statement that, if the space $\mathbb {P}_{p-1}\left( \mathcal {T}_{z}\right)$ is large enough, then a few tiny elements can be still harmless. We can see that these numerical examples confirm our hypothesis that $c_{\pi }$ depends on the shape regularity of our meshes but does not depend on $p$.

1.2 $h$-Refinement

In this section we study the similar problem as in previous section but with $h$-refinement instead of $p$-refinement. In other word, we apply one level of regular $h$-refinement on each mesh and observe the behaviour of the constant $c_{\pi }$ with respect to $p$ on the refined mesh. To be able to make a comparison between the results, we take the same patches as in previous section. From the definition of $\varPhi _{z}$ for this case we have

$$\begin{aligned}&c_{\pi }=\\&\inf _{v\in \mathbb {P}_{p-1}(K)}\frac{\sum _{K\subset \omega _{z}}\int _{K}v\varPhi _{K,K}^{(1)}v+\int _{K}v\left( \varPhi _{K,K}^{(1)}+\varPhi _{K} ^{(2)}\right) \prod _{K}^{p-1}v}{{\large \sum _{K\subset \omega _{z}}\int _{K}v\left( \varPhi _{K,K}^{(1)}+\varPhi _{K}^{(2)}\right) v}}, \end{aligned}$$

where $\varPhi _{K,K}^{(1)}$ and $\varPhi _{K}^{(2)}$ are piecewise linear and quadratic functions defined as in (5.4). Figure 7 shows the behaviour of $c_{\pi }$ for the same patches with respect to $p$. It supports our hypothesis and shows the similar behaviour as in $p$-version. Also here we observe that $c_{\pi }$ does not depend on $p$, but it only depends on the shape regularity of the mesh.

Fig. 7

Behaviour of $c_{\pi }$ versus $p$ with one level of $h$-refinement

Appendix 2: Polynomial inverse estimates

We start with a one-dimensional estimate.

Lemma 2

For $a<b$, let $\varPhi _{\left[ a,b\right] }\left( x\right) =\frac{\left( x-a\right) \left( b-x\right) }{\left( b-a\right) ^{2}}$ denote the one-dimensional bubble function. Then,

$$\begin{aligned} \left\| \left( \varPhi _{\left[ a,b\right] }v\right) ^{\prime }\right\| _{L^{2}\left( \left[ a,b\right] \right) }\le&C\frac{p+1}{b-a}\left\| \varPhi _{\left[ a,b\right] }^{1/2}v\right\| _{L^{2}\left( \left[ a,b\right] \right) }\\&\qquad \quad \qquad \qquad \forall v\in {\mathbb {P}}_{p}\left( \left[ a,b\right] \right) . \end{aligned}$$

Proof

We first prove the result for $\left( a,b\right) =\left( 0,1\right)$. Observe that $\left\| \varPhi _{\left[ 0,1\right] }^{\prime }\right\| _{L^{\infty }\left( \left[ 0,1\right] \right) }=1$ so that Leibniz rule gives us

$$\begin{aligned} \left\| \left( \varPhi _{\left[ 0,1\right] }v\right) ^{\prime }\right\| _{L^{2}\left( \left[ 0,1\right] \right) }&\le \left\| \varPhi _{\left[ 0,1\right] }^{\prime }v\right\| _{L^{2}\left( \left[ 0,1\right] \right) }+\left\| \varPhi _{\left[ 0,1\right] }v^{\prime }\right\| _{L^{2}\left( \left[ 0,1\right] \right) }\\&\le \left\| v\right\| _{L^{2}\left( \left[ 0,1\right] \right) }+\left\| \varPhi _{\left[ 0,1\right] } v^{\prime }\right\| _{L^{2}\left( \left[ 0,1\right] \right) }. \end{aligned}$$

For the first term, we apply [27, Lemma 2.4 with $\alpha =0$ and $\beta =1$ ] and for the second term [27, Lemma 2.4 with $\delta =1$ ] to obtain

$$\begin{aligned} \left\| \left( \varPhi _{\left[ 0,1\right] }v\right) ^{\prime }\right\| _{L^{2}\left( \left[ 0,1\right] \right) }\le C\left( p+1\right) \left\| \varPhi _{\left[ 0,1\right] }^{1/2}v\right\| _{L^{2}\left( \left[ 0,1\right] \right) }. \end{aligned}$$

(8.1)

The result then follows via a scaling argument.$\square$

Corollary 5

Let $a<b$ and $\varPhi _{\left[ a,b\right] }$ be as in Lemma 2. Let $\Psi _{\left[ a,b\right] }\in W^{1,\infty }\left( \left[ a,b\right] \right)$ be a function with the properties

$$\begin{aligned}&\left| \Psi _{\left[ a,b\right] }\right| \le C_{11}\varPhi _{\left[ a,b\right] }\quad \text {pointwise}\quad \text {and}\\&\quad \left\| \Psi _{\left[ a,b\right] }^{\prime }\right\| _{L^{\infty }\left( \left[ a,b\right] \right) }\le \frac{C_{12}}{b-a}. \end{aligned}$$

Then

$$\begin{aligned}&\left\| \left( \Psi _{\left[ a,b\right] }v\right) ^{\prime }\right\| _{L^{2}\left( \left[ a,b\right] \right) }\!\le \! C\left( C_{11} +C_{12}\right) \frac{p+1}{b-a}\left\| \varPhi _{\left[ a,b\right] } ^{1/2}v\right\| _{L^{2}\left( \left[ a,b\right] \right) }\\&\quad \forall \, v\in {\mathbb {P}}_{p}\left( \left[ a,b\right] \right) . \end{aligned}$$

Proof

Leibniz’ rule gives us

$$\begin{aligned} \left\| \left( \Psi _{\left[ a,b\right] }v\right) ^{\prime }\right\| _{L^{2}\left( \left[ a,b\right] \right) }&\le \left\| \Psi _{\left[ a,b\right] }^{\prime }v\right\| _{L^{2}\left( \left[ a,b\right] \right) }+\left\| \Psi _{\left[ a,b\right] }v^{\prime }\right\| _{L^{2}\left( \left[ a,b\right] \right) }\\ \le&\frac{C_{12}}{b{-}a}\left\| v\right\| _{L^{2}\left( \left[ a,b\right] \right) }+C_{11}\left\| \varPhi _{\left[ a,b\right] }v^{\prime }\right\| _{L^{2}\left( \left[ a,b\right] \right) }\\ \le&C\left( C_{11}+C_{12}\right) \frac{p+1}{b-a}\left\| \varPhi _{\left[ a,b\right] }^{1/2}v\right\| _{L^{2}\left( \left[ a,b\right] \right) }, \end{aligned}$$

where the last inequality follows as (8.1).$\square$

The two-dimensional version is formulated next. The estimates are similar to those in [39, Sec. 3.6] but differ by powers of the weight functions in the right-hand side and also by the choice of the weight function in Lemma 4. The proofs follow the lines of the proofs in [39, Prop. 3.46] and also employs tools from [28, Appendix D].

Lemma 3

Let $K$ denote a triangle and let $\varPhi _{K}$ be the cubic bubble function as defined in (5.3). Then, it holds for all $v\in {\mathbb {P}}_{p}\left( K\right)$

$$\begin{aligned} \left\| \nabla \left( \varPhi _{K}v\right) \right\| _{L^{2}\left( K\right) }\le C\frac{p+1}{h_{K}}\left\| \varPhi _{K}^{1/2}v\right\| _{L^{2}\left( K\right) }. \end{aligned}$$

Proof

Let $K=\widehat{K}$ be the two-dimensional reference triangle. Note that

$$\begin{aligned} \varPhi _{\widehat{K}}\left( x_{1},x_{2}\right) =\varPhi _{\left[ 0,1-x_{1}\right] }\left( x_{2}\right) \left( 1-x_{1}\right) \varPhi _{\left[ 0,1\right] }\left( x_{1}\right) \end{aligned}$$

with $\varPhi _{\left[ a,b\right] }$ as in Lemma 2. First, we consider the derivative with respect to $x_{2}$ and obtain

$$\begin{aligned}&\left\| \partial _{2}\left( \varPhi _{\widehat{K}}v\right) \right\| _{L^{2}\left( \widehat{K}\right) }^{2} \\&\quad =\int _{0}^{1}\left( \int _{0}^{1-x_{1}}\left( \partial _{2}\left( \varPhi _{\widehat{K}}\left( x_{1},x_{2}\right) v\left( x_{1},x_{2}\right) \right) \right) ^{2} dx_{2}\right) dx_{1}\\&\quad =\int _{0}^{1}\varPhi _{\left[ 0,1\right] }^{2}\left( x_{1}\right) \left( 1-x_{1}\right) ^{2}\\&\quad \times \left( \int _{0}^{1-x_{1}}\left( \frac{\partial }{\partial x_{2}}\left( \varPhi _{\left[ 0,1-x_{1}\right] }\left( x_{2}\right) v\left( x_{1},x_{2}\right) \right) \right) ^{2}dx_{2}\right) dx_{1}. \end{aligned}$$

We then get

$$\begin{aligned}&\left( 1-x_{1}\right) ^{2}\int _{0}^{1-x_{1}}\left( \frac{\partial }{\partial x_{2}}\left( \varPhi _{\left[ 0,1-x_{1}\right] }\left( x_{2}\right) v\left( x_{1},x_{2}\right) \right) \right) ^{2}dx_{2}\\&\quad =\left( 1-x_{1}\right) ^{2}\left\| \left( \varPhi _{\left[ 0,1-x_{1}\right] }v\left( x_{1},\cdot \right) \right) ^{\prime }\right\| _{L^{2}\left( 0,1-x_{1}\right) }^{2}\\&\overset{\text {Lem. 2}}{\le }C\left( p+1\right) ^{2}\left\| \varPhi _{\left[ 0,1-x_{1}\right] }^{1/2}v\left( x_{1} ,\cdot \right) \right\| _{L^{2}\left( 0,1-x_{1}\right) }^{2}. \end{aligned}$$

Since $\varPhi _{\left[ 0,1\right] }^{2}\left( x_{1}\right) \varPhi _{\left[ 0,1-x_{1}\right] }\left( x_{2}\right) \le \varPhi _{\widehat{K}}\left( x_{1},x_{2}\right)$ we end up with

$$\begin{aligned}&\left\| \partial _{2}\left( \varPhi _{\widehat{K}}v\right) \right\| _{L^{2}\left( \widehat{K}\right) }^{2} \le C\left( p+1\right) ^{2} \\&\quad \times \int _{0}^{1}\int _{0}^{1-x_{1}}\varPhi _{\left[ 0,1\right] }^{2}\left( x_{1}\right) \varPhi _{\left[ 0,1-x_{1}\right] }\left( x_{2}\right) v^{2}\left( x_{1},x_{2}\right) dx_{2}dx_{1}\\&\quad \le C\left( p+1\right) ^{2}\left\| \varPhi _{\widehat{K}}^{1/2} v\right\| _{L^{2}\left( \widehat{K}\right) }^{2}. \end{aligned}$$

Since $\widehat{K}$, $\psi _{\widehat{K}}$, and the integral are invariant under permutations of the coordinates, the same estimate holds for the other partial derivatives.$\square$

Lemma 4

Let $K$ be regularly $h$-refined and let $\varPhi _{K}^{\left( 2\right) }$ be as explained in Definition 3 and illustrated in Fig. 3. Then, it holds for all $v\in {\mathbb {P}}_{p}\left( K\right)$

$$\begin{aligned} \left\| \nabla \left( \varPhi _{K}^{\left( 2\right) }v\right) \right\| _{L^{2}\left( K\right) }\le C\frac{p+1}{h_{K}}\left\| \sqrt{\varPhi _{K}^{\left( 2\right) }}v\right\| _{L^{2}\left( K\right) }. \end{aligned}$$

Proof

Via an affine transformation it suffices to prove the result for the reference element $K_{1}=\widehat{K}$ and $K_{2}=\mathrm{conv}\left\{ \genfrac(){0.0pt}1{0}{0}\right.$, $\left. \genfrac(){0.0pt}1{0}{1},\genfrac(){0.0pt}1{-1}{0}\right\}$. The common edge is $E=\left\{ 0\right\} \times \left( 0,1\right)$. Let $K=K_{1}\cup K_{2}$. The edge bubble $\varPhi _{K}^{\left( 2\right) }$ [cf. (5.4)] is given by

$$\begin{aligned} \varPhi _{K}^{\left( 2\right) }\left( x_{1},x_{2}\right) =x_{2}\left( 1-\left| x_{1}\right| -x_{2}\right) . \end{aligned}$$

We first consider the derivative with respect to $x_{2}$. Let

$$\begin{aligned} \varPhi _{\left[ 0,1-\left| x_{1}\right| \right] }\left( x_{2}\right) =\frac{x_{2}\left( 1-\left| x_{1}\right| -x_{2}\right) }{\left( 1-\left| x_{1}\right| \right) ^{2}}, \end{aligned}$$

i.e., $\varPhi _{\left[ 0,1-\left| x_{1}\right| \right] }$ is the one-dimensional bubble function for $\left[ 0,1-\left| x_{1}\right| \right]$ and satisfies $\varPhi _{K}^{\left( 2\right) }=\left( 1-\left| x_{1}\right| \right) ^{2}\varPhi _{\left[ 0,1-\left| x_{1}\right| \right] }$. Hence,

$$\begin{aligned}&\left\| \partial _{2}\left( \varPhi _{K}^{\left( 2\right) }v\right) \right\| _{L^{2}\left( K\right) }^{2} =\int _{-1}^{1}\left( 1-\left| x_{1}\right| \right) ^{4}\nonumber \\&\quad \quad \qquad \times \int _{0}^{1-\left| x_{1}\right| }\left( \partial _{2}\left( \varPhi _{\left[ 0,1-\left| x_{1}\right| \right] }\left( x_{2}\right) v\left( x_{1},x_{2}\right) \right) \right) ^{2}dx_{2}dx_{1}\nonumber \\&\quad \quad \overset{\text {Lem. 2}}{\le }C\left( p+1\right) ^{2} \int _{-1}^{1}\left( 1-\left| x_{1}\right| \right) ^{2}\nonumber \\&\quad \quad \qquad \times \int _{0}^{1-\left| x_{1}\right| }\varPhi _{\left[ 0,1-\left| x_{1}\right| \right] }\left( x_{2}\right) v^{2}\left( x_{1} ,x_{2}\right) dx_{2}dx_{1}\nonumber \\&\qquad \quad =C\left( p+1\right) ^{2}\int _{K}\varPhi _{K}^{\left( 2\right) }v^{2}. \end{aligned}$$

(8.2)

Next, we will estimate the derivative with respect to $x_{1}$. We split the triangle into the two regions

$$\begin{aligned} D_{1}&:= \left\{ \genfrac(){0.0pt}1{x_{1}}{x_{2}}\in K:x_{2}\le \frac{1}{2}\right\} \quad \text {and}\\ D_{2}&:= \mathrm{conv}\left\{ \genfrac(){0.0pt}1{1/2}{1/2},\genfrac(){0.0pt}1{0}{1},\genfrac(){0.0pt}1{-1/2}{1/2}\right\} . \end{aligned}$$

In addition, we will need

$$\begin{aligned} D_{3}:=\mathrm{conv}\left\{ \genfrac(){0.0pt}1{0}{0},\genfrac(){0.0pt}1{1/2}{1/2},\genfrac(){0.0pt}1{0}{1},\genfrac(){0.0pt}1{-1/2}{1/2}\right\} . \end{aligned}$$

On $D_{1}$ we obtain

$$\begin{aligned}&\left\| \partial _{1}\left( \varPhi _{K}^{\left( 2\right) }v\right) \right\| _{L^{2}\left( D_{1}\right) }^{2} \le \int _{0}^{1/2}x_{2} ^{2}\left( 1-x_{2}\right) ^{2}\nonumber \\&\qquad \quad \times \int _{x_{2}-1}^{1-x_{2}}\left( \partial _{1}\left( \frac{1-\left| x_{1}\right| -x_{2}}{1-x_{2}}v\left( x_{1},x_{2}\right) \right) \right) ^{2}dx_{1}dx_{2}\nonumber \\&\overset{\text {Lem. 5}}{\le }C\left( p+1\right) ^{2}\int _{0}^{1/2}\int _{x_{2}-1}^{1-x_{2}}\frac{x_{2}}{1-x_{2}}\nonumber \\&\qquad \quad \times \varPhi _{K}^{\left( 2\right) }\left( x_{1},x_{2}\right) v^{2}\left( x_{1},x_{2}\right) dx_{1}dx_{2}\nonumber \\&\quad \le C\left( p+1\right) ^{2}\left\| \sqrt{\varPhi _{K}^{\left( 2\right) }}v\right\| _{L^{2}\left( D_{1}\right) }^{2}, \end{aligned}$$

(8.3)

since $x_{2}/\left( 1-x_{2}\right) \le 1$ on $D_{1}$.

On $D_{2}$, we observe that

$$\begin{aligned} \frac{1}{2}\left\| \partial _{1}\left( \varPhi _{K}^{\left( 2\right) }v\right) \right\| _{L^{2}\left( D_{2}\right) }^{2}\le \left\| v\partial _{1}\varPhi _{K}^{\left( 2\right) }\right\| _{L^{2}\left( D_{2}\right) }^{2}+\left\| \varPhi _{K}^{\left( 2\right) }\partial _{1}v\right\| _{L^{2}\left( D_{2}\right) }^{2}. \nonumber \\ \end{aligned}$$

(8.4)

Let $d:D_{3}\rightarrow {\mathbb {R}}$ be defined by

$$\begin{aligned} d\left( x_{1},x_{2}\right) =c_{3}\mathrm{dist}\,\left( \left( x_{1},x_{2}\right) ^{\intercal },\partial D_{3}\right) \end{aligned}$$

where the scaling $c_{3}$ is chosen such that $d$ interpolates $\varPhi _{K}^{\left( 2\right) }$ at the vertices of the two triangles $K_{m}\cap D_{3}$, $m=1,2$. Note that

$$\begin{aligned} d\le \varPhi _{K}^{\left( 2\right) }\!\le \!1\,\text {pointwise in}\, D_{3}\, \text {and}\, \varPhi _{K}^{\left( 2\right) }\!\le \!2d\,\text {pointwise in}\, D_{2}. \end{aligned}$$

Since $\left\| \partial _{1}\varPhi _{K}^{\left( 2\right) }\right\| _{L^{\infty }\left( K\right) }\le C$ we obtain for the first term in (8.4) as in (5.16b)

$$\begin{aligned} \left\| v\partial _{1}\varPhi _{K}^{\left( 2\right) }\right\| _{L^{2}\left( D_{2}\right) }&\le C\left\| v\right\| _{L^{2}\left( D_{2}\right) }\nonumber \\&\le C\left\| v\right\| _{L^{2}\left( D_{3}\right) }\le C\left( p+1\right) \left\| d^{1/2}v\right\| _{L^{2}\left( D_{3}\right) }\nonumber \\&\le C\left( p+1\right) \left\| \sqrt{\varPhi _{K}^{\left( 2\right) }}v\right\| _{L^{2}\left( D_{3}\right) }. \end{aligned}$$

(8.5a)

For the second term in (8.4) we get, again, as in (5.16b)

$$\begin{aligned}&\left\| \varPhi _{K}^{\left( 2\right) }\partial _{1}v\right\| _{L^{2}\left( D_{2}\right) }\le 2\left\| d\partial _{1}v\right\| _{L^{2}\left( D_{3}\right) }\nonumber \\&\,\,\,\qquad \quad \,\overset{\text {[26, (23) with}\, \delta =1]}{\le }C\left( p+1\right) \left\| \sqrt{d}v\right\| _{L^{2}\left( D_{3}\right) }\nonumber \\&\qquad \qquad \qquad \quad \,\,\le C\left( p+1\right) \left\| \sqrt{\varPhi _{K}^{\left( 2\right) }}v\right\| _{L^{2}\left( D_{3}\right) }. \end{aligned}$$

(8.5b)

The combination of (8.4) and (8.5) yields

$$\begin{aligned} \left\| \partial _{1}\left( \varPhi _{K}^{\left( 2\right) }v\right) \right\| _{L^{2}\left( D_{2}\right) }^{2}\le C\left( p+1\right) ^{2}\left\| \sqrt{\varPhi _{K}^{\left( 2\right) }}v\right\| _{L^{2}\left( D_{3}\right) }^{2}. \end{aligned}$$

(8.6)

The combination of (8.2), (8.3), and (8.6) yields the assertion.$\square$

The following lemma is illustrated in Fig. 8.

Fig. 8

Reference triangle $\widehat{K}$ which is split into $\widehat{K} _{1}$ and $\widehat{K}_{2}$. The shaded regions illustrate the integration domains in the splitting of the integral in (8.8)

Lemma 5

Let $\widehat{K}$ be the reference triangle split into $\widehat{K}_{1}=\mathrm{conv}\left( \genfrac(){0.0pt}1{0}{0},\genfrac(){0.0pt}1{a}{0},\genfrac(){0.0pt}1{0}{1}\right)$ and $\widehat{K}_{2}=\mathrm{conv}\left( \genfrac(){0.0pt}1{a}{0},\genfrac(){0.0pt}1{1}{0},\genfrac(){0.0pt}1{0}{1}\right)$ for some $a\in \left] 0,1\right[$. Let $\varphi _{E}^{\mathrm{lin}\,}$ denote the continuous, piecewise linear function which has value $1$ at $\genfrac(){0.0pt}1{a}{0}$ and vanishes at $\partial \widehat{K}\backslash E_{1}$ with $E_{1}=\left[ 0,1\right] \times \left\{ 0\right\}$. Then, for any polynomial $v\in {\mathbb {P}}_{p}$ which is constant with respect to $x_{2}$ it holds

$$\begin{aligned} \left\| \sqrt{\varphi _{E}^{\mathrm{lin}\,}}v\right\| _{L^{2}\left( \widehat{K}\right) }&\le C\left\| \sqrt{\varphi _{E} ^{\mathrm{lin}\,}}v\right\| _{L^{2}\left( E_{1}\right) },\\ \left\| \nabla \left( \varphi _{E}^{\mathrm{lin}\,}v\right) \right\| _{L^{2}\left( \widehat{K}\right) }&\le C\left( p+1\right) \left\| \sqrt{\varphi _{E}^{\mathrm{lin}\,}}v\right\| _{L^{2}\left( E_{1}\right) }. \end{aligned}$$

Proof

We prove this lemma only for $a=1/2$ to reduce technicalities. The arguments apply verbatim for the general case. The function $\varphi _{E} ^{\mathrm{lin}\,}$ and its partial derivatives are given by

$$\begin{aligned} \varphi _{E}^{\mathrm{lin}\,}\left( x_{1},x_{2}\right)&=\left\{ \begin{array}{ll} 2x_{1} &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{1},\\ 2\left( 1-x_{1}-x_{2}\right) &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{2}, \end{array} \right. \\ \partial _{1}\varphi _{E}^{\mathrm{lin}\,}\left( x_{1},x_{2}\right)&=\left\{ \begin{array}{ll} 2 &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{1},\\ -2 &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{2}. \end{array} \right. \\ \quad \partial _{2}\varphi _{E}^{\mathrm{lin}\,}\left( x_{1} ,x_{2}\right)&=\left\{ \begin{array}{ll} 0 &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{1},\\ -2 &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{2}, \end{array} \right. \end{aligned}$$

Since $v\in {\mathbb {P}}_{p}$ is constant with respect to $x_{2}$ we write, with a slight abuse of notation, $v\left( x_{1},x_{2}\right) =v\left( x_{1}\right)$. Hence,

$$\begin{aligned} \left\| \sqrt{\varphi _{E}^{\mathrm{lin}\,}}v\right\| _{L^{2}\left( \widehat{K}\right) }^{2}=\int _{0}^{1}v^{2}\left( x_{1}\right) \left( \int _{0}^{1-x_{1}}\varphi _{E}^{\mathrm{lin}\,}\left( x_{1},x_{2}\right) dx_{2}\right) dx_{1}. \end{aligned}$$

The result of the inner integration is

$$\begin{aligned} r\left( x_{1}\right)&:=\int _{0}^{1-x_{1}}\varphi _{E}^{\mathrm{lin}\, }\left( x_{1},x_{2}\right) dx_{2}\\&\,=\left\{ \begin{array}{l} 2\left( \int _{0}^{1-2x_{1}}x_{1}dx_{2}+\int _{1-2x_{1}}^{1-x_{1}}\left( 1-x_{1}-x_{2}\right) dx_{2}\right) \\ 2\int _{0}^{1-x_{1}}\left( 1-x_{1}-x_{2}\right) dx_{2} \end{array} \right. \\&\,=\left\{ \begin{array}{lc} x_{1}\left( 2-3x_{1}\right) &{} x_{1}\le 1/2,\\ \left( 1-x_{1}\right) ^{2} &{} x_{1}>1/2. \end{array} \right. \end{aligned}$$

Since $r\le \varphi _{E}^{\mathrm{lin}\,}\left( \cdot ,0\right)$ pointwise on $\left[ 0,1\right]$, the first assertion follows.

Next, we investigate the derivative with respect to $x_{2}$. It holds

$$\begin{aligned} \partial _{2}\left( \varphi _{E}^{\mathrm{lin}\,}v\right) =v\partial _{2}\varphi _{E}^{\mathrm{lin}\,}=v\times \left\{ \begin{array}{ll} 0 &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{1},\\ -2 &{} \left( x_{1},x_{2}\right) \in \widehat{K}_{2}. \end{array} \right. \end{aligned}$$

Thus,

$$\begin{aligned}&\int _{\widehat{K}}\left( \partial _{2}\left( \varphi _{E}^{\mathrm{lin}\, }v\right) \right) ^{2} =\int _{\widehat{K}}\left( v\partial _{2} \varphi _{E}^{\mathrm{lin}\,}\right) ^{2}\le 4\int _{\widehat{K}}v^{2} \le 4\int _{0}^{1}v^{2}\nonumber \\&\overset{\text {[26, Lemma 2.4 with}\, \alpha =0\, \text {and}\, \beta =1]}{\le }4\left( p+1\right) ^{2}\int _{0}^{1}\varPhi _{\left[ 0,1\right] }v^{2}\nonumber \\&\le 4C\left( p+1\right) ^{2}\int _{0}^{1}\varphi _{E}^{\mathrm{lin}\,}v^{2}. \end{aligned}$$

(8.7)

For the derivative with respect to $x_{1}$, we get

$$\begin{aligned} q\left( x_{1},x_{2}\right)&:= \partial _{1}\left( \varphi _{E} ^{\mathrm{lin}\,}\left( x_{1},x_{2}\right) v\left( x_{1}\right) \right) \\&= 2\left\{ \begin{array}{ll} \left( x_{1}v\left( x_{1}\right) \right) ^{\prime } &{} \text {in}\, \widehat{K}_{1},\\ \left( 1-x_{1}-x_{2}\right) v^{\prime }\left( x_{1}\right) -v\left( x_{1}\right) &{} \text {in}\, \widehat{K}_{2}. \end{array}\right. \end{aligned}$$

The function $q$ is on $\widehat{K}_{1}$ and on $\widehat{K}_{2}$, an affine function with respect to $x_{2}$. We split the integral into

$$\begin{aligned}&\int _{0}^{1}\int _{0}^{1-x_{1}}\cdots \nonumber \\&\quad =\int _{0}^{1/2}\int _{0}^{1-2x_{1}} \cdots +\int _{0}^{1/2}\int _{1-2x_{1}}^{1-x_{1}}\cdots +\int _{1/2}^{1}\int _{0}^{1-x_{1}}\nonumber \\&\quad =:W_{1}+W_{2}+W_{3} \end{aligned}$$

(8.8)

and obtain for the summands

$$\begin{aligned}&W_{1} =4\int _{0}^{1/2}\left( 1-2x_{1}\right) \left( \left( x_{1}v\left( x_{1}\right) \right) ^{\prime }\right) ^{2}dx_{1}\\&\quad \,\,\le \int _{0}^{1/2}\left( \left( 2x_{1}v\left( x_{1}\right) \right) ^{\prime }\right) ^{2}dx_{1}\\&\quad \,\, \le \int _{0}^{1}\left( \left( \varphi _{E}^{\mathrm{lin}\,}\left( x_{1},0\right) v\left( x_{1}\right) \right) ^{\prime }\right) ^{2} dx_{1}\\&\,\,\,\overset{\text {Cor. 5}}{\le }C\left( p+1\right) ^{2}\left\| \sqrt{\varphi _{E}^{\mathrm{lin}\,}}v\right\| _{L^{2}\left( \left[ 0,1\right] \right) }^{2}. \end{aligned}$$

For $W_{2}$ and $W_{3}$ we use the fact that the Simpson rule is exact for quadratic polynomials and (8.7) to obtain

$$\begin{aligned} W_{2}&=4\int _{0}^{1/2}\int _{1-2x_{1}}^{1-x_{1}}\left( \left( 1-x_{1}-x_{2}\right) v^{\prime }\left( x_{1}\right) -v\left( x_{1}\right) \right) ^{2}dx_{2}dx_{1}\\&=\frac{2}{3}\int _{0}^{1/2}x_{1}\left( \left( \underset{=\left( x_{1}v\left( x_{1}\right) \right) ^{\prime }-2v\left( x_{1}\right) }{\underbrace{x_{1}v^{\prime }\left( x_{1}\right) -v\left( x_{1}\right) } }\right) ^{2}\!+\!4\left( \underset{\frac{\left( x_{1}v\left( x_{1}\right) \right) ^{\prime }}{2}-\frac{3}{2}v\left( x_{1}\right) }{\underbrace{\frac{x_{1}}{2}v^{\prime }\left( x_{1}\right) -v\left( x_{1}\right) } }\right) ^{2}\right. \\&\qquad \left. \quad +\,v^{2}\left( x_{1}\right) \right) dx_{1}\\&\quad \le \frac{2}{3}\int _{0}^{1/2}x_{1}\left( 4\left( \left( x_{1}v\left( x_{1}\right) \right) ^{\prime }\right) ^{2}+27v^{2}\left( x_{1}\right) \right) dx_{1}\\&\quad \le \frac{4}{3}\int _{0}^{1}\left( \left( \left( \varphi _{E} ^{\mathrm{lin}\,}\left( x_{1},0\right) v\left( x_{1}\right) \right) ^{\prime }\right) ^{2}\right) dx_{1}+9\left\| v\right\| _{L^{2}\left( \left[ 0,1\right] \right) }^{2}\\&\quad \overset{\text {Cor. 5}}{\le }C\left( p+1\right) ^{2}\left\| \sqrt{\varphi _{E} ^{\mathrm{lin}\,}}v\right\| _{L^{2}\left( \left[ 0,1\right] \right) }^{2}. \end{aligned}$$

Finally, for $W_{3}$ we obtain

$$\begin{aligned} W_{3}&=4\int _{1/2}^{1}\int _{0}^{1{-}x_{1}}\left( \left( 1{-}x_{1} {-}x_{2}\right) v^{\prime }\left( x_{1}\right) {-}v\left( x_{1}\right) \right) ^{2}dx_{2}dx_{1}\\&=4\int _{1/2}^{1}\frac{1-x_{1}}{6}\left( \left( \underset{\left( \left( 1-x_{1}\right) v\left( x_{1}\right) \right) ^{\prime }}{\underbrace{\left( 1-x_{1}\right) v^{\prime }\left( x_{1}\right) -v\left( x_{1}\right) } }\right) ^{2}\right. \\&\quad \left. +\,4\left( \underset{\frac{1}{2}\left( \left( \left( 1-x_{1}\right) v\left( x_{1}\right) \right) ^{\prime }-v\left( x_{1}\right) \right) }{\underbrace{\left( \frac{1-x_{1}}{2}\right) v^{\prime }\left( x_{1}\right) -v\left( x_{1}\right) }}\right) ^{2} +v^{2}\left( x_{1}\right) \right) dx_{1}\\&\le 4\int _{1/2}^{1}\frac{1-x_{1}}{6}\left( 3\left( \left( \left( 1-x_{1}\right) v\left( x_{1}\right) \right) ^{\prime }\right) ^{2}+9v\left( x_{1}\right) \right) dx_{1}\\&\le \frac{1}{12}\int _{1/2}^{1}3\left( \left( 2\left( 1-x_{1}\right) v\left( x_{1}\right) \right) ^{\prime }\right) ^{2}+36v^{2}\left( x_{1}\right) dx_{1}\\&\le \frac{1}{4}\int _{0}^{1}\left( \left( \varphi _{E}^{\mathrm{lin}\,}\left( x_{1}\right) v\left( x_{1}\right) \right) ^{\prime }\right) ^{2}dx_{1}+3\left\| v\right\| _{L^{2}\left[ 0,1\right] }^{2}\\&\overset{\text {Cor. 5}}{\le }C\left( p+1\right) ^{2}\left\| \sqrt{\varphi _{E}^{\mathrm{lin}\,}}v\right\| _{L^{2}\left( \left[ 0,1\right] \right) }^{2}. \end{aligned}$$

$\square$

Corollary 6

Let $\widehat{K}$ be the reference triangle and let $\varphi _{E}\left( x_{1},x_{2}\right)$ $=x_{1}\left( 1-x_{1} -x_{2}\right)$ denote the quadratic edge bubble on $\widehat{K}$ for the edge $E_{1}=\left[ 0,1\right] \times \left\{ 0\right\}$. Then, for any polynomial $v\in {\mathbb {P}}_{p}$ which is constant with respect to $x_{2}$ it holds

$$\begin{aligned}&\left\| \sqrt{\varphi _{E}}v\right\| _{L^{2}\left( \widehat{K}\right) } \le C\left\| \sqrt{\varphi _{E}}v\right\| _{L^{2}\left( E_{1}\right) },\\&\left\| \nabla \left( \varphi _{E}v\right) \right\| _{L^{2}\left( \widehat{K}\right) } \le C\left( p+1\right) \left\| \sqrt{\varphi _{E}}v\right\| _{L^{2}\left( E_{1}\right) }. \end{aligned}$$

The proof follows by a simple repetition of the arguments of the proof of Lemma 5.

Lemma 6

Let $K$ be a triangle and $E$ one of its edges. Then, for any of the functions $\varphi _{E}$ (6.4) and corresponding version $\varPhi _{K}$ as in (5.4) it holds

$$\begin{aligned} \left\| \varphi _{E}^{1/2}v\right\| _{L^{2}\left( D\right) }\le C\left( p+1\right) \left\| \varPhi _{K}^{1/2}v\right\| _{L^{2}\left( D\right) }\qquad \forall v\in {\mathbb {P}}_{p}\left( D\right) . \end{aligned}$$

The proof requires two preparatory lemmata and follows the ideas in [28, Appendix D].

Lemma 7

Let $I=\left[ a,b\right]$ for some $a<b$ and let $\omega :I\rightarrow {\mathbb {R}}$ be a weight function which satisfies

$$\begin{aligned}&\exists A,B,D\ge 0\, \text {with}\nonumber \\&\quad \, \left\{ \begin{array}{l} \omega \, \text {is positive in}\, \left] a,b\right[,\\ \omega \left( x\right) \le A\varphi _{a}\left( x\right) +B\varphi _{b}\left( x\right) +D\varPhi _{\left[ a,b\right] }\left( x\right) , \end{array}\right. \,\,P_I \end{aligned}$$

where $\varphi _{b}\left( x\right) =\frac{x-a}{b-a}$, $\varphi _{a} =1-\varphi _{b}$, and $\varPhi _{\left[ a,b\right] }=\varphi _{a}\varphi _{b}$ as in (2). Then, it holds

$$\begin{aligned} \left\| \omega ^{1/2}v\right\| _{L^{2}\left( I\right) }\le C\left( p+1\right) \left\| \sqrt{\omega \varPhi _{\left[ a,b\right] }}v\right\| _{L^{2}\left( I\right) }\qquad \forall v\in {\mathbb {P}}_{p}\left( I\right) , \end{aligned}$$

where $C$ is independent of $p,v,\omega ,a,b$.

Proof

By employing an affine transform it is sufficient to prove the assertion for the unit interval $I=\left[ 0,1\right]$.

• (a) $\omega \left( x\right) =\varPhi _{\left[ 0,1\right] }\left( x\right) =x\left( 1-x\right)$. We may apply standard inverse estimates to obtain

  $$\begin{aligned} \left\| \varPhi _{\left[ 0,1\right] }^{1/2}v\right\| _{L^{2}\left( I\right) }&\overset{\text {[26, with}\; \alpha =1, \beta =2]}{\le }C\left( p+1\right) \left\| \varPhi _{\left[ 0,1\right] }v\right\| _{L^{2}\left( I\right) }\\&\qquad \qquad =C\left( p+1\right) \left\| \sqrt{\omega \varPhi _{\left[ 0,1\right] }}v\right\| _{L^{2}\left( I\right) }. \end{aligned}$$

• (b) For $\omega \left( x\right) =\varphi _{b}\left( x\right) =x$ we observe that $\omega \left( x\right) \le 2\varPhi _{\left[ 0,1\right] }\left( x\right)$ holds for all $0\le x\le 1/2$ so that

  $$\begin{aligned}&\left\| \omega ^{1/2}v\right\| _{L^{2}\left( I\right) }^{2} \le 2\left\| \varPhi _{\left[ 0,1\right] }^{1/2}v\right\| _{L^{2}\left( I\right) }^{2}+\left\| v\right\| _{L^{2}\left( \left[ \frac{1}{2},1\right] \right) }^{2}\\&\qquad \quad \overset{\text {[26, Lem. 2.4]} }{\le }C\left( p+1\right) ^{2}\\&\qquad \qquad \qquad \times \, \left( \left\| \varPhi _{\left[ 0,1\right] }v\right\| _{L^{2}\left( I\right) }^{2}+\left\| \varPhi _{\left[ \frac{1}{2},1\right] }^{1/2}v\right\| _{L^{2}\left( \left[ \frac{1}{2},1\right] \right) }^{2}\right) . \end{aligned}$$

  The result now follows from $\varPhi _{\left[ 0,1\right] }^{1/2}\le \omega ^{1/2}$ pointwise in $\left[ 0,1\right]$ and $\sqrt{2\omega }\ge 1$ pointwise on $\left[ \frac{1}{2},1\right]$.

• (c) The $\omega \left( x\right) =\varphi _{a}\left( x\right)$ follows from Case b by symmetry.

• (d) Let $\omega$ be a general weight function which satisfies the assumptions of the lemma. Hence, from Part a,b,c we conclude that

  $$\begin{aligned}&\left\| \omega ^{1/2}v\right\| _{L^{2}\left( I\right) }^{2}=A\left\| \sqrt{\varphi _{0}}v\right\| _{L^{2}\left( I\right) }^{2}+B\left\| \sqrt{\varphi _{1}}v\right\| _{L^{2}\left( I\right) }^{2}\\&\quad +\,D\left\| \sqrt{\varPhi _{\left[ 0,1\right] }}v\right\| _{L^{2}\left( I\right) } ^{2}\le C^{\prime }\left( p+1\right) ^{2}\left\| \sqrt{\omega \varPhi _{\left[ 0,1\right] }}v\right\| _{L^{2}\left( I\right) }^{2} \end{aligned}$$

holds.$\square$

The following lemma is a weighted version of [28, Lem. D3].

Lemma 8

Let $d\in \left( 0,1\right)$, $a,b$ be given such that $-1+ad<1+bd$ and define the trapezoid

$$\begin{aligned}&D:=D\left( a,b,d\right) :=\left\{ \left( x_{1},x_{2}\right) \in {\mathbb {R}}^{2}\mid x_{2}\in \left( 0,d\right) \right. \\&\quad \left. \quad \text {and}\quad -1+ax_{2}<x_{1}<1+bx_{2}\right\} . \end{aligned}$$

Let $\omega \in {\mathbb {P}}_{2}\left( D\right)$ be a polynomial such that for any $0\le x_{2}\le d$, $\omega \left( \cdot ,x_{2}\right)$ has property $P_{\left[ -1+ax_{2},1+bx_{2}\right] }$.

On $D$ we define the weight function

$$\begin{aligned} \varPhi _{a,b,d}\left( x_{1},x_{2}\right) :=\min \left\{ \left| x_{1}-\left( -1+ax_{2}\right) \right| ,\left| x_{1}-\left( 1+bx_{2}\right) \right| \right\} \end{aligned}$$

which measures the distance of the point $\left( x_{1},x_{2}\right)$ from the lateral edges of $D$. Then, there exists a constant $C=C\left( a,b,d\right)$ such that for all $p\in \mathbb {N}$ and all polynomials $v\in {\mathbb {P}}_{p}\left( D\right)$ it holds

$$\begin{aligned} \left\| \omega ^{1/2}v\right\| _{L^{2}\left( D\right) }\le C\left( p+1\right) \left\| \sqrt{\omega \varPhi _{a,b,d}}v\right\| _{L^{2}\left( D\right) }. \end{aligned}$$

Proof

Note that

$$\begin{aligned} C_{13}\varPhi _{\left[ -1+ax_{2},1+bx_{2}\right] }\left( x_{1}\right)&\le \varPhi _{a,b,d}\left( x_{1},x_{2}\right) \\&\le C_{14}\varPhi _{\left[ -1+ax_{2},1+bx_{2}\right] }\left( x_{1}\right) \end{aligned}$$

for positive constants $C_{13},C_{14}$ which only depends on $a,b,d$. Hence, the one-dimensional case (Lemma 7) implies

$$\begin{aligned}&\int _{-1+ax_{2}}^{1+bx_{2}}\omega ^{1/2}\left( x_{1},x_{2}\right) v^{2}\left( x_{1},x_{2}\right) dx_{1}\le C\left( p+1\right) ^{2}\\&\quad \times \, \int _{-1+ax_{2}}^{1+bx_{2}}\sqrt{\omega \left( x_{1},x_{2}\right) \varPhi _{a,b,d}\left( x_{1},x_{2}\right) }v^{2}\left( x_{1},x_{2}\right) dx_{1}. \end{aligned}$$

Integrating this estimate over $x_{2}\in \left( 0,d\right)$ completes the proof.$\square$

Proof of Lemma 6

By using an affine pullback we may restrict to the case that $K$ is the equi-sided triangle $\mathrm{conv} \left( \genfrac(){0.0pt}1{0}{0},\genfrac(){0.0pt}1{1}{0},\frac{1}{2}\genfrac(){0.0pt}1{1}{\sqrt{3}}\right)$ and $E=\left( 0,1\right) \times \left\{ 0\right\}$.

It turns out that the proofs for the different cases in (6.4) for $\varphi _{E}$ and in (5.4) for $\varPhi _{K}$ uses the same arguments and we work them out exemplarily for the case of the quadratic edge bubble

$$\begin{aligned} \varphi _{E}\left( x_{1},x_{2}\right) =\left( x_{1}-\frac{x_{2}}{\sqrt{3} }\right) \left( 1-x_{1}-\frac{x_{2}}{\sqrt{3}}\right) \end{aligned}$$

and for $\varPhi _{K}=\varPhi _{K}^{\left( 3\right) }$ being the cubic bubble on $K$.

First, we will cover $K$ with $4$ trapezoids and one triangle: Let $v=\left( \cos \frac{\pi }{4},\sin \frac{\pi }{4}\right) ^{\intercal }=2^{-1/2}\left( 1,1\right) ^{\intercal }$. Then,

• 1. $T_{1}:=\left\{ \genfrac(){0.0pt}1{\hat{x}_{1}}{0}+sv:\left( \begin{array}{l} 0\le \hat{x}_{1}\le 1/2\\ 0\le s\le L_{1}\left( \hat{x}_{1}\right) \end{array} \right) \right\}$ with

  $$\begin{aligned} L_{1}\left( \hat{x}_{1}\right) :=\frac{\sqrt{6} }{1+\sqrt{3}}\left( 1-\hat{x}_{1}\right) . \end{aligned}$$

• 2. $T_{2}:$ mirror image of $T_{1}$ with respect to the angle bisector at $\left( 0,0\right) ^{\intercal }.$

• 3. $T_{3}:$ counter-clockwise rotations of $T_{1}$ by $\frac{3\pi }{4}$about the barycenter of $K.$

• 4. $T_{4}:$ mirror image of $T_{3}$ with respect to the angle bisector at $\left( 1,0\right) ^{\intercal }$.

• 5. $T_{5}:=\left\{ \left( x_{1},x_{2}\right) ^{\intercal }\in K\mid x_{2}\ge 1/2\right\}$.

Case $T_{1}$: We introduce

$$\begin{aligned} \chi :\left( \begin{array}{c} 0\le \hat{x}_{1}\le 1/2\\ 0\le s\le L_{1}\left( \hat{x}_{1}\right) \end{array} \right) \rightarrow T_{1}\quad \text {by}\quad \chi \left( \hat{x}_{1},s\right) :=\genfrac(){0.0pt}1{\hat{x}_{1}}{0}+sv \end{aligned}$$

The bubble function $\varphi _{E}$ restricted to the line $\genfrac(){0.0pt}1{\hat{x}_{1} }{0}+sv$ results in

$$\begin{aligned} \psi _{\hat{x}_{1}}\left( s\right) \!:=\!\varphi _{E}\circ \chi \left( \hat{x} _{1},s\right) \!&=\!\left( L_{1}\left( \hat{x}_{1}\right) -s\right) \left( \frac{\sqrt{3}\!+\!1}{\sqrt{6}}\hat{x}_{1}\!+\!\frac{s}{3}\right) \\&\qquad \qquad \qquad \forall \left( \begin{array}{l} 0\le \hat{x}_{1}\le 1/2\\ 0\le s\le L_{1}\left( \hat{x}_{1}\right) \end{array} \right) . \end{aligned}$$

Note that the function $\psi _{\hat{x}_{1}}$ satisfies the assumptions of Lemma 7 and $\hat{v}:=v\circ \chi$ is a polynomial of maximal degree $p$. Hence,

$$\begin{aligned}&\int _{T_{1}}\varphi _{E}v^{2} =\int _{0}^{1/2}\left( \int _{0}^{L_{1}\left( \hat{x}_{1}\right) }\psi _{\hat{x}_{1}}\left( s\right) \hat{v}^{2}ds\right) dx_{1}\\&\le C\left( p+1\right) ^{2}\int _{0}^{1/2}\left( \int _{0}^{L_{1}\left( \hat{x}_{1}\right) }\psi _{\hat{x}_{1}}\left( s\right) \varPhi _{\left[ 0,L_{1}\left( \hat{x}_{1}\right) \right] }\left( s\right) \hat{v} ^{2}ds\right) d\hat{x}_{1}. \end{aligned}$$

Composing $\varPhi _{\left[ 0,L_{1}\left( \hat{x}_{1}\right) \right] }\left( s\right)$ with $\chi ^{-1}$ yields the function

$$\begin{aligned} d\left( x_{1},x_{2}\right) =\frac{1+\sqrt{3}}{\sqrt{3}}\frac{x_{2}\left( 1-x_{1}-\frac{x_{2}}{\sqrt{3}}\right) }{\left( 1-x_{1}+x_{2}\right) ^{2}}. \end{aligned}$$

Note that the distance function

$$\begin{aligned} \varPhi _{K}^{1}\left( x_{1},x_{2}\right) =\mathrm{dist}\,\left( \left( x_{1},x_{2}\right) ^{\intercal },\partial K\right) \end{aligned}$$

is piecewise linear on $K$. It is easy to verify that $d\left( x_{1},x_{2}\right) \le C\varPhi _{K}^{1}\left( x_{1},x_{2}\right)$ pointwise on $T_{1}$ for some $C=O\left( 1\right)$ so that

$$\begin{aligned} \int _{T_{1}}\varphi _{E}v^{2}\le C^{\prime }\left( p+1\right) ^{2}\int _{T_{1}}\varphi _{E}\varPhi _{K}^{1}v^{2}. \end{aligned}$$

Since $\varphi _{E}\varPhi _{K}^{1}\le \tilde{C}\varPhi _{K}$ pointwise on $K$ we have proved the assertion for $T_{1}$.

Case $T_{3}$: The proof for the trapezoid $T_{3}$ follows by symmetry.

Case $T_{2}$: Next, we will consider the trapezoid $T_{2}$ and first note that by interchanging the $x_{1},x_{2}$-variables the case becomes equivalent to the estimate

$$\begin{aligned} \int _{T_{1}}\varphi _{\tilde{E}}v^{2}\le C^{\prime }\left( p+1\right) ^{2}\int _{T_{1}}\varPhi _{K}^{\left( 3\right) }v^{2}\qquad v\in {\mathbb {P}} _{p}\left( T_{1}\right) , \end{aligned}$$

where $\varphi _{\tilde{E}}$ is the qudratic edge bubble for the edge $\tilde{E}=\overline{\genfrac(){0.0pt}1{0}{0},\genfrac(){0.0pt}1{1/2}{\sqrt{3}/2}}$ with explicit form

$$\begin{aligned} \varphi _{\tilde{E}}\left( x_{1},x_{2}\right) =\frac{2}{\sqrt{3}}x_{2}\left( 1-x_{1}-\frac{x_{2}}{\sqrt{3}}\right) . \end{aligned}$$

This time, the bubble function $\varphi _{\tilde{E}}$, restricted to the line $\genfrac(){0.0pt}1{\hat{x}_{1}}{0}+sv$, is given by

$$\begin{aligned} \tilde{\psi }_{\hat{x}_{1}}\left( s\right) :=\varphi _{\tilde{E}}\circ \chi \left( \hat{x}_{1},s\right)&=\frac{\sqrt{3}+1}{3}s\left( L\left( \hat{x}_{1}\right) -s\right) \\&\qquad \forall \left( \begin{array}{l} 0\le \hat{x}_{1}\le 1/2\\ 0\le s\le L_{1}\left( \hat{x}_{1}\right) \end{array} \right) . \end{aligned}$$

The function $\tilde{\psi }_{\hat{x}_{1}}$ satisfies the assumptions of Lemma 7 so that

$$\begin{aligned}&\int _{T_{1}}\varphi _{\tilde{E}}v^{2} =\int _{0}^{1/2}\left( \int _{0}^{L_{1}\left( \hat{x}_{1}\right) }\tilde{\psi }_{\hat{x}_{1}}\left( s\right) \hat{v}^{2}ds\right) dx_{1}\\&\le C\left( p+1\right) ^{2}\int _{0}^{1/2}\left( \int _{0}^{L_{1}\left( \hat{x}_{1}\right) }\tilde{\psi }_{\hat{x}_{1}}\left( s\right) \varPhi _{\left[ 0,L_{1}\left( \hat{x}_{1}\right) \right] }\left( s\right) \hat{v} ^{2}ds\right) d\hat{x}_{1}. \end{aligned}$$

Now we can argue as for the Case of $T_{1}$ to obtain

$$\begin{aligned} \int _{T_{1}}\varphi _{\tilde{E}}v^{2}\le C^{\prime }\left( p+1\right) ^{2}\int _{T_{1}}\varphi _{\tilde{E}}\varPhi _{K}^{1}v^{2}. \end{aligned}$$

Since $\varphi _{\tilde{E}}\varPhi _{K}^{1}\le \tilde{C}\varPhi _{K}$ pointwise on $K$ the assertion follows for $T_{2}$.

Case $T_{4}$: The proof for the trapezoid $T_{4}$ again follows by symmetry from the case $T_{2}$.

Case $T_{5}$: On $T_{5}$ we have the pointwise estimate $\varphi _{E}\le C\varPhi _{K}^{\left( 3\right) }$ and the estimate for $T_{5}$ is trivial.$\square$