Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions

Abstract

We analyze the approximation of the solutions of second-order elliptic problems, which have point singularities and belong to a countably normed space of analytic functions, with a first-order, h-version finite element (FE) method based on uniform tensor-product meshes. The FE solutions are well known to converge with algebraic rate at most 1 / 2 in terms of the number of degrees of freedom, and even slower in the presence of singularities. We analyze the compression of the FE coefficient vectors represented in the so-called quantized-tensor-train format. We prove, in a reference square, that the resulting FE approximations converge exponentially in terms of the effective number N of degrees of freedom involved in the representation: $N={\mathcal {O}} ( \log ^{5} \varepsilon ^{-1} )$, where $\varepsilon \in (0,1)$ is the accuracy measured in the energy norm. Numerically we show for solutions from the same class that the entire process of solving the tensor-structured Galerkin first-order FE discretization can achieve accuracy $\varepsilon$ in the energy norm with $N={\mathcal {O}} ( \log ^{\kappa } \varepsilon ^{-1} )$ parameters, where $\kappa <3$.

Appendices

A Auxiliary results

Lemma A.1

Let $n\in {\mathbb {N}}$. Then, for the sum of the inverses of the corresponding $n+1$ binomial coefficients, one has

$$\begin{aligned} \sum _{k=0}^{n} \left( {\begin{array}{c}n\\ k\end{array}}\right) ^{-1} \le \frac{8}{3} \, . \end{aligned}$$

(A.1)

Proof

For $n\in {\mathbb {N}}$, let us denote the left-hand side of (A.1) by $I_n$. For every $n\in {\mathbb {N}}$, we have the recurrence relation $I_{n+1} = ( n+2 ) / ( 2n+2 ) \, I_n + 1$, see [69, Theorem 1]. By induction, starting from $n=3$, it follows that $I_n \ge 2 ( 1+n^{-1} )$ for all $n \ge 3$. Then $I_{n+1} \le I_n + ( 2n+2 ) ^{-1} I_n - 1$ for all $n \ge 3$. Again by induction, we obtain $I_n \le I_3 = 8/3$ for all $n > 3$. Since $I_1,I_2 \le I_3$, this proves the claim. $\square$

As above, for all $p\in {\mathbb {N}}$ and $s=0,1,\ldots ,p$, we use the notation

$$\begin{aligned} \varUpsilon _{p {{\,}\!}s}=\frac{ ( p-s ) !}{ ( p+s ) !} . \end{aligned}$$

Lemma A.2

For all $\varrho > 1$ and $p\in {\mathbb {N}}$ such that $p \ge \varrho$, let us set $s= \bigl \lfloor p / \varrho \bigr \rfloor$, so that $1 \le s \le p$. Then the bound

$$\begin{aligned} ( \varrho -1 ) ^{2s} ( s! ) ^{2} \, \varUpsilon _{p {{\,}\!}s} \le c^2 \, s \, \exp \biggl ( -\frac{2p}{\varrho } \biggr ) \end{aligned}$$

holds with $c^{2} = e^{5}/\sqrt{2\pi }$.

Proof

Using Stirling’s bound for Euler’s Gamma function, we obtain the bounds

$$\begin{aligned} \varUpsilon _{p {{\,}\!}s}&\le \frac{e \, ( p-s ) ^{p-s+\frac{1}{2}} \, e^{- ( p-s ) }}{\sqrt{2 \pi } \, ( p+s ) ^{p+s+\frac{1}{2}} \, e^{- ( p+s ) }} = \frac{e}{\sqrt{2 \pi }} \biggl ( \frac{p-s}{p+s} \biggr ) ^{p-s+\frac{1}{2}} \frac{e^{2s}}{ ( p+s ) ^{2s}}, \\ ( s! ) ^2&\le e^2 \, s^{2s+1} \, e^{-2s} , \end{aligned}$$

which yield together

$$\begin{aligned} ( \varrho -1 ) ^{2s} ( s! ) ^{2} \, \varUpsilon _{p {{\,}\!}s} \le \frac{e^{3}}{\sqrt{2\pi }} \, s \, \biggl ( \frac{p-s}{p+s} \biggr ) ^{p-s+\frac{1}{2}} \biggl [ ( \varrho -1 ) \frac{s}{p+s} \biggr ] ^{2s} . \end{aligned}$$

For $s= \lfloor p/\varrho \rfloor$, we have $p/\varrho - 1 < s \le p/\varrho$, so that $( \varrho - 1 ) s \le p-s$. We denote $t=2s / ( p+s )$ and, using that $( 1-t ) ^{1/t} < e^{-1} \sqrt{1-t}$ holds for $0 < t \le 1$, obtain

$$\begin{aligned} ( \varrho -1 ) ^{2s} ( s! ) ^{2} \, \varUpsilon _{p {{\,}\!}s}\le & {} \frac{e^{3}}{\sqrt{2\pi }} \, s \, \biggl ( \frac{p-s}{p+s} \biggr ) ^{ p +s + \frac{1}{2} } \le \frac{e^{3}}{\sqrt{2\pi }} \sqrt{\frac{p-s}{p+s}} \, s \, ( 1-t ) ^{ 2s/t }\\\le & {} \frac{e^{3}}{\sqrt{2\pi }} \sqrt{\frac{p-s}{p+s}} \, s \, \exp ( -2s ) \, \le \frac{e^{5}}{\sqrt{2\pi }} \, s \, \exp \biggl ( -\frac{2p}{\varrho } \biggr ) . \end{aligned}$$

$\square$

B Bounds for univariate quasi-interpolation

We shall use the following bound, for which one may refer to either of [62, Corollary 3.15] or [67, Corollary 2].

Proposition B.1

Assume that $p\in {\mathbb {N}}$ and $s\in {{\mathbb {N}}}_{0}$ are such that $s \le p$. Then, for any function ${\hat{u}} \in \mathbb {H}^{s+1}_{} ( \hat{\mathrm {J}} )$, the interpolant $\hat{\pi }_{p} {\hat{u}}$ satisfies the error bounds

$$\begin{aligned} \begin{aligned} | {\hat{u}} - \hat{\pi }_{p} {\hat{u}} | ^2_{\mathbb {H}^{1}_{} ( \hat{\mathrm {J}} ) }&\le \varUpsilon _{p {{\,}\!}s} \, | {\hat{u}} | ^2_{\mathbb {H}^{s+1}_{} ( \hat{\mathrm {J}} ) } ,\\ || {\hat{u}} - \hat{\pi }_{p} {\hat{u}} || ^2_{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }&\le \frac{1}{p\, ( p+1 ) } \varUpsilon _{p {{\,}\!}s} \, | {\hat{u}} | ^2_{\mathbb {H}^{s+1}_{} ( \hat{\mathrm {J}} ) } . \end{aligned} \end{aligned}$$

(B.1)

We shall also use the following stability bound on the second derivative of the interpolant.

Lemma B.2

For every $p\in {\mathbb {N}}$ and for every $u\in \mathbb {H}^{2}_{} ( \hat{\mathrm {J}} )$, the following bounds hold:

$$\begin{aligned} | \hat{\pi }_{p} {\hat{u}} | _{\mathbb {H}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \le \frac{1}{4} \, p^2 ( p^2 - 1 ) | {\hat{u}} | _{\mathbb {H}^{1}_{} ( \hat{\mathrm {J}} ) }^{2} ,\quad | \hat{\pi }_{p} {\hat{u}} | _{\mathbb {H}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \le \frac{1}{2} ( p^2 - 1 ) | {\hat{u}} | _{\mathbb {H}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} . \end{aligned}$$

(B.2)

Proof

Consider $\mathbb {L}^{2}_{w} ( \hat{\mathrm {J}} )$, a weighted space of square-integrable functions defined on $\hat{\mathrm {J}}$, with the weight w given by $w ( x ) =1-x^{2}$ for all $x\in \hat{\mathrm {J}}$. For this weight, we have ${\hat{u}}''\in \mathbb {L}^{2}_{w} ( \hat{\mathrm {J}} )$ with $|| {\hat{u}}'' || _{\mathbb {L}^{2}_{w} ( \hat{\mathrm {J}} ) } \le || {\hat{u}}'' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }$. Note that $L_{i}'$, $i\in {\mathbb {N}}$, are orthogonal in $\mathbb {L}^{2}_{w} ( \hat{\mathrm {J}} )$, namely

$$\begin{aligned} \langle L_{i}',L_{i'}' \rangle _{\mathbb {L}^{2}_{w} ( \hat{\mathrm {J}} ) } = \frac{\delta _{i i'}}{i + \frac{1}{2}} \frac{ ( i+1 ) !}{ ( i-1 ) !} \quad \text {for all}\quad i,i'\in {\mathbb {N}}. \end{aligned}$$

Below, we shall also use that $|| L_{i}' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}=i ( i+1 )$ for every $i\in {{\mathbb {N}}}_{0}$. This follows readily from integration by parts and the orthogonality of the Legendre polynomials, as in [62, theorem 3.91]:

$$\begin{aligned} \int \limits _{-1}^{1} L_{i}'L_{i}' = L_{i}L_{i}'\vert _{-1}^{1} - \int \limits _{-1}^{1} L_{i} L_{i}'' = L_{i}L_{i}'\vert _{-1}^{1} = 2 \, L_{i} ( 1 ) \, L_{i}' ( 1 ) = i ( i+1 ) . \end{aligned}$$

Since ${\hat{u}}'\in \mathbb {L}^{2}_{} ( \hat{\mathrm {J}} )$, we have a Legendre representation ${\hat{u}}'=\sum _{i=0}^{\infty } c_{i} L_{i}$ in $\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} )$ with coefficients $c_{i}$, $i\in {{\mathbb {N}}}_{0}$. Then ${\hat{u}}''=\sum _{i=1}^{\infty } c_{i} L_{i}'$ holds in $\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} )$ and, thus, also in $\mathbb {L}^{2}_{w} ( \hat{\mathrm {J}} )$. This results in the bound

$$\begin{aligned} \sum _{i=1}^{\infty } \frac{1}{i+\frac{1}{2}} \frac{ ( i+1 ) !}{ ( i-1 ) !} | c_{i} | ^{2} = || {\hat{u}}'' || _{\mathbb {L}^{2}_{w} ( \hat{\mathrm {J}} ) }^{2} \le || {\hat{u}}'' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} . \end{aligned}$$

(B.3)

By Definition 5.3, we have $( \hat{\pi }_{p} {\hat{u}} ) '=\sum _{i=0}^{p-1} c_{i} L_{i}$, and by the triangle inequality we obtain

$$\begin{aligned} || ( \hat{\pi }_{p} {\hat{u}} ) '' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) } = \Bigl || \sum _{i=1}^{p-1} c_{i} L_{i}' \Bigr || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }&\le \sum _{i=1}^{p-1} \sqrt{i \Bigl ( i+\frac{1}{2} \Bigr ) ( i+1 ) } \cdot \Biggl [ \frac{c_{i}^{2}}{i+\frac{1}{2}} \Biggr ] ^{\frac{1}{2}} \end{aligned}$$

(B.4)

$$\begin{aligned}&= \sum _{i=1}^{p-1} \sqrt{i+\frac{1}{2}} \cdot \Biggl [ \frac{c_{i}^{2}}{i+\frac{1}{2}} \frac{ ( i+1 ) !}{ ( i-1 ) !} \Biggr ] ^{\frac{1}{2}} . \end{aligned}$$

(B.5)

Finally, we apply the Cauchy–Bunyakovsky–Schwarz inequality to (B.4) and (B.5) to arrive at the bounds

$$\begin{aligned} || ( \hat{\pi }_{p} {\hat{u}} ) '' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}&\le || {\hat{u}}' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \sum _{i=1}^{p-1} i \Bigl ( i+\frac{1}{2} \Bigr ) ( i+1 ) = \frac{1}{4} \, p^2 ( p^2 - 1 ) || {\hat{u}}' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} , \\ || ( \hat{\pi }_{p} {\hat{u}} ) '' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}&\le || {\hat{u}}'' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \sum _{i=1}^{p-1} \Bigl ( i+\frac{1}{2} \Bigr ) = \frac{1}{2} ( p^2 - 1 ) || {\hat{u}}'' || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} . \end{aligned}$$

$\square$

By rescaling from an interval $\mathrm {I}$ to $\hat{\mathrm {J}}=(-1,1)$, from Proposition B.1 and Lemma B.2 we obtain the following statement.

Corollary B.3

Consider an interval $\mathrm {I}= (2a,2a+2 h)$ with $0< h < \infty$. Let $\varphi$ be an affine map from $\hat{\mathrm {J}}$ onto $\mathrm {I}$. Then, for all $p,s\in {\mathbb {N}}$ such that $s \le p$, for every $u \in \mathbb {H}^{s+1}_{} ( \mathrm {G} )$ and for $v\in \mathbb {Q}_{p ,{\,}\!p}$ given by $v {{\mathrm{\circ }}}\varphi = \hat{\pi }_{k ,{\,}\!p} \, ( u {{\mathrm{\circ }}}\varphi )$, the following inequalities hold:

$$\begin{aligned} || u - v || _{\mathbb {L}^{2}_{} ( \mathrm {I} ) }^{2}\le & {} \frac{1}{p\, ( p+1 ) } \, h^{2 ( s+1 ) } \, \varUpsilon _{p {{\,}\!}s} \, | u | _{\mathbb {H}^{s+1}_{} ( \mathrm {I} ) }^2 ,\\ | u - v | _{\mathbb {H}^{1}_{} ( \mathrm {I} ) }^{2}\le & {} h^{2s} \, \varUpsilon _{p {{\,}\!}s} \, | u | _{\mathbb {H}^{s+1}_{} ( \mathrm {I} ) }^2 ,\\ | v | _{\mathbb {H}^{2}_{} ( \mathrm {I} ) }^{2}\le & {} \frac{1}{4} \, \frac{ p^2 ( p^2-1 ) }{2h} \, | u | _{\mathbb {H}^{1}_{} ( \mathrm {I} ) }^{2} , \quad | v | _{\mathbb {H}^{2}_{} ( \mathrm {I} ) }^{2} \le \frac{1}{2} \, ( p^2-1 ) \, | u | _{\mathbb {H}^{2}_{} ( \mathrm {I} ) }^{2} . \nonumber \end{aligned}$$

C Bounds for tensor-product bivariate quasi-interpolation

We shall use the following error bound for the projection $\hat{\Pi }_{p_1,p_2}$. For similar bounds for tensor-product interpolation operators, we also refer to [62, lemma 4.67] and [67, theorem 5].

Lemma C.1

Let $p= ( p_1,p_2 ) \in {\mathbb {N}}^2$ and $q_1,q_2,r_1,r_2,s_1,s_2\in {{\mathbb {N}}}_{0}$ be such that $q_1,r_1,s_1 \le p_1$ and $q_2,r_2,s_2 \le p_2$. Then the following bounds hold true for every ${\hat{u}} \in \mathbb {H}^{q_1+1,2}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} ) \cap \mathbb {H}^{2,q_2+1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} ) \cap \mathbb {H}^{r_1+1,r_2+1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} ) \cap \mathbb {H}^{s_1+1 ,{\,}\!1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} ) \cap \mathbb {H}^{1,s_2+1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} )$:

$$\begin{aligned} || {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \le&3 \sum _{k=1}^{2} \frac{1}{p_k ( p_k + 1 ) } \varUpsilon _{p_k {{\,}\!}s_k} \, || \partial _{k}^{s_k + 1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^2 \\&+ 3 \biggl [ \prod _{k=1}^{2} \frac{1}{p_k ( p_k + 1 ) } \varUpsilon _{p_k {{\,}\!}r_k} \, \biggr ] \; || \partial _{1}^{r_1 + 1} \partial _{2}^{r_2 + 1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} , \\ | {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} | _{\mathbb {H}^{1}_{} ( \hat{\mathrm {Q}} ) }^{2} \le&2 \sum _{k=1}^{2} \varUpsilon _{p_k {{\,}\!}s_k} \, || \partial _{k}^{s_k + 1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^2\\&+ \frac{2}{p_1 ( p_1 + 1 ) } \varUpsilon _{p_1 {{\,}\!}q_1} \, || \partial _{1}^{q_1 + 1} \partial _{2}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^2 \\&+ \frac{2}{p_2 ( p_2 + 1 ) } \varUpsilon _{p_2 {{\,}\!}q_2} \, || \partial _{1}^{}\partial _{2}^{q_2 + 1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^2 , \\ | \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} | _{\mathbb {H}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \le&( p_1^2 - 1 ) || \partial _{1}^{2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + ( p_2^2 - 1 ) || \partial _{2}^{2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \\&+ \biggl \{ 1 + \frac{1}{2} \frac{p_1^2}{p_2} \frac{p_1^2 - 1}{p_2 + 1} + \frac{1}{2} \frac{p_2^2}{p_1} \frac{p_2^2 - 1}{p_1 + 1} \biggr \} || \partial _{1}^{}\partial _{2}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} . \end{aligned}$$

Proof

Note that the inequality

$$\begin{aligned} || \partial _{k}^{} \hat{\pi }_{k ,{\,}\!p_k} \hat{v} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \le || \partial _{k}^{} \hat{v} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \quad \text {for every}\quad \hat{v}\in \mathbb {H}^{1 ,{\,}\!1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} ) \end{aligned}$$

(C.1)

follows from Definition 5.3.

Let ${\hat{u}} \in \mathbb {H}^{s_1+1 ,{\,}\!1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} ) \cap \mathbb {H}^{1,s_2+1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} )$. Since $\hat{\Pi }_{p_1 {{\,}\!}p_2} = \hat{\pi }_{1 ,{\,}\!p_1} {{\mathrm{\circ }}}\, \hat{\pi }_{2 ,{\,}\!p_2}$, we have the decomposition

$$\begin{aligned} {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}}= & {} ( \mathsf {id}_{}- \hat{\pi }_{1 ,{\,}\!p_1} ) {\hat{u}} + \hat{\pi }_{1 ,{\,}\!p_1} ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) {\hat{u}}\nonumber \\= & {} ( \mathsf {id}_{}- \hat{\pi }_{1 ,{\,}\!p_1} ) {\hat{u}} + ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) {\hat{u}} - ( \mathsf {id}_{}- \hat{\pi }_{1 ,{\,}\!p_1} ) ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) {\hat{u}}.\qquad \quad \end{aligned}$$

(C.2)

$\underline{\mathbb {L}^{2}_{}\text { error bound.}}$ By the triangle inequality, from (C.2) we obtain

$$\begin{aligned} || {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2}\le & {} 3 || {\hat{u}} - \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + 3 || {\hat{u}} - \hat{\pi }_{2 ,{\,}\!p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2}\\&+ 3 || ( \mathsf {id}_{}- \hat{\pi }_{1 ,{\,}\!p_1} ) ( {\hat{u}} - \hat{\pi }_{2 ,{\,}\!p_2}{\hat{u}} ) || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} . \end{aligned}$$

Using Proposition B.1, we bound the first two terms as follows: for every $k\in \{ 1,2 \}$, we have

$$\begin{aligned} || {\hat{u}} - \hat{\pi }_{k ,{\,}\!p_k} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \le \frac{1}{p_k ( p_k + 1 ) } \, \varUpsilon _{p_k {{\,}\!}s_k} \, || \partial _{k}^{s_k + 1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^2 \end{aligned}$$

(C.3)

for every $s_k\in {{\mathbb {N}}}_{0}$ such that $s_k \le p_k$. Similarly, for the third term we obtain

$$\begin{aligned}&|| ( \mathsf {id}_{}- \hat{\pi }_{1 ,{\,}\!p_1} ) ( {\hat{u}} - \hat{\pi }_{2 ,{\,}\!p_2}{\hat{u}} ) || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \le \\&\quad \frac{1}{p_1 ( p_1 + 1 ) } \varUpsilon _{p_1 {{\,}\!}r_1} \, || \partial _{1}^{r_1 + 1} ( {\hat{u}} - \hat{\pi }_{2 ,{\,}\!p_2}{\hat{u}} ) || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2}\\&\quad \le \biggl [ \prod _{k=1}^{2} \frac{1}{p_k ( p_k + 1 ) } \varUpsilon _{p_k {{\,}\!}r_k} \, \biggr ] \; || \partial _{1}^{r_1 + 1} \partial _{2}^{r_2 + 1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \end{aligned}$$

for all $r_1,r_2\in {{\mathbb {N}}}_{0}$ such that $r_1\le p_1$ and $r_2\le p_2$. By combining the bounds for the three terms, we obtain the $\mathbb {L}^{2}_{}$-norm estimate claimed.

$\underline{\mathbb {H}^{1}_{}\text { error bound.}}$ We use (C.2), the triangle inequality, (C.1) and Proposition B.1 to arrive at

$$\begin{aligned}&|| \partial _{1}^{} ( {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} ) || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2}\\&\quad \le 2 \, || \partial _{1}^{} ( \mathsf {id}_{}- \hat{\pi }_{1 ,{\,}\!p_1} ) {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + 2 \, || \partial _{1}^{}\hat{\pi }_{1 ,{\,}\!p_1} ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2}\\&\quad \le 2 \, || \partial _{1}^{} ( \mathsf {id}_{}- \hat{\pi }_{1 ,{\,}\!p_1} ) {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + 2 \, || ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) \partial _{1}^{} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2}\\&\quad \le 2 \, \varUpsilon _{p_1 {{\,}\!}s_1} \, || \partial _{1}^{s_1 + 1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^2 + \frac{2}{p_2 ( p_2 + 1 ) } \, \varUpsilon _{p_2 {{\,}\!}q_2} \, || \partial _{2}^{q_2+1} \partial _{1}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^2 . \end{aligned}$$

An analogous bound holds for $|| \partial _{2}^{} ( {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} ) || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2}$. Together, the two bounds prove the $\mathbb {H}^{1}_{}$-norm error estimate claimed.

$\underline{\mathbb {H}^{2}_{}\text {-stability estimate.}}$ Let us decompose the interpolant as follows:

$$\begin{aligned} \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} = \hat{\pi }_{1 ,{\,}\!p_1}{\hat{u}} - ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} . \end{aligned}$$

(C.4)

Using the triangle inequality and the properties of the interpolation operators, we obtain

$$\begin{aligned} || \partial _{1}^{2} \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }\le & {} || \partial _{1}^{2} ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) } + || \partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }\\= & {} || ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) \partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) } + || \partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) } \end{aligned}$$

For these two terms, let us use the corresponding bounds of Lemma B.2. For the second, we obtain $|| \partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \le \frac{1}{2} ( p_1^2 - 1 ) || \partial _{1}^{2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}$. For the first term, we use the error bound of Proposition B.1: $|| ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) \partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \le p_2^{-1} ( p_2 + 1 ) ^{-1} || \partial _{2}^{}\partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} = p_2^{-1} ( p_2 + 1 ) ^{-1} || \partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} \partial _{2}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}$. Then, by the first bound of Lemma B.2, the inequality $|| ( \mathsf {id}_{}- \hat{\pi }_{2 ,{\,}\!p_2} ) \partial _{1}^{2} \hat{\pi }_{1 ,{\,}\!p_1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \le p_2^{-1} ( p_2 + 1 ) ^{-1} p_1^2 / 4 \, ( p_1^2 - 1 ) || \partial _{1}^{}\partial _{2}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}$ holds true. By combining the bounds for the two terms, we arrive at

$$\begin{aligned} || \partial _{1}^{2} \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \le ( p_1^2 - 1 ) || \partial _{1}^{2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} + \frac{1}{2} \frac{p_1^2}{p_2} \frac{p_1^2 - 1}{p_2 + 1} || \partial _{1}^{}\partial _{2}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} . \end{aligned}$$

An analogous estimate follows for $|| \partial _{2}^{2} \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}$. For the mixed derivative, (C.1) allows to obtain

$$\begin{aligned} || \partial _{1}^{}\partial _{2}^{}\hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} = || \partial _{1}^{}\hat{\pi }_{1 ,{\,}\!p_1} \partial _{2}^{}\hat{\pi }_{2 ,{\,}\!p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}\le & {} || \partial _{1}^{}\partial _{2}^{}\hat{\pi }_{2 ,{\,}\!p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}\\= & {} || \partial _{2}^{}\hat{\pi }_{2 ,{\,}\!p_2} \partial _{1}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}\\\le & {} || \partial _{2}^{}\partial _{1}^{}{\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} . \end{aligned}$$

Together, the bounds on the second-order derivatives yield the stability estimate. $\square$

Corollary C.2

Consider a rectangle $\mathrm {G}= (2a_1,2a_1+2 h_1) {{\mathrm{\times }}}(2a_2,2a_2+2 h_2)$ with $\lambda h \le h_1,h_2 \le \Lambda h$, where $h>0$, $0 < \lambda \le 1$ and $\Lambda \ge 1$. Let $\varphi$ be an affine map from $\hat{\mathrm {Q}}$ onto $\mathrm {G}$. Then, for all $p,s\in {\mathbb {N}}$ such that $s \le p$, for every $u \in \mathbb {H}^{s+2}_{} ( \mathrm {G} )$ and for $v\in \mathbb {Q}_{p ,{\,}\!p}$ given by $v {{\mathrm{\circ }}}\varphi = \hat{\Pi }_{p ,{\,}\!p} \, ( u {{\mathrm{\circ }}}\varphi )$, the following inequalities hold:

$$\begin{aligned} || u - v || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^{2}\le & {} 3 \, ( \Lambda h ) ^{2 ( s+1 ) } \, \frac{\varUpsilon _{p {{\,}\!}s}}{p\, ( p+1 ) } \, | u | _{\mathbb {H}^{s+1}_{} ( \mathrm {G} ) }^2 ,\\ | u - v | _{\mathbb {H}^{1}_{} ( \mathrm {G} ) }^{2}\le & {} 4 \, \frac{\Lambda ^2}{\lambda ^2} \, ( \Lambda h ) ^{2s} \, \varUpsilon _{p {{\,}\!}s} \, | u | _{\mathbb {H}^{s+1}_{} ( \mathrm {G} ) }^2 ,\\ | v | _{\mathbb {H}^{2}_{} ( \mathrm {G} ) }^{2}\le & {} \frac{\Lambda ^4}{\lambda ^4} \, ( p^2-1 ) \, | u | _{\mathbb {H}^{2}_{} ( \mathrm {G} ) }^{2} . \end{aligned}$$

Proof

The statement follows by a rescaling argument from Lemma C.1 with $s_1=s_2=s$ and $r_1=s-1$, $r_2=0$. $\square$

Lemma C.3

Consider $p_1,p_2\in {\mathbb {N}}$ and $s_2\in {{\mathbb {N}}}_{0}$ such that $s_2\le p_2$. Then the following bounds on $\hat{\Gamma }_{}$, the left edge of $\hat{\mathrm {Q}}$, hold for every ${\hat{u}} \in \mathbb {H}^{1,s_2+1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} )$:

$$\begin{aligned} || {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\Gamma }_{} ) }^{2}&\le \frac{3}{2} \frac{ \varUpsilon _{p_2 {{\,}\!}s_2} }{p_{2} ( p_{2} + 1 ) } \, \Bigl \{ || \partial _{2}^{s_2+1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + || \partial _{1}^{}\partial _{2}^{s_2+1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \Bigr \} , \\ | {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} | _{\mathbb {H}^{1}_{} ( \hat{\Gamma }_{} ) }^{2}&\le \frac{3}{2} \, \varUpsilon _{p_2 {{\,}\!}s_2} \, \Bigl \{ || \partial _{2}^{s_2+1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + || \partial _{1}^{}\partial _{2}^{s_2+1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \Bigr \} , \\ | \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} | _{\mathbb {H}^{2}_{} ( \hat{\Gamma }_{} ) }^{2}&\le \frac{3}{4} \, ( p_2^2 - 1 ) \Bigl \{ || \partial _{2}^{2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + || \partial _{1}^{}\partial _{2}^{2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \Bigr \} . \end{aligned}$$

Proof

First, we note that

$$\begin{aligned} || \hat{v} || _{\mathbb {L}^{2}_{} ( \hat{\Gamma }_{} ) }^{2} \le \frac{1}{2} ( 1+\varkappa ^{-1} ) || \hat{v} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} + ( 1+\varkappa ) || \partial _{1}^{}\hat{v} || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {Q}} ) }^{2} \quad \text {for all}\quad \hat{v}\in \mathbb {H}^{1 ,{\,}\!1}_{{\mathrm{mix}}} ( \hat{\mathrm {Q}} ) \nonumber \\ \end{aligned}$$

(C.5)

holds with every $\varkappa >0$. This explicit form of the trace theorem follows immediately from the formula $\hat{v} ( -1,y ) = \hat{v} ( x,y ) - \int _{-1}^{x} \partial _{1}^{}\hat{v} ( t,y ) \,{\mathrm{d}}t$, valid for all $( x,y ) \in \hat{\mathrm {Q}}$, and from the Cauchy–Bunyakovsky–Schwarz inequality for ${\mathbb {R}}^{2}$ and $\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} )$.

The interpolation property of $\hat{\pi }_{1 ,{\,}\!p_1}$ and $\hat{\pi }_{2 ,{\,}\!p_2}$ implies the relations

$$\begin{aligned} \partial _{2}^{\alpha _2} \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} ( -1,y )= & {} \hat{\pi }_{1 ,{\,}\!p_1} \partial _{2}^{\alpha _2} \hat{\pi }_{2 ,{\,}\!p_2} {\hat{u}} ( -1,y ) \\= & {} \partial _{2}^{\alpha _2} \hat{\pi }_{2 ,{\,}\!p_2} {\hat{u}} ( -1,y ) = ( \hat{\pi }_{p_2} [ {\hat{u}} ( -1,\cdot ) ] ) ^{ ( \alpha _2 ) } ( y ) \end{aligned}$$

for all $y\in \hat{\mathrm {J}}$ and $\alpha _2\in \{ 0,1,2 \}$. Therefore, by Proposition B.1 and Lemma B.2, the following inequalities hold:

$$\begin{aligned} || {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\Gamma }_{} ) }^{2}&= || {\hat{u}} ( -1,\cdot ) - \hat{\pi }_{p_2} {\hat{u}} ( -1,\cdot ) || _{\mathbb {L}^{2}_{} ( \hat{\mathrm {J}} ) }^{2}\\&\le \frac{\varUpsilon _{p_2 {{\,}\!}s_2}}{p_2 ( p_2 + 1 ) } \, || \partial _{2}^{s_2+1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\Gamma }_{} ) }^{2} ,\\ | {\hat{u}} - \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} | _{\mathbb {H}^{1}_{} ( \hat{\Gamma }_{} ) }^{2}&= | {\hat{u}} ( -1,\cdot ) - \hat{\pi }_{p_2} {\hat{u}} ( -1,\cdot ) | _{\mathbb {H}^{1}_{} ( \hat{\mathrm {J}} ) }^{2} \le \varUpsilon _{p_2 {{\,}\!}s_2} \, || \partial _{2}^{s_2+1} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\Gamma }_{} ) }^{2} , \\ | \hat{\Pi }_{p_1 {{\,}\!}p_2} {\hat{u}} | _{\mathbb {H}^{2}_{} ( \hat{\Gamma }_{} ) }^{2}&= | \hat{\pi }_{p_2} {\hat{u}} ( -1,\cdot ) | _{\mathbb {H}^{2}_{} ( \hat{\mathrm {J}} ) }^{2} \le \frac{1}{2} \, ( p_2^2 - 1 ) || \partial _{2}^{2} {\hat{u}} || _{\mathbb {L}^{2}_{} ( \hat{\Gamma }_{} ) }^{2} . \end{aligned}$$

Then, applying inequality (C.5) with $\varkappa =\frac{1}{2}$, we obtain the claim. $\square$

Corollary C.4

Consider a rectangle $\mathrm {G}= (2a_1,2a_1+2 h_1) {{\mathrm{\times }}}(2a_2,2a_2+2 h_2) \subset \mathrm {Q}$. Let $\gamma$ denote the left edge of $\mathrm {G}$ and $\varphi$ be an affine map from $\hat{\mathrm {Q}}$ onto $\mathrm {G}$.

Then, for all $p\in {\mathbb {N}}$ and $s\in {{\mathbb {N}}}_{0}$ such that $s\le p$, for every $u \in \mathbb {H}^{s+3}_{} ( \mathrm {G} )$ and for $v\in \mathbb {Q}_{p ,{\,}\!p}$ given by $v {{\mathrm{\circ }}}\varphi = \hat{\Pi }_{p ,{\,}\!p} \, ( u {{\mathrm{\circ }}}\varphi )$, the following inequalities hold:

$$\begin{aligned} || u - v || _{\mathbb {L}^{2}_{} ( \gamma ) }^{2}&\le \frac{3}{2} \frac{h_2^{2 ( s+1 ) }}{h_1} \, \frac{\varUpsilon _{p {{\,}\!}s}}{p ( p + 1 ) } \, \Bigl \{ || \partial _{2}^{s+1} u || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^{2} + h_1^2 \, || \partial _{1}^{} \partial _{2}^{s+1} u || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^{2} \Bigr \} , \\ | u - v | _{\mathbb {H}^{1}_{} ( \gamma ) }^{2}&\le \frac{3}{2} \, \frac{h_2^{2s}}{h_1} \, \varUpsilon _{p {{\,}\!}s} \, \Bigl \{ || \partial _{2}^{s+1} u || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^{2} + h_1^2 \, || \partial _{1}^{} \partial _{2}^{s+1} u || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^{2} \Bigr \} , \\ | v | _{\mathbb {H}^{2}_{} ( \gamma ) }^{2}&\le \frac{3}{4} \, \frac{p^2 - 1}{h_1} \Bigl \{ || \partial _{2}^{2} u || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^{2} + h_1^2 \, || \partial _{1}^{}\partial _{2}^{2} u || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^{2} \Bigr \} . \end{aligned}$$

Proof

Follows from Lemma C.3 by a rescaling argument. $\square$

Proposition C.5

Consider a rectangle $\mathrm {G}= (2a_1,2a_1+2 h_1) {{\mathrm{\times }}}(2a_2,2a_2+2 h_2) \subset \mathrm {Q}$. Let $w\in \mathbb {P}_{1}$ be the polynomial satisfying $w ( 2a_1 ) =1$ and $w ( 2a_1+2h_1 ) =0$.

Then, for every $v\in \mathbb {P}_{p}$ with $p\in {\mathbb {N}}$, the polynomial $\xi =w {{\mathrm{\otimes }}}v \in \mathbb {Q}_{1 ,{\,}\!p}$ satisfies

$$\begin{aligned} || \xi || _{\mathbb {L}^{2}_{} ( \mathrm {G} ) }^2 = \frac{2 h_1}{3} \, || v || _{\mathbb {L}^{2}_{} ( \gamma ) }^2 \quad \text {and}\quad | \xi | _{\mathbb {H}^{m+1}_{} ( \mathrm {G} ) }^2 = \frac{2 h_1}{3} \, | v | _{\mathbb {H}^{m+1}_{} ( \gamma ) }^2 + \frac{1}{2 h_1} \, | v | _{\mathbb {H}^{m}_{} ( \gamma ) }^2 , \end{aligned}$$

with $m=0,1$.

Proposition C.6

Consider a rectangle $\mathrm {Q}=(0,1)^{2}$ and $\beta \in [0,1)$. Then there exist positive constants $D_{0}$ and $D_{1}$ such that, for every $u\in \mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {Q} )$, the polynomial $v \in \mathbb {Q}_{1 ,{\,}\!1}$ interpolating u at the vertices of $\mathrm {Q}$ is well-defined and satisfies the error bounds

$$\begin{aligned} || u - v || _{\mathbb {L}^{2}_{} ( \mathrm {Q} ) }^2 \le D_{0}^2 \, | u | _{\mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {Q} ) }^2 \quad \text {and}\quad | u - v | _{\mathbb {H}^{1}_{} ( \mathrm {Q} ) }^2 \le D_{1}^2 \, | u | _{\mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {Q} ) }^2 . \end{aligned}$$

Proof

By Proposition 3.1, the function u admits continuous extension to ${{\mathrm{\mathbf{cl}\,}}}\mathrm {Q}$. This ensures that the interpolant is well defined. The error estimates follow from [29, Lemma 3.6], see also [62, Lemma 4.25]. $\square$

D Proof of Proposition 5.2

Proof

Let $u\in \mathbb {H}^{2}_{\beta } ( \mathrm {Q} )$ and $l\ge 2$. Applying Corollary C.2 in every except with $i_{0}= ( \mu ^{}_{1 {{\,}\!}1}-1 ,{\,}\!\mu ^{}_{2 {{\,}\!}1}-1 )$ and Proposition C.6 with a rescaling argument in , we obtain

where $D_{0}$ and $D_{1}$ are positive constants depending only on $\beta$. The claim follows immediately. $\square$

E Proofs of theorems for hp quasi-interpolation

1.1 E.1 Proof of Lemma 5.8

Proof

For an arbitrary function $u\in \mathbb {H}^{1 ,{\,}\!1}_{{\mathrm{mix}}} ( \mathrm {Q} )$, we shall define a lifting term so that and the mapping $u \mapsto w^{l}$ is linear.

For $1 \le j \le l$, let $\gamma ^{{{\,}\!}j}_{1}$ and $\tilde{\gamma }^{{{\,}\!}j}_{1}$ denote the right edges of $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}1}$ and $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}0}$ respectively, $\gamma ^{{{\,}\!}j}_{2}$ and $\tilde{\gamma }^{{{\,}\!}j}_{2}$, the top edges of $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}1}$ and $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{0 {{\,}\!}1}$ respectively, $\tilde{\Gamma }^{{{\,}\!}j}_{1}$ and $\tilde{\Gamma }^{{{\,}\!}j}_{2}$, the left edge of $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}0}$ and the bottom edge of $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{0 ,{\,}\!1}$ respectively.

Let us set consider the polynomial interpolants

given in Definition 5.7. For $1 \le j < l$ and $k\in \{ 1,2 \}$, we define linear univariate polynomials $\psi ^{{{\,}\!}j}_{k}\in \mathbb {P}_{1}$ by requiring

$$\begin{aligned} \psi ^{{{\,}\!}j}_{k} ( x^{{{\,}\!}l ,{\,}\!j}_{k} ) = 0 \quad \text {and}\quad \psi ^{{{\,}\!}j}_{k} ( x^{{{\,}\!}l ,{\,}\!j+1}_{k} ) = 1 . \end{aligned}$$

Using these as factors, we introduce bivariate lifting polynomials $\eta ^{{{\,}\!}j}_{1},\tilde{\eta }^{{{\,}\!}j}_{1}\in \mathbb {P}_{1 ,{\,}\!p}$ and $\eta ^{{{\,}\!}j}_{2},\tilde{\eta }^{{{\,}\!}j}_{2}\in \mathbb {P}_{p ,{\,}\!1}$ with $1 \le j < l$ by setting

(E.1a)

Additionally, we introduce lifting polynomials $\zeta _{1}\in \mathbb {P}_{1 ,{\,}\!p}$ and $\zeta _{2}\in \mathbb {P}_{p ,{\,}\!1}$:

(E.1b)

From the lifting polynomials defined in (E.1a)–(E.1b), we construct the lifting term :

$$\begin{aligned} w^{l} = \mathbbm {1}_{\mathrm {G}^{{{\,}\!}l ,{\,}\!1}_{1 {{\,}\!}0}} \zeta _{1} + \mathbbm {1}_{\mathrm {G}^{{{\,}\!}l ,{\,}\!1}_{0 {{\,}\!}1}} \zeta _{2} + \sum _{j=1}^{l-1} \Bigl \{ \mathbbm {1}_{\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}1}} \eta ^{{{\,}\!}j}_{1} + \mathbbm {1}_{\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}1}} \eta ^{{{\,}\!}j}_{2} + \mathbbm {1}_{\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}0}} \tilde{\eta }^{{{\,}\!}j}_{1} + \mathbbm {1}_{\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{0 {{\,}\!}1}} \tilde{\eta }^{{{\,}\!}j}_{2} \Bigr \} . \end{aligned}$$

(E.2)

By Definition 5.5, the interpolants of u in any two elements sharing an entire edge coincide on that edge. Then, due to the nodal exactness of interpolation in each element, the construction of (E.1a)–(E.1b) ensures that, first, $w^{l}$ vanishes in $\mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{}$ and on ${{\mathrm{\varvec{\partial }}}}\mathrm {Q}$ and, second, the lifted interpolant extends to a function continuous across the edges of the elements of : .

Since the operator is linear, so is the mapping $u \mapsto w^{l}$. This allows to define a linear operator on u by setting . $\square$

1.2 E.2 Preliminary bounds

In this section, using the auxiliary results presented in Appendices B and C below, we prove preliminary approximation and stability results of hp approximation, which are specified for the analyticity class in Theorem 5.10.

For all $r\in {\mathbb {N}}$, $\sigma \in {\mathbb {R}}$, we introduce $[ \,\cdot \, ] _{r+1 ,{\,}\!\sigma }$, a broken Sobolev seminorm on $\mathbb {H}^{r+1}_{} ( \mathrm {Q}\setminus \mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{} )$:

$$\begin{aligned}{}[ u ] ^{2}_{r+1 ,{\,}\!\sigma } = \sum _{j=1}^{l} \sum _{\nu \in \mathcal {N}} ( 2^{j-2-l} \Lambda _{l} ) ^{2 ( r+1-\sigma ) } \, | u | _{\mathbb {H}^{r+1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2 \quad \text {for all}\quad u\in \mathbb {H}^{r+1}_{} ( \mathrm {Q}\setminus \mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{} ) .\nonumber \\ \end{aligned}$$

(E.3)

We note that $[ \cdot ] _{r+1 ,{\,}\!\sigma }$ depends on $\mu ^{}$ and l, which define . For the sake of brevity, we do not indicate this dependence explicitly in the notation.

Lemma E.1

(Estimates for hp quasi-interpolation in terms of the broken Sobolev seminorms) Let $\beta \in [0,1)$. Then there exist positive constants $D_{0}$ and $D_{1}$ such that, for every $\mu ^{}\in \{ 0,1 \} ^{2\times 2}$, for all $l,p,s\in {\mathbb {N}}$ such that $l \ge 2$ and $s \le p$ and for every $u\in \mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {Q} ) \cap \mathbb {H}^{s+2 ,{\,}\!2}_{\beta } ( \mathrm {G}^{l} )$, the hp approximation satisfies the following error and stability bounds:

$$\begin{aligned} || u - v^{l} || _{\mathbb {L}^{2}_{} ( \mathrm {Q} ) }^2&\le 3 \, \frac{ \varUpsilon _{p {{\,}\!}s} }{ p ( p+1 ) } \, [ u ] ^{2}_{s+1 ,{\,}\!0} + D_{0}^2 \, \frac{\Lambda _{l}^{4}}{\lambda _{l}^{2}} \, 2^{-2 ( 2-\beta ) l} \, | u | _{\mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{} ) }^2 , \end{aligned}$$

(E.4a)

(E.4b)

$$\begin{aligned} \sum _{j {{\,}\!}\nu } | v^{l} | _{\mathbb {H}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^{2}&\le \frac{\Lambda _{l}^{4}}{\lambda _{l}^{4}} \, ( p^2 - 1 ) \, [ u ] ^{2}_{2 ,{\,}\!2} \,.\qquad \end{aligned}$$

(E.4c)

Furthermore, on the boundary we have the inequalities

$$\begin{aligned} || u-v^{l} || _{\mathbb {L}^{2}_{} ( \Gamma ^{{{\,}\!}l} ) }^2&\le \frac{3}{2} \, \frac{\Lambda _{l}}{\lambda _{l}} \, \frac{\varUpsilon _{p {{\,}\!}s}}{p ( p+1 ) } \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!\frac{1}{2}}+ \, [ u ] ^{2}_{s+2 ,{\,}\!\frac{1}{2}} \Bigr \} , \end{aligned}$$

(E.5a)

$$\begin{aligned} | u-v^{l} | _{\mathbb {H}^{1}_{} ( \Gamma ^{{{\,}\!}l} ) }^2&\le \frac{3}{2} \, \frac{\Lambda _{l}}{\lambda _{l}} \, \varUpsilon _{p {{\,}\!}s} \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!\frac{3}{2}}+ \, [ u ] ^{2}_{s+2 ,{\,}\!\frac{3}{2}} \Bigr \} , \end{aligned}$$

(E.5b)

(E.5c)

where with $j=1,\ldots ,l$ and $\nu \in \mathcal {N}$ are given by (5.8a).

Proof

By Proposition C.6 and a rescaling argument, the interpolant $v^{l}$ satisfies the error bounds

$$\begin{aligned} || u - v^{l} || _{\mathbb {L}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{} ) }^2&\le D_{0}^2 \, \frac{\Lambda _{l}^{4}}{\lambda _{l}^{2}} \, 2^{-2 ( 2-\beta ) l} \, | u | _{\mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{} ) }^2 , \end{aligned}$$

(E.6a)

$$\begin{aligned} | u - v^{l} | _{\mathbb {H}^{1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{} ) }^2&\le D_{1}^2 \, \frac{\Lambda _{l}^{4}}{\lambda _{l}^{4}} \, 2^{-2 ( 1-\beta ) l} \, | u | _{\mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{} ) }^2 \end{aligned}$$

(E.6b)

with positive constants $D_{0}$ and $D_{1}$ depending only on $\beta$.

For all $j=1,\ldots ,l$, $\nu \in \mathcal {N}$ and $s\in {\mathbb {N}}$ such that $s \le p$, Corollary C.2 yields the following:

$$\begin{aligned} \begin{aligned} || u - v^{l} || _{\mathbb {L}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^{2}&\le 3 \, ( 2^{j-2-l} \Lambda _{l} ) ^{2s+2} \, \frac{\varUpsilon _{p {{\,}\!}s}}{p\, ( p+1 ) } \, | u | _{\mathbb {H}^{s+1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2 ,\\ | u - v^{l} | _{\mathbb {H}^{1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^{2}&\le 4 \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \, ( 2^{j-2-l} \Lambda _{l} ) ^{2s} \, \varUpsilon _{p {{\,}\!}s} \, | u | _{\mathbb {H}^{s+1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2 ,\\ | v^{l} | _{\mathbb {H}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^{2}&\le \frac{\Lambda _{l}^{4}}{\lambda _{l}^{4}} \, ( p^2 - 1 ) \, | u | _{\mathbb {H}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^{2} . \end{aligned} \end{aligned}$$

(E.7)

By combining the inequalities (E.6a)–(E.6b) and (E.7), we obtain the bounds of (E.4a)–(E.4c).

Finally, for all $j=1,\ldots ,l$ and $\nu \in \mathcal {N}$ such that of (5.8a) is nonempty, we apply Corollary C.4 to $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu }$ and :

By summing over all $j=1,\ldots ,l$ and $\nu \in \mathcal {N}$, we obtain (E.5a)–(E.5c). $\square$

Let us denote the right and upper halves of $\mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{}$ as follows:

$$\begin{aligned} \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{1} = (\frac{1}{2} x^{{{\,}\!}l ,{\,}\!1}_{1},x^{{{\,}\!}l ,{\,}\!1}_{1}) {{\mathrm{\times }}}(0,x^{{{\,}\!}l ,{\,}\!1}_{2}) , \quad \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{2} = (0,x^{{{\,}\!}l ,{\,}\!1}_{1}) {{\mathrm{\times }}}(\frac{1}{2} x^{{{\,}\!}l ,{\,}\!1}_{2},x^{{{\,}\!}l ,{\,}\!1}_{2}) . \end{aligned}$$

(E.8)

Lemma E.2

(Estimates for trace lifting in terms of the broken Sobolev seminorms) Let $\beta \in [0,1)$. Then, for every $\mu ^{}\in \{ 0,1 \} ^{2\times 2}$, for all $l,p,s\in {\mathbb {N}}$ such that $l \ge 2$ and $s \le p$ and for every $u\in \mathbb {H}^{3 ,{\,}\!2}_{} ( \mathrm {Q} ) \cap \mathbb {H}^{s+2 ,{\,}\!2}_{\beta } ( \mathrm {G}^{l} )$, the trace-lifting term of Lemma 5.8 satisfies the bounds

$$\begin{aligned} || w^{l} || _{\mathbb {L}^{2}_{} ( \mathrm {Q} ) }^2&\le \frac{1}{4} \, \frac{\Lambda _{l}^5}{\lambda _{l}} \, 2^{-4l} \, Z_0^2 + 3 \, \frac{\Lambda _{l}}{\lambda _{l}} \, \frac{\varUpsilon _{p {{\,}\!}s}}{p ( p+1 ) } \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!0} + [ u ] ^{2}_{s+2 ,{\,}\!0} \Bigr \} , \end{aligned}$$

(E.9a)

$$\begin{aligned} \sum _{j {{\,}\!}\nu } | w^{l} | _{\mathbb {H}^{1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2&\le \frac{11}{4} \, \frac{\Lambda _{l}^4}{\lambda _{l}^2} \, 2^{-2l} \, Z_0^2 + \frac{15}{2} \, \frac{\Lambda _{l}^2}{\lambda _{l}^2} \, \varUpsilon _{p {{\,}\!}s} \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!1} + [ u ] ^{2}_{s+2 ,{\,}\!1} \Bigr \} , \end{aligned}$$

(E.9b)

$$\begin{aligned} \sum _{j {{\,}\!}\nu } | w^{l} | _{\mathbb {H}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2&\le 6 \, \frac{\Lambda _{l}^2}{\lambda _{l}^2} Z_0^2 + 3 p^2 \Bigl \{ [ u ] ^{2}_{2 ,{\,}\!2} + [ u ] ^{2}_{3 ,{\,}\!2} \Bigr \} , \end{aligned}$$

(E.9c)

where

$$\begin{aligned} Z_0^2 = \sum _{k=1,2} \Bigl \{ | u | _{\mathbb {H}^{2}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 + ( 2^{-l-2} \Lambda _{l} ) ^2 \, | u | _{\mathbb {H}^{3}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 \Bigr \} \end{aligned}$$

(E.10)

with $\tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{1}$ and $\tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{2}$ given by (E.8).

Proof

Using the Cauchy–Bunyakovsky–Schwarz inequality in each $\mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu }$, we may bound the lifting term $w^{l}$ of Lemma 5.8 as follows:

$$\begin{aligned} \sum _{j=1}^{l-1} \sum _{\nu \in \mathcal {N}} | w^{l} | _{\mathbb {H}^{m}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2\le & {} 2 \, \Bigl \{ | \zeta _{1} | _{\mathbb {H}^{m}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!1}_{1 {{\,}\!}0} ) }^2 + | \zeta _{2} | _{\mathbb {H}^{m}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!1}_{0 {{\,}\!}1} ) }^2 \Bigr \} \nonumber \\&+ 2 \sum _{j=1}^{l-1} \sum _{\nu \in \mathcal {N}} \left\{ | \tilde{\eta }^{{{\,}\!}j}_{1} | _{\mathbb {H}^{m}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}0} ) }^2 + | \tilde{\eta }^{{{\,}\!}j}_{2} | _{\mathbb {H}^{m}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{0 {{\,}\!}1} ) }^2 \right. \nonumber \\&\left. + | \eta ^{{{\,}\!}j}_{1} | _{\mathbb {H}^{m}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}1} ) }^2 + | \eta ^{{{\,}\!}j}_{2} | _{\mathbb {H}^{m}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{1 {{\,}\!}1} ) }^2 \right\} \end{aligned}$$

(E.11)

for $m=0,1,2$.

Applying Proposition C.5 and the Cauchy–Bunyakovsky–Schwarz inequality on edges of the elements, we obtain from (E.2) and (E.11) that

$$\begin{aligned} || w^{l} || _{\mathbb {L}^{2}_{} ( \mathrm {Q} ) }^2\le & {} \frac{4}{3} \, 2^{-l} \Lambda _{l} \, \left\{ || u-v^{{{\,}\!}0} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{1} ) }^2 + || u-v^{{{\,}\!}1}_{1 ,{\,}\!0} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{1} ) }^2\right. \\&\left. + || u-v^{{{\,}\!}0} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{2} ) }^2 + || u-v^{{{\,}\!}1}_{0 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{2} ) }^2 \right\} \\&+ \sum _{j=1}^{l-1} \frac{4}{3} \, 2^{j-1-l} \Lambda _{l} \, \left\{ || u-v^{{{\,}\!}j}_{1 ,{\,}\!0} || _{\mathbb {L}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{1} ) }^2 + || u-v^{{{\,}\!}j}_{1 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \gamma ^{{{\,}\!}j}_{1} ) }^2 \right. \nonumber \\&+ || u-v^{{{\,}\!}j+1}_{1 ,{\,}\!0} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j+1}_{1} ) }^2\\&\left. + || u-v^{{{\,}\!}j}_{0 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{2} ) }^2 + || u-v^{{{\,}\!}j}_{1 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \gamma ^{{{\,}\!}j}_{2} ) }^2 + || u-v^{{{\,}\!}j+1}_{0 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j+1}_{2} ) }^2 \right\} . \end{aligned}$$

Rearranging the terms and using Corollary C.4, we arrive at

$$\begin{aligned} || w^{l} || _{\mathbb {L}^{2}_{} ( \mathrm {Q} ) }^2\le & {} \frac{4}{3} \, 2^{-l} \Lambda _{l} \, || u-v^{{{\,}\!}0} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{1} \cup \, \tilde{\Gamma }^{{{\,}\!}1}_{2} ) }^2\nonumber \\&+ \frac{4}{3} \, \sum _{j=1}^{l} 2^{j-1-l} \Lambda _{l} \, \bigg \{ \frac{1}{2} \, || u-v^{{{\,}\!}j}_{1 ,{\,}\!0} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{1} ) }^2 + || u-v^{{{\,}\!}j}_{1 ,{\,}\!0} || _{\mathbb {L}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{1} ) }^2\nonumber \\&+ \frac{1}{2} \, || u-v^{{{\,}\!}j}_{0 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{2} ) }^2 + || u-v^{{{\,}\!}j}_{0 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{2} ) }^2 + || u-v^{{{\,}\!}j}_{1 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \gamma ^{{{\,}\!}j}_{1} \cup \, \gamma ^{{{\,}\!}j}_{2} ) }^2 \bigg \}\nonumber \\\le & {} \frac{1}{4} \, \frac{\Lambda _{l}}{\lambda _{l}} \, ( 2^{-l} \Lambda _{l} ) ^{4} \, \sum _{k=1}^{2} \Bigl \{ | u | _{\mathbb {H}^{2}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 + ( 2^{-l-2} \Lambda _{l} ) ^2 \, | u | _{\mathbb {H}^{3}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 \Bigr \} \nonumber \\&+ 3 \, \frac{\Lambda _{l}}{\lambda _{l}} \, \frac{\varUpsilon _{p {{\,}\!}s}}{p ( p+1 ) } \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!0} + [ u ] ^{2}_{s+2 ,{\,}\!0} \Bigr \} , \end{aligned}$$

(E.12a)

where $\tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{1}$ and $\tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{2}$ are given by (E.8).

Analogously to (E.12a), we obtain the bounds

$$\begin{aligned} \sum _{j=1}^{l-1} \sum _{\nu \in \mathcal {N}} | w^{l} | _{\mathbb {H}^{1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2\le & {} \frac{4}{3} \, 2^{-l} \Lambda _{l} \, | u-v^{{{\,}\!}0} | _{\mathbb {H}^{1}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{1} \cup \, \tilde{\Gamma }^{{{\,}\!}1}_{2} ) }^2 + \frac{4}{2^{-l} \lambda _{l}} \, || u-v^{{{\,}\!}0} || _{ \mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{1} \cup \, \tilde{\Gamma }^{{{\,}\!}1}_{2} ) }\nonumber \\&+ \frac{4}{3} \, \sum _{j=1}^{l} 2^{j-1-l} \Lambda _{l} \, \bigg \{ \frac{1}{2} \, | u-v^{{{\,}\!}j}_{1 ,{\,}\!0} | _{\mathbb {H}^{1}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{1} ) }^2 + | u-v^{{{\,}\!}j}_{1 ,{\,}\!0} | _{\mathbb {H}^{1}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{1} ) }^2\nonumber \\&+ \frac{1}{2} \, | u-v^{{{\,}\!}j}_{0 ,{\,}\!1} | _{\mathbb {H}^{1}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{2} ) }^2 + | u-v^{{{\,}\!}j}_{0 ,{\,}\!1} | _{\mathbb {H}^{1}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{2} ) }^2 + | u-v^{{{\,}\!}j}_{1 ,{\,}\!1} | _{\mathbb {H}^{1}_{} ( \gamma ^{{{\,}\!}j}_{1} \cup \, \gamma ^{{{\,}\!}j}_{2} ) }^2 \bigg \}\nonumber \\&+ 4 \, \sum _{j=1}^{l} \frac{1}{2^{j-1-l} \lambda _{l}} \, \bigg \{ 2 \, || u-v^{{{\,}\!}j}_{1 ,{\,}\!0} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{1} ) }^2 + || u-v^{{{\,}\!}j}_{1 ,{\,}\!0} || _{\mathbb {L}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{1} ) }^2\nonumber \\&+ 2 \, || u-v^{{{\,}\!}j}_{0 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{2} ) }^2 + || u-v^{{{\,}\!}j}_{0 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{2} ) }^2 + || u-v^{{{\,}\!}j}_{1 ,{\,}\!1} || _{\mathbb {L}^{2}_{} ( \gamma ^{{{\,}\!}j}_{1} \cup \, \gamma ^{{{\,}\!}j}_{2} ) }^2 \bigg \}\nonumber \\\le & {} \frac{\Lambda _{l}}{\lambda _{l}} \biggl [ 2 + \frac{3}{4} \, \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] \, ( 2^{-l} \Lambda _{l} ) ^{2} \, \sum _{k=1}^{2} \Bigl \{ | u | _{\mathbb {H}^{2}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 + ( 2^{-l-2} \Lambda _{l} ) ^2 \, | u | _{\mathbb {H}^{3}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 \Bigr \} \nonumber \\&+\, 6 \, \frac{\Lambda _{l}}{\lambda _{l}} \biggl [ 1 + \frac{3}{2} \, \frac{\Lambda _{l}}{\lambda _{l}} \frac{1}{p ( p+1 ) } \biggr ] \, \varUpsilon _{p {{\,}\!}s} \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!1} + [ u ] ^{2}_{s+2 ,{\,}\!1} \Bigr \} \end{aligned}$$

(E.12b)

and

$$\begin{aligned} \sum _{j=1}^{l-1} \sum _{\nu \in \mathcal {N}} | w^{l} | _{\mathbb {H}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2\le & {} \frac{4}{2^{-l} \lambda _{l}} \, | u-v^{{{\,}\!}0} | _{\mathbb {H}^{1}_{} ( \tilde{\Gamma }^{{{\,}\!}1}_{1} \cup \, \tilde{\Gamma }^{{{\,}\!}1}_{2} ) }^2\nonumber \\&+ \frac{4}{3} \, \sum _{j=1}^{l} 2^{j-1-l} \Lambda _{l} \, \bigg \{ \frac{1}{2} \, | v^{{{\,}\!}j}_{1 ,{\,}\!0} | _{\mathbb {H}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{1} ) }^2 + | v^{{{\,}\!}j}_{1 ,{\,}\!0} | _{\mathbb {H}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{1} ) }^2\nonumber \\&+ \frac{1}{2} \, | v^{{{\,}\!}j}_{0 ,{\,}\!1} | _{\mathbb {H}^{2}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{2} ) }^2 + | v^{{{\,}\!}j}_{0 ,{\,}\!1} | _{\mathbb {H}^{2}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{2} ) }^2 + | v^{{{\,}\!}j}_{1 ,{\,}\!1} | _{\mathbb {H}^{2}_{} ( \gamma ^{{{\,}\!}j}_{1} \cup \, \gamma ^{{{\,}\!}j}_{2} ) }^2 \bigg \}\nonumber \\&+ 4 \, \sum _{j=1}^{l} \frac{1}{2^{j-1-l} \lambda _{l}} \, \bigg \{ 2 \, | u-v^{{{\,}\!}j}_{1 ,{\,}\!0} | _{\mathbb {H}^{1}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{1} ) }^2 + | u-v^{{{\,}\!}j}_{1 ,{\,}\!0} | _{\mathbb {H}^{1}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{1} ) }^2\nonumber \\&+ 2 \, | u-v^{{{\,}\!}j}_{0 ,{\,}\!1} | _{\mathbb {H}^{1}_{} ( \tilde{\Gamma }^{{{\,}\!}j}_{2} ) }^2 + | u-v^{{{\,}\!}j}_{0 ,{\,}\!1} | _{\mathbb {H}^{1}_{} ( \tilde{\gamma }^{{{\,}\!}j}_{2} ) }^2 \nonumber \\&+ | u-v^{{{\,}\!}j}_{1 ,{\,}\!1} | _{\mathbb {H}^{1}_{} ( \gamma ^{{{\,}\!}j}_{1} \cup \, \gamma ^{{{\,}\!}j}_{2} ) }^2 \bigg \}\nonumber \\\le & {} 6 \, \frac{\Lambda _{l}^2}{\lambda _{l}^2} \sum _{k=1}^{2} \Bigl \{ | u | _{\mathbb {H}^{2}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 + ( 2^{-l-2} \Lambda _{l} ) ^{2} \, | u | _{\mathbb {H}^{3}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2 \Bigr \} \nonumber \\&+ 3 \, \frac{\Lambda _{l}}{\lambda _{l}} \biggl [ p^2 - 1 + 3 \, \frac{\Lambda _{l}}{\lambda _{l}} \, \frac{1}{p ( p+1 ) } \biggr ] \Bigl \{ [ u ] ^{2}_{2 ,{\,}\!2} + [ u ] ^{2}_{3 ,{\,}\!2} \Bigr \} .\nonumber \\ \end{aligned}$$

(E.12c)

The final inequalities of (E.12a)–(E.12c) prove the bounds of (E.9a)–(E.9b). $\square$

Let $\mathrm {G}\subset \mathrm {Q}$ be a rectangle. Assume that in the sense of Definition 3.3 with positive constants $C_{u}$ and $\delta _{u}$. Then, using Lemma A.1, we obtain for every $r\in {\mathbb {N}}$ that

(E.13a)

Let $l\in {\mathbb {N}}$, $j=1,\ldots ,l$ and $\nu \in \mathcal {N}$. Using (5.4), we obtain the bound for all $x\in \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu }$ and $r\in {\mathbb {N}}$. The corresponding Sobolev seminorm is thus bounded as follows:

$$\begin{aligned} | u | _{\mathbb {H}^{r+1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2\le & {} \frac{64}{9} \, \bigl \{ C_{u} \, \delta _{u}^{r+1} \, ( r+1 ) ! \bigr \} ^2 \, \lambda _{l}^{-2 ( r+1 ) } \, 2^{-2 ( r+\beta ) ( j-1-l ) } \, \Lambda _{l}^2 \, 2^{2 ( j-1-l ) }\nonumber \\= & {} \frac{64}{9} \, \frac{\Lambda _{l}^2}{\lambda _{l}^{2 ( r+1 ) }} \, \bigl \{ C_{u} \, \delta _{u}^{r+1} \, ( r+1 ) ! \bigr \} ^2 \, 2^{-2 ( r+\beta -1 ) ( j-1-l ) } \end{aligned}$$

(E.13b)

for all $r\in {\mathbb {N}}$, $l\in {\mathbb {N}}$, $j=1,\ldots ,l$ and $\nu \in \mathcal {N}$.

For $\tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k}$ with $k=1,2$ given by (E.8), we obtain a similar bound: for all $x\in \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu }$ and $r\in {\mathbb {N}}$, and, therefore,

$$\begin{aligned} | u | _{\mathbb {H}^{r+1}_{} ( \tilde{\mathrm {G}}^{{{\,}\!}l ,{\,}\!0}_{k} ) }^2\le & {} \frac{64}{9} \, \bigl \{ C_{u} \, \delta _{u}^{r+1} \, ( r+1 ) ! \bigr \} ^2 \, \lambda _{l}^{-2 ( r+1 ) } \, 2^{2 ( r+\beta ) ( l+1 ) } \, \frac{1}{2} \, \Lambda _{l}^2 \, 2^{-2l}\nonumber \\= & {} \frac{128}{9} \, \frac{\Lambda _{l}^2}{\lambda _{l}^{2 ( r+1 ) }} \, \bigl \{ C_{u} \, \delta _{u}^{r+1} \, ( r+1 ) ! \bigr \} ^2 \, 2^{2 ( r+\beta -1 ) ( l+1 ) } \end{aligned}$$

(E.13c)

holds for all $r\in {\mathbb {N}}$, $l\in {\mathbb {N}}$, and $k=1,2$.

Lemma E.3

(estimates for the broken Sobolev seminorms) Assume $\beta \in [0,1)$, $\alpha \ge 1$, $\mu ^{}\in \{ 0,1 \} ^{2\times 2}$ and in the sense of Definition 3.3 with positive constants $C_{u}$ and $\delta _{u}$. Then, with p and s given by (5.10), the following bound is satisfied for all $l,r\in {\mathbb {N}}$ and $\sigma \in {\mathbb {R}}$:

$$\begin{aligned} \varUpsilon _{p {{\,}\!}s} \, [ u ] ^{2}_{r+1 ,{\,}\!\sigma } \le \frac{64}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, \frac{\Lambda _{l}^{2 ( r+2-\sigma ) }}{\lambda _{l}^{2 ( r+1 ) }} \, \frac{M_{l {{\,}\!}\sigma }^2 \, C_{u}^2 \, \delta _{u}^{2 ( r+1-s ) }}{ 2^{2 ( r+1-\sigma -s ) } } \, s \, \biggl \{ \frac{ ( r+1 ) !}{s!} \biggr \} ^2 \, 2^{-2 ( \alpha -\beta ) l} , \end{aligned}$$

where

$$\begin{aligned} M_{l {{\,}\!}\sigma }^2 = \max _{1 \le j \le l} 2^{2 ( 2-\beta -\sigma ) ( j-1-l ) } . \end{aligned}$$

Proof

Applying (E.13b) for all $j=1,\ldots ,l$ and $\nu \in \mathcal {N}$, we deduce the following bound for $[ u ] ^{2}_{r+1 ,{\,}\!\sigma }$ of (E.3) with arbitrary $r\in {\mathbb {N}}$ and $\sigma \in {\mathbb {R}}$:

$$\begin{aligned}{}[ u ] ^{2}_{r+1 ,{\,}\!\sigma }\le & {} \frac{64}{9} \, \bigl \{ C_{u} \, \delta _{u}^{r+1} \, ( r+1 ) ! \bigr \} ^2 \, \sum _{j {{\,}\!}\nu } \frac{ \Lambda _{l}^{2 ( r+2-\sigma ) } \, 2^{2 ( r+1-\sigma ) ( j-2-l ) } }{ \lambda _{l}^{2 ( r+1 ) } \, 2^{2 ( r+\beta -1 ) ( j-1-l ) } }\nonumber \\= & {} \frac{64}{9} \, 2^{-2 ( r+1-\sigma ) } \, \frac{\Lambda _{l}^{2 ( r+2-\sigma ) }}{\lambda _{l}^{2 ( r+1 ) }} \, \bigl \{ C_{u} \, \delta _{u}^{r+1} \, ( r+1 ) ! \bigr \} ^2 \, 3 \sum _{j=1}^{l} 2^{2 ( 2-\beta -\sigma ) ( j-1-l ) }\nonumber \\= & {} \frac{64}{3} \, \frac{\Lambda _{l}^{2 ( r+2-\sigma ) }}{\lambda _{l}^{2 ( r+1 ) }} \, 2^{-2 ( r+1-\sigma ) } \, l \, M_{l {{\,}\!}\sigma }^2 \bigl \{ C_{u} \, \delta _{u}^{r+1} \, ( r+1 ) ! \bigr \} ^2 . \end{aligned}$$

For p and s given by (5.10), using Lemma A.2, we arrive at

$$\begin{aligned} \varUpsilon _{p {{\,}\!}s} \, [ u ] ^{2}_{r+1 ,{\,}\!\sigma }\le & {} ( \varrho -1 ) ^{2s} \, \varUpsilon _{p {{\,}\!}s} \, ( s! ) ^2 \, \frac{64}{3} \, \frac{\Lambda _{l}^{2 ( r+2-\sigma ) }}{\lambda _{l}^{2 ( r+1 ) }} \, \frac{l \, M_{l {{\,}\!}\sigma }^2 \, C_{u}^2 \, \delta _{u}^{2 ( r+1-s ) }}{ 2^{2 ( r+1-\sigma -s ) } } \, \biggl \{ \frac{ ( r+1 ) !}{s!} \biggr \} ^2\nonumber \\\le & {} \frac{64}{3} \, \frac{e^{5}}{\sqrt{2\pi }} \, s \, \exp \biggl ( -\frac{2p}{\varrho } \biggr ) \, \frac{\Lambda _{l}^{2 ( r+2-\sigma ) }}{\lambda _{l}^{2 ( r+1 ) }} \, \frac{l \, M_{l {{\,}\!}\sigma }^2 \, C_{u}^2 \, \delta _{u}^{2 ( r+1-s ) }}{ 2^{2 ( r+1-\sigma -s ) } } \, \biggl \{ \frac{ ( r+1 ) !}{s!} \biggr \} ^2\nonumber \\\le & {} \frac{64}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, \frac{\Lambda _{l}^{2 ( r+2-\sigma ) }}{\lambda _{l}^{2 ( r+1 ) }} \, \frac{l \, M_{l {{\,}\!}\sigma }^2 \, C_{u}^2 \, \delta _{u}^{2 ( r+1-s ) }}{ 2^{2 ( r+1-\sigma -s ) } } \, s \, \biggl \{ \frac{ ( r+1 ) !}{s!} \biggr \} ^2 \, 2^{-2 ( \alpha -\beta ) l} . \end{aligned}$$

$\square$

1.3 E.3 Proof of Theorem 5.10

Proof

For p and s given by (5.10), Lemma E.3 yields the following bounds:

$$\begin{aligned}&\varUpsilon _{p {{\,}\!}s} \, [ u ] ^{2}_{s+1 ,{\,}\!1} \le \frac{64}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, l \, C_{u}^2\, \delta _{u}^{2} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2 ( s+1 ) } \, s \, ( s+1 ) ^2 \; 2^{-2 ( \alpha -\beta ) l}\nonumber \\&\quad \le \frac{64}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, C_{u}^2\, \delta _{u}^{2} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2 ( \chi _{l}+2 ) } \frac{\chi _{l}+1}{\chi _{l}-1} \, ( \chi _{l}+2 ) ^2 \, l \; 2^{-2 ( \alpha -\beta ) l} , \end{aligned}$$

(E.14a)

$$\begin{aligned}&\varUpsilon _{p {{\,}\!}s} \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!\frac{1}{2}} + [ u ] ^{2}_{s+2 ,{\,}\!\frac{1}{2}} \Bigr \} \nonumber \\&\quad \le \frac{64}{3} \, \biggl \{ \frac{1}{s^2} + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \frac{e^{3}}{\sqrt{2\pi }} \, \frac{\Lambda _{l}^{2s+3}}{\lambda _{l}^{2s+2}} \, \frac{ C_{u}^2 }{2} \, \delta _{u}^4 \, l \, s \, ( s+1 ) ^2 \, ( s+2 ) ^2 \, 2^{-2 ( \alpha -\beta ) l}\nonumber \\&\quad \le \frac{32}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, C_{u}^2 \, \delta _{u}^4 \, \biggl \{ \frac{1}{ ( \chi _{l}-1 ) ^2} + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \Lambda _{l} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2 ( \chi _{l}+3 ) }\nonumber \\&\qquad \cdot ( \chi _{l}+1 ) ( \chi _{l}+2 ) ^2 ( \chi _{l}+3 ) ^2 \, l \, 2^{-2 ( \alpha -\beta ) l} , \end{aligned}$$

(E.14b)

$$\begin{aligned}&\varUpsilon _{p {{\,}\!}s} \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!\frac{3}{2}} + [ u ] ^{2}_{s+2 ,{\,}\!\frac{3}{2}} \Bigr \} \le \frac{64}{3} \, \biggl \{ \frac{1}{s^2} + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \frac{e^{3}}{\sqrt{2\pi }} \, \frac{\Lambda _{l}^{2s+1}}{\lambda _{l}^{2s}} \, 2^{-2 ( \tilde{\beta }-\beta ) l} \, 2 \, C_{u}^2 \, \delta _{u}^4 \, l\nonumber \\&\quad \cdot s \, ( s+1 ) ^2 \, ( s+2 ) ^2 \, 2^{-2 ( \alpha -\beta ) l}\nonumber \\&\quad \le \frac{128}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, C_{u}^2 \, \delta _{u}^4 \, \biggl \{ \frac{1}{ ( \chi _{l}-1 ) ^2} + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \Lambda _{l} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2\chi _{l}+2}\nonumber \\&\qquad \cdot ( \chi _{l}+1 ) ( \chi _{l}+2 ) ^2 ( \chi _{l}+3 ) ^2 \, l \, 2^{-2 ( \alpha +\tilde{\beta }-2\beta ) l} \, , \end{aligned}$$

(E.14c)

where $\tilde{\beta }=\min \bigl \{ \frac{1}{2} ,{\,}\!\beta \bigr \}$. Also, using (E.3) and (E.13b), we obtain the estimates

$$\begin{aligned}{}[ u ] ^{2}_{2 ,{\,}\!2}&\le \frac{256}{9} \, C_{u}^2 \, \delta _{u}^4 \, \frac{\Lambda _{l}^2}{\lambda _{l}^{4}} \, 2^{2\beta ( l+1 ) } , \end{aligned}$$

(E.15a)

$$\begin{aligned}{}[ u ] ^{2}_{2 ,{\,}\!\frac{5}{2}} + [ u ] ^{2}_{3 ,{\,}\!\frac{5}{2}}&\le \frac{256}{9} \, C_{u}^2 \, \delta _{u}^4 \, \frac{\Lambda _{l}^{4}}{\lambda _{l}^{4}} \, \Lambda _{l}^{-1} \lambda _{l}^{-2} \, \biggl \{ 1 + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, 2^{ ( 2\beta +1 ) ( l+1 ) } . \end{aligned}$$

(E.15b)

Combining the bounds of Lemma E.1 with (E.14a)–(E.15b) and applying Definition 3.3 again, we obtain

where $\tilde{\beta }=\min \bigl \{ \frac{1}{2} ,{\,}\!\beta \bigr \}$,

with any positive $C_{1},C_{2},c_{0},c_{1},c_{2}$ such that

$$\begin{aligned} C_{1}^2\ge & {} \frac{64}{3} \frac{e^{3}}{\sqrt{2\pi }} \, C_{u}^2 \, \delta _{u}^{2} \, \biggl \{ \frac{1}{32} \, \frac{1}{\varrho _{\delta }^{2} \chi _{l}^{2}} + 4 \biggr \} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2\chi _{l}+6} \frac{\chi _{l}+1}{\chi _{l}-1} \, \frac{ ( \chi _{l}+2 ) ^2}{l^2} \; 2^{-2 ( \alpha -1 ) l}\\&+ \frac{1}{l^3} \, \Bigl \{ 2^{-2l} \, \lambda _{l}^2 \, D_{0}^2 + D_{1}^2 \Bigr \} \, \frac{\Lambda _{l}^{4}}{\lambda _{l}^{4}} \, | u | _{\mathbb {H}^{2 ,{\,}\!2}_{\beta } ( \mathrm {Q} ) }^2 ,\\ C_{2}^2\ge & {} \frac{256}{9} \, 2^{2\beta } \, C_{u}^2 \, \delta _{u}^4 \, \frac{\Lambda _{l}^{6}}{\lambda _{l}^{8}} \, \frac{ ( \varrho _{\delta } \chi _{l} + 1 ) ^2}{l^2} , \\ c_{0}^2\ge & {} 16 \, \frac{e^{3}}{\sqrt{2\pi }} \, \frac{ C_{u}^2 \, \delta _{u}^4}{\varrho _{\delta }^2} \, \frac{ ( \chi _{l}+1 ) ( \chi _{l}+2 ) ^2 ( \chi _{l}+3 ) ^2}{\chi _{l}^2\, l^3}\\&\cdot \biggl \{ \frac{1}{ ( \chi _{l}-1 ) ^2} + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \Lambda _{l} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2\chi _{l}+5} ,\\ c_{1}^2\ge & {} 64 \, \frac{e^{3}}{\sqrt{2\pi }} \, C_{u}^2 \, \delta _{u}^4 \, \frac{ ( \chi _{l}+1 ) ( \chi _{l}+2 ) ^2 ( \chi _{l}+3 ) ^2}{l^5}\\&\cdot \biggl \{ \frac{1}{ ( \chi _{l}-1 ) ^2} + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \Lambda _{l} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2\chi _{l}+3} ,\\ c_{2}^2\ge & {} \frac{128}{3} \, 2^{2\beta } \, C_{u}^2 \, \delta _{u}^4 \, \frac{\Lambda _{l}^{4}}{\lambda _{l}^{7}} \, \frac{ ( \varrho _{\delta } \chi _{l} + 1 ) ^2}{l^2} \, \biggl \{ 1 + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} , \end{aligned}$$

where the expressions on the right-hand side are monotonically decreasing with respect to l. $\square$

1.4 E.4 Proof of Theorem 5.11

Proof

Let $l\in {\mathbb {N}}$ be greater than one. Consider p and s given by (5.10). According to Lemma E.2, there exists vanishing in $\mathrm {G}^{{{\,}\!}l ,{\,}\!0}_{}$ and on ${{\mathrm{\varvec{\partial }}}}\mathrm {Q}$ and such that and the bounds (E.9a)–(E.9b) are satisfied. From those bounds, we obtain

$$\begin{aligned} \sum _{j {{\,}\!}\nu } || w^{l} || _{\mathbb {H}^{1}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2&\le 3 \, \frac{\Lambda _{l}^4}{\lambda _{l}^2} \, 2^{-2l} \, Z_0^2 + 9 \, \frac{\Lambda _{l}^2}{\lambda _{l}^2} \, \varUpsilon _{p {{\,}\!}s} \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!1} + [ u ] ^{2}_{s+2 ,{\,}\!1} \Bigr \} , \end{aligned}$$

(E.16a)

$$\begin{aligned} \sum _{j {{\,}\!}\nu } | w^{l} | _{\mathbb {H}^{2}_{} ( \mathrm {G}^{{{\,}\!}l ,{\,}\!j}_{\nu } ) }^2&\le 6 \, \frac{\Lambda _{l}^2}{\lambda _{l}^2} Z_0^2 + 3 p^2 \Bigl \{ [ u ] ^{2}_{2 ,{\,}\!2} + [ u ] ^{2}_{3 ,{\,}\!2} \Bigr \} , \end{aligned}$$

(E.16b)

where $Z_0^2$ is given by (E.10) and can be estimated using (E.13c) as follows:

$$\begin{aligned} Z_0^2 \le \frac{512}{9} \frac{\Lambda _{l}^2}{\lambda _{l}^4} \, C_{u}^2 \, \delta _{u}^4 \, \biggl \{ 1 + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, 2^{2\beta ( l+1 ) } . \end{aligned}$$

(E.17)

By Lemma E.3, there holds a bound

$$\begin{aligned} \varUpsilon _{p {{\,}\!}s} \, \Bigl \{ [ u ] ^{2}_{s+1 ,{\,}\!1} + [ u ] ^{2}_{s+2 ,{\,}\!1} \Bigr \}\le & {} \frac{64}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, \frac{\Lambda _{l}^{2s+2}}{\lambda _{l}^{2s+2}} \, C_{u}^2 \, \delta _{u}^{4} \, \biggl \{ \frac{1}{s^2 ( s+1 ) ^2} + \frac{\delta _{u}^{2}}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, l\nonumber \\&\cdot s \, ( s+1 ) ^2 \, ( s+2 ) ^2 \, 2^{-2 ( \alpha -\beta ) l}\nonumber \\\le & {} \frac{64}{3} \, \frac{e^{3}}{\sqrt{2\pi }} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2\chi _{l}+4} C_{u}^2 \, \delta _{u}^4 \, \biggl \{ 1 + \frac{\delta _{u}^{2}}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \nonumber \\&\times ( \chi _{l}+1 ) ( \chi _{l}+2 ) ^2 ( \chi _{l}+3 ) ^2 \, l \, 2^{-2 ( \alpha -\beta ) l} . \end{aligned}$$

(E.18a)

From (E.3), (E.13c) and Definition 3.2, we obtain also that

$$\begin{aligned}{}[ u ] ^{2}_{2 ,{\,}\!2} + [ u ] ^{2}_{3 ,{\,}\!2} \le \frac{256}{9} \, \frac{\Lambda _{l}^2}{\lambda _{l}^4} \, C_{u}^2 \, \delta _{u}^4 \, \biggl \{ 1 + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, 2^{2\beta l} . \end{aligned}$$

(E.18b)

Combining the bounds (E.17)–(E.18b) with (E.16a)–(E.16b), we obtain the inequalities (5.13) with any positive $\tilde{C}_{1},\tilde{C}_{2}$ such that

$$\begin{aligned} C_{1}^2 \ge&\frac{64}{9} \, C_{u}^2 \, \delta _{u}^4 \, \biggl \{ 1 + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \left\{ 48 \cdot 2^{2\beta } \, \frac{1}{l^6} \, \frac{\Lambda _{l}^{6}}{\lambda _{l}^{6}}\right. \\&\left. + 27 \, \frac{e^{3}}{\sqrt{2\pi }} \, \biggl [ \frac{\Lambda _{l}}{\lambda _{l}} \biggr ] ^{2\chi _{l}+6} \, \frac{ ( \chi _{l}+1 ) ( \chi _{l}+2 ) ^2 ( \chi _{l}+3 ) ^2}{l^5} \, 2^{-2 ( \alpha -1 ) l} \right\} ,\\ C_{2}^2 \ge&\frac{64}{9} \, C_{u}^2 \, \delta _{u}^4 \, \frac{\Lambda _{l}^2}{\lambda _{l}^{4}} \, \biggl \{ 1 + \frac{\delta _{u}^2}{4} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} \biggr \} \, \biggl \{ 96 \cdot 2^{2\beta } \, \frac{1}{l^2} \, \frac{\Lambda _{l}^{2}}{\lambda _{l}^{2}} + 9 \, \frac{ ( \varrho _{\delta } \chi _{l} + 2 ) ^2}{l^2} \biggr \} , \end{aligned}$$

where the expressions on the right-hand side are monotonically decreasing with respect to l. $\square$