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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 \(\kappa <3\).

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Notes

  1. I. V. Oseledets. QTT decomposition of the characteristic function of a simplex. September 2010, private communication.

  2. We use the master branch of the GitHub version 2.2+ of July 24, 2014 (git tag http://github.com/oseledets/TT-Toolbox/tree/v2.3-4-ge1a3f2c).

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Authors

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Correspondence to Vladimir Kazeev.

Additional information

Dedicated to Prof. Reinhold Schneider on the occasion of his 60th birthday.

The research of VK was performed mainly at the Seminar for Applied Mathematics, ETH Zurich. During the preparation of this work, CS was supported in part by the European Research Council through the FP7 Advanced Grant AdG247277.

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

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 . \(\square \)

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

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

\(\square \)

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. \(\square \)

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. \(\square \)

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Kazeev, V., Schwab, C. Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions. Numer. Math. 138, 133–190 (2018). https://doi.org/10.1007/s00211-017-0899-1

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  • DOI: https://doi.org/10.1007/s00211-017-0899-1

Mathematics Subject Classification

  • 15A69
  • 35C99
  • 35J25
  • 65N12
  • 65N30
  • 65N35