An optimal adaptive tensor product wavelet solver of a space-time FOSLS formulation of parabolic evolution problems

Abstract

In this work, we construct a well-posed first-order system least squares (FOSLS) simultaneously space-time formulation of parabolic PDEs. Using an adaptive wavelet solver, this problem is solved with the best possible rate in linear complexity. Thanks to the use of a basis that consists of tensor products of wavelets in space and time, this rate is equal to that when solving the corresponding stationary problem. Our findings are illustrated by numerical results.

Appendix: Decay estimates

In this Appendix we prove the technical results Theorem A.3, Corollaries A.7, A.9, and A.11 that were used in the proof of Theorem 4.18.

The following lemma is an application of Schur’s lemma that is often used to bound the spectral norm of a matrix whose row and column indices run over index sets of multi-level bases.

Lemma A.1

For index sets$J,J^{\prime }$,let$|\cdot | \colon J \cup J^{\prime } \rightarrow \mathbb {N}_{0}$,and let$M:=[m_{\lambda ^{\prime },\lambda }]_{(\lambda ^{\prime },\lambda ) \in J^{\prime } \times J}$besuch that for someξ ≥ 0,ρ > 0,

$$\begin{array}{@{}rcl@{}} \#\{\lambda^{\prime}\colon m_{\lambda^{\prime},\lambda}\neq 0,|\lambda^{\prime}|=|\lambda|+k\} &\lesssim 2^{\xi k} \quad(\lambda \in J,k \in \mathbb{N}_{0}),\\ \#\{\lambda\colon m_{\lambda^{\prime},\lambda}\neq 0,|\lambda^{\prime}|=|\lambda|+k\} &\lesssim 1 \quad(\lambda^{\prime} \in J^{\prime},k \in \mathbb{N}_{0}), \end{array} $$

and

$$|m_{\lambda^{\prime},\lambda}| \lesssim 2^{(|\lambda|-|\lambda^{\prime}|)(\rho+\frac{\xi}{2})} \quad(|\lambda^{\prime}|\geq |\lambda|). $$

Then

$$\||M|_{\{(\lambda^{\prime},\lambda)\colon |\lambda^{\prime}|>|\lambda|+k\}}|\| \lesssim 2^{-\rho k}, $$

where $(M|_{\{(\lambda ^{\prime },\lambda )\colon |\lambda ^{\prime }|>|\lambda |+k\}})_{\lambda ^{\prime },\lambda }:=\left \{\begin {array}{cl} m_{\lambda ^{\prime },\lambda } & \text {when} |\lambda ^{\prime }|>|\lambda |+k\\ 0 & \text {otherwise} \end {array} \right .$, and ∥⋅∥ denotes the matrix spectral norm, i.e., here the norm on $\mathcal {L}(\ell _{2}(J),\ell _{2}(J^{\prime }))$. The absolute value refers to taking entry-wise absolute value. (Similar notations will be used at other occasions.)

Proof

With $I_{\ell ^{\prime },\ell }:=[|m_{\lambda ^{\prime },\lambda }|]_{\{(\lambda ^{\prime },\lambda )\colon |\lambda ^{\prime }|=\ell ^{\prime }, |\lambda |=\ell \}}$, we have

$$\||M|_{\{(\lambda^{\prime},\lambda)\colon |\lambda^{\prime}|>|\lambda|+k\}}|\|^{2} \lesssim \max_{\ell^{\prime}} \sum\limits_{\ell<\ell^{\prime}-k} \|I_{\ell^{\prime},\ell}\| \times \max_{\ell} \sum\limits_{\ell^{\prime}>\ell+k} \|I_{\ell^{\prime},\ell}\|, $$

where $\ell , \ell ^{\prime }$ run over $\mathbb {N}_{0}$.

The number of non-zero entries in each column or row of $I_{\ell ^{\prime },\ell }$ is $\lesssim 2^{\xi (\ell ^{\prime }-\ell )}$ or $\lesssim 1$, respectively. Using $\|\cdot \|^{2} \leq \|\cdot \|_{1}\|\cdot \|_{\infty }$, we infer that $\|I_{\ell ^{\prime },\ell }\|^{2} \lesssim 2^{\xi (\ell ^{\prime }-\ell )} \cdot 2^{(\frac {\xi }{2}+\rho )(\ell -\ell ^{\prime })} \cdot 1 \cdot 2^{(\frac {\xi }{2}+\rho )(\ell -\ell ^{\prime })} = 4^{\rho (\ell -\ell ^{\prime })}$. □

The next lemma concerns near-sparsity of a generalized mass matrix corresponding to two temporal wavelet bases.

Lemma A.2

For $k \in \mathbb {N}_{0}$ , ${\Theta }^{*}, {\Theta }^{\circ } \in \{{\Theta }^{\mathscr{V}_{1}}, {\Theta }^{\mathscr{P}}, {\Theta }^{a}, {\Theta }^{\mathscr{U}}/\|{\Theta }^{\mathscr{U}}\|_{L_{2}(\mathrm {I})}\}$ we have

$$\||\langle {\Theta}^{*}, {\Theta}^{\circ}\rangle_{L_{2}(\mathrm{I})}|_{\{(\lambda^{\prime},\lambda) \colon |\lambda^{\prime}|>|\lambda|+k\}}|\|\lesssim 2^{-k/2}. $$

Proof

Using that Θ^∗ satisfies (t₁)–(t₄), being the counterparts of (s₁)–(s₄) for the spatial wavelets, we split the matrix into B^r + B^s, where B^r contains all its entries $\langle \theta ^{*}_{\lambda ^{\prime }}, \theta ^{\circ }_{\lambda }\rangle _{L_{2}(\mathrm {I})}$ for which $\text {supp} \theta ^{*}_{\lambda ^{\prime }}$ is contained in ω for some $\omega \in \mathcal {O}_{\mathrm {I}}$ with |ω| = |λ| (the ‘regular’ entries), and where B^s contains the remaining (‘singular’) entries.

The number of non-zero entries with $|\lambda ^{\prime }|=\ell ^{\prime }$ and |λ| = ℓ in each column or row of B^r is $\lesssim 2^{\ell ^{\prime }-\ell }$ or $\lesssim 1$, respectively. Thanks to (t₄), for each of these entries we have $|\langle \theta ^{*}_{\lambda ^{\prime }}, \theta ^{\circ }_{\lambda }\rangle _{L_{2}(\mathrm {I})}| \leq \|\theta ^{*}_{\lambda ^{\prime }}\|_{L_{1}(\mathrm {I})} 2^{-\ell ^{\prime }} |\theta ^{\circ }_{\lambda }|_{W^{1}_{\infty }(\text {supp} \theta ^{*}_{\lambda ^{\prime }})} \lesssim 2^{3(\ell -\ell ^{\prime })/2}$. An application of Lemma A.1 with ξ = ρ = 1 shows that $\||B^{r}|\| \lesssim 2^{-k}$.

The number of non-zero entries with $|\lambda ^{\prime }|=\ell ^{\prime }$ and |λ| = ℓ in each column or row of B^s is $\lesssim 1$. For each of these entries, we have $|\langle \theta ^{*}_{\lambda ^{\prime }}, \theta ^{\circ }_{\lambda }\rangle _{L_{2}(\mathrm {I})}| \leq \|\theta ^{*}_{\lambda ^{\prime }}\|_{L_{1}(\mathrm {I})} \|\theta ^{\circ }_{\lambda }\|_{L_{\infty }(\mathrm {I})} \lesssim 2^{(\ell -\ell ^{\prime })/2}$. An application of Lemma A.1 with ξ = 0, $\rho =\frac {1}{2}$ shows that $\||B^{s}|\| \lesssim 2^{-k/2}$. □

The following theorem provides the main ingredient for bounding $\|\textbf {r}_{\frac {1}{2}}-{\tilde {\textbf {r}}}_{\frac {1}{2}}\|$.

Theorem A.3

Let Λ^a ⊂∨_abea multi-tree, and$r \in \text {span} {\Psi }^{a}\|_{{\Lambda }^{a}}$For$k \in \mathbb {N}_{0}$,it holds that

$$ \|\langle {\Psi}^{\mathscr{V}_{1}},r\rangle_{L_{2}(\mathrm{I} \times {\Omega})}|_{\vee_{\mathscr{V}_{1}} \setminus \vee_{\mathscr{V}_{1}}({\Lambda}^{a},k)}\| \lesssim 2^{-k/2} \|r\|_{L_{2}(\mathrm{I};H^{-1}({\Omega}))}. $$

(A.1)

Proof

We write $r={\sum }_{(\lambda ,\mu ) \in {\Lambda }^{a}} r_{\lambda \mu } \theta ^{a}_{\lambda } \otimes \sigma ^{a}_{\mu }$, and let

$$\delta_{\lambda}(\mu^{\prime}):=\left\{ \begin{array}{ll} 0 & |\mu^{\prime}| \leq \max\{|\mu| \colon \mu \in {{\Lambda}^{a}_{2}}(\lambda),\text{meas}(\mathcal{S}(\sigma_{\mu^{\prime}}^{\mathscr{V}_{1}}) \cap \text{supp}\sigma^{a}_{\mu})>0\}+k\\ 1 & \text{elsewhere} \end{array} \right., $$

Writing ${\Lambda }^{\mathscr{V}_{1}}:=\vee _{\mathscr{V}_{1}}({\Lambda }^{a},k)$, from $\frac {1}{2}(a+b)^{2} \leq (a^{2}+b^{2})$, we have that

$$\begin{array}{@{}rcl@{}} &&{}\frac{1}{2}\|\langle {\Psi}^{\mathscr{V}_{1}},r\rangle_{L_{2}(\mathrm{I} \times {\Omega})}|_{\vee_{\mathscr{V}_{1}} \setminus {\Lambda}^{\mathscr{V}_{1}}}\|^{2} \\ &&{}\leq{}\sum\limits_{\mu^{\prime} \in \Diamond_{\mathscr{V}_{1}}} \sum\limits_{\lambda^{\prime} \in \lhd_{\mathscr{V}_{1}}\setminus {\Lambda}^{\mathscr{V}_{1}}_{1}(\mu^{\prime})} |\sum\limits_{|\lambda^{\prime}|>|\lambda|+k} \langle \theta_{\lambda^{\prime}}^{\mathscr{V}_{1}},\theta^{a}_{\lambda}\rangle_{L_{2}(\mathrm{I})} \langle \sigma^{\mathscr{V}_{1}}_{\mu^{\prime}},\sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\rangle_{L_{2}({\Omega})}|^{2}\\ &&{}+{}\sum\limits_{\mu^{\prime} \in \Diamond_{\mathscr{V}_{1}}} \sum\limits_{\lambda^{\prime} \in \lhd_{\mathscr{V}_{1}}\setminus {\Lambda}^{\mathscr{V}_{1}}_{1}(\mu^{\prime})} |{}\sum\limits_{|\lambda| \geq |\lambda^{\prime}|-k} \langle \theta_{\lambda^{\prime}}^{\mathscr{V}_{1}},\theta^{a}_{\lambda}\rangle_{L_{2}(\mathrm{I})} \delta_{\lambda}(\mu^{\prime}) \langle \sigma^{\mathscr{V}_{1}}_{\mu^{\prime}},{}\sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\rangle_{L_{2}({\Omega})}|^{2}\\ \end{array} $$

(A.2)

Here, we could insert the factor $\delta _{\lambda }(\mu ^{\prime })$ in the second sum because of the following reason: Let $(\lambda ^{\prime },\mu ^{\prime }) \in \vee _{\mathscr{V}_{1}} \setminus {\Lambda }^{\mathscr{V}_{1}}$ and $\lambda \in {{\Lambda }^{a}_{1}}$ with $|\lambda |\geq |\lambda ^{\prime }|-k$. If $\text {meas}(\mathcal {S}(\theta _{\lambda ^{\prime }}^{\mathscr{V}_{1}}) \cap \text {supp} \theta ^{a}_{\lambda })= 0$, then the value of $\delta _{\lambda }(\mu ^{\prime })$ is irrelevant. If $\text {meas}(\mathcal {S}(\theta _{\lambda ^{\prime }}^{\mathscr{V}_{1}}) \cap \text {supp} \theta ^{a}_{\lambda })>0$, then the definition of ${\Lambda }^{\mathscr{V}_{1}}=\vee _{\mathscr{V}_{1}}({\Lambda }^{a},k)$ shows that $|\mu ^{\prime }|>|\mu |+k$ for all $\mu \in {{\Lambda }^{a}_{2}}(\lambda )$ with $\text {meas}(\mathcal {S}(\sigma _{\mu ^{\prime }}^{\mathscr{V}_{1}}) \cap \text {supp} \sigma ^{a}_{\mu })>0$, meaning that $\delta _{\lambda }(\mu ^{\prime })= 1$.

Using Lemma A.2 for $({\Theta }^{*},{\Theta }^{\circ })=({\Theta }^{\mathscr{V}_{1}},{\Theta }^{a})$, the first sum can be bounded on a multiple of

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{\mu^{\prime} \in \Diamond_{\mathscr{V}_{1}}} 2^{-k} \sum\limits_{\lambda \in \lhd_{a}} |\langle \sigma_{\mu^{\prime}}^{\mathscr{V}_{1}}, \sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\rangle_{L_{2}({\Omega})}|^{2}\\ &&= 2^{-k}\sum\limits_{\lambda \in \lhd_{a}} \sum\limits_{\mu^{\prime} \in \Diamond_{\mathscr{V}_{1}}} |\langle \sigma_{\mu^{\prime}}^{\mathscr{V}_{1}}, \sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\rangle_{L_{2}({\Omega})}|^{2}\\&&\eqsim 2^{-k} \sum\limits_{\lambda \in \lhd_{a}} \|\sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\|_{H^{-1}({\Omega})}^{2}\\ && \eqsim 2^{-k} \|\sum\limits_{\lambda \in \lhd_{a}} \theta^{a}_{\lambda} \otimes \sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\|_{L_{2}(\mathrm{I};H^{-1}({\Omega}))}^{2} = 2^{-k}\| r\|_{L_{2}(\mathrm{I};H^{-1}({\Omega}))}^{2}, \end{array} $$

where we used that ${\Sigma }^{\mathscr{V}_{1}}$ is a Riesz basis for ${H^{1}_{0}}({\Omega })$, and that Θ^a is a Riesz basis for L₂(I).

To bound the second sum, recall that for μ ∈◇_a, it holds that $\text {supp} \sigma ^{a}_{\mu }=\omega _{\mu }$ for some $\omega _{\mu } \in \mathcal {O}_{\Omega }$ with $|\omega _{\mu }|=\max (|\mu |-1,0)$. Define the tiling $\mathcal {T}(\lambda ) \in \mathcal {O}_{\Omega }$ as the union, over the leaves μ of the tree ${{\Lambda }^{a}_{2}}(\lambda )$, of the children of ω_μ when |μ| > 0, or of ω_μ itself when |μ| = 0. Then $\text {span}\{\sigma ^{a}_{\mu } \colon \mu \in {{\Lambda }^{a}_{2}}(\lambda )\}=\mathcal {P}_{m}(\mathcal {T}(\lambda ))$, and $\{\mu ^{\prime } \in \Diamond _{\mathscr{V}_{1}}\colon \delta _{\lambda }(\mu ^{\prime })= 1\}=\Diamond _{\mathscr{V}_{1}} \setminus \Diamond _{\mathscr{V}_{1}}(\mathcal {T}(\lambda ),k)$, cf. Definition 4.7.

Since Θ^a and ${\Theta }^{\mathscr{V}_{1}}$ are Riesz bases for L₂(I), and so $\langle {\Theta }^{\mathscr{V}_{1}},{\Theta }^{a} \rangle _{L_{2}(\mathrm {I})} \in \mathcal {L}(\ell _{2}(\lhd _{a}),\ell _{2}(\lhd _{\mathscr{V}_{1}}))$, invoking [34, Prop. A.1] using that ${\Sigma }^{\mathscr{V}_{1}}$ satisfies (s₁)–(s₄), the second sum can be bounded on a multiple of

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{\mu^{\prime} \in \Diamond_{\mathscr{V}_{1}}} \sum\limits_{\lambda \in \lhd_{a}} | \delta_{\lambda}(\mu^{\prime}) \langle \sigma_{\mu^{\prime}}^{\mathscr{V}_{1}}, \sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\rangle_{L_{2}({\Omega})}|^{2}\\ &&= \sum\limits_{\lambda \in \lhd_{a}} \sum\limits_{\mu^{\prime} \in \Diamond_{\mathscr{V}_{1}}} | \delta_{\lambda}(\mu^{\prime}) \langle \sigma_{\mu^{\prime}}^{\mathscr{V}_{1}}, \sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\rangle_{L_{2}({\Omega})}|^{2} \\&&\lesssim \sum\limits_{\lambda \in \lhd_{a}} 4^{-k} \|\sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\|_{H^{-1}({\Omega})}^{2}\\ && \eqsim 4^{-k} \|\sum\limits_{\lambda \in \lhd_{a}} \theta^{a}_{\lambda} \otimes \sum\limits_{\mu \in {{\Lambda}^{a}_{2}}(\lambda)} r_{\lambda \mu} \sigma^{a}_{\mu}\|_{L_{2}(\mathrm{I};H^{-1}({\Omega}))}^{2} = 4^{-k}\| r\|_{L_{2}(\mathrm{I};H^{-1}({\Omega}))}^{2}, \end{array} $$

where we used that Θ^a is a Riesz basis for L₂(I). □

If Θ^a was a Riesz basis for H^− 1(Ω), then in the proof of Theorem A.3 it would have been natural to write $\langle \sigma ^{\mathscr{V}_{1}}_{\mu ^{\prime }},{\sum }_{\mu \in {{\Lambda }^{a}_{2}}(\lambda )} r_{\lambda \mu } \sigma ^{a}_{\mu }\rangle _{L_{2}({\Omega })}$ as $\langle {\Sigma }^{\mathscr{V}_{1}},{\Sigma }^{a}\rangle _{L_{2}({\Omega })}$$[ r_{\lambda \mu }]_{\mu \in {{\Lambda }^{a}_{2}}(\lambda )}$. In this case the approach of the insertion of the factor $\delta _{\lambda }(\mu ^{\prime })$ would have given the bound

$$\begin{array}{@{}rcl@{}} {\textstyle \frac{1}{2}}\sqrt{2} &&\|\langle {\Psi}^{\mathscr{V}_{1}},{\Psi}^{a}\rangle_{L_{2}(\mathrm{I} \times {\Omega})}|_{\vee_{\mathscr{V}_{1}} \setminus {\Lambda}^{\mathscr{V}_{1}}({\Lambda}^{a},k)\times {\Lambda}^{a}}\| \leq \\ && \|\langle{\Theta}^{\mathscr{V}_{1}},{\Theta}^{a}\rangle_{L_{2}(\mathrm{I})}|_{\{(\lambda^{\prime},\lambda)\colon |\lambda^{\prime}|>|\lambda|+k\}}\| \|\langle {\Sigma}^{\mathscr{V}_{1}}, {\Sigma}^{a} \rangle_{L_{2}({\Omega})} \| +\\ && \|\langle{\Theta}^{\mathscr{V}_{1}},{\Theta}^{a}\rangle_{L_{2}(\mathrm{I})}\| \|\langle{\Sigma}^{\mathscr{V}_{1}},{\Sigma}^{a}\rangle_{L_{2}({\Omega})}|_{\{(\mu^{\prime},\mu)\colon |\mu^{\prime}|>|\mu|+k\}}\|. \end{array} $$

Although in the current setting where $\langle {\Sigma }^{\mathscr{V}_{1}}, {\Sigma }^{a} \rangle _{L_{2}({\Omega })} \not \in \mathcal {L}(\ell _{2}(\Diamond _{a}),\ell _{2}(\Diamond _{\mathscr{V}_{1}}))$, this estimate makes not much sense, for other collections this result, formulated in the next proposition, is going to be useful.

Proposition A.4

For$*,\circ \in \{\mathscr{U},\mathscr{V}_{1},\mathscr{P},a\}$,let$M_{\lhd }:=[m^{\lhd }_{\lambda ^{\prime }, \lambda }]_{(\lambda ^{\prime },\lambda )} \in \mathcal {L}(\ell _{2}(\lhd _{*}),\ell _{2}(\lhd _{\circ }))$,$M_{\Diamond }:=[m^{\Diamond }_{\mu ^{\prime }, \mu }]_{(\mu ^{\prime },\mu )} \in \mathcal {L}(\ell _{2}(\Diamond _{*}),\ell _{2}(\Diamond _{\circ }))$,where$m^{\lhd }_{\lambda ^{\prime }, \lambda }= 0$when$\text {meas}(\mathcal {S}(\theta ^{\circ }_{\lambda ^{\prime }}) \cap \text {supp} \theta ^{*}_{\lambda }) = 0$,and$m^{\Diamond }_{\mu ^{\prime }, \mu }= 0$when$\text {meas}(\mathcal {S}(\sigma ^{\circ }_{\mu ^{\prime }}) \cap \text {supp} \sigma ^{*}_{\mu }) = 0$.Then for a multi-tree Λ^∗⊂∨_∗,and$k \in \mathbb {N}_{0}$,it holds that

$$\begin{array}{@{}rcl@{}} {\textstyle \frac{1}{2}}\sqrt{2}&&\|M_{\lhd} \otimes M_{\Diamond} |_{(\vee_{\circ} \setminus {\Lambda}^{\circ}({\Lambda}^{*},k)) \times {\Lambda}^{*}}\| \leq\\ &&\|M_{\lhd}|_{\{(\lambda^{\prime},\lambda)\colon |\lambda^{\prime}|>|\lambda|+k\}}\| \|M_{\Diamond} \| +\|M_{\lhd}\| \|M_{\Diamond}|_{\{(\mu^{\prime},\mu)\colon |\mu^{\prime}|>|\mu|+k\}}\|. \end{array} $$

The remaining of this Appendix will consist of various applications of Proposition A.4 for which in several lemmas we estimate norms of type $\|M_{\lhd }|_{\{(\lambda ^{\prime },\lambda )\colon |\lambda ^{\prime }|>|\lambda |+k\}}\|$ or $\|M_{\lhd }|_{\{(\lambda ^{\prime },\lambda )\colon |\lambda ^{\prime }|>|\lambda |+k\}}\|$. The next lemma deals with the first task.

Lemma A.5

For $k \in \mathbb {N}_{0}$ , it holds that

$$\begin{array}{@{}rcl@{}} \left\| \left| \left\langle \frac{{\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{-1}({\Omega})}},{\Sigma}^{\mathscr{V}_{1}}\right\rangle_{L_{2}({\Omega})}|_{\{(\mu^{\prime},\mu)\colon |\mu^{\prime}|>|\mu|+k\}}\right| \right\| \lesssim 2^{-k/2},\\ \left\| \left| \left\langle \frac{\frac{\partial}{\partial x_{i}}{\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{1}({\Omega})}},{\Sigma}^{a}\right\rangle_{L_{2}({\Omega})}|_{\{(\mu^{\prime},\mu)\colon |\mu^{\prime}|>|\mu|+k\}}\right| \right\| \lesssim 2^{-k/2}. \end{array} $$

Proof

For proving the first inequality, we split the matrix into B^r + B^s, where B^r contains all its entries $\left \langle \frac {\sigma _{\mu ^{\prime }}^{\mathscr{U}}}{\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{-1}({\Omega })}},\sigma _{\mu }^{\mathscr{V}_{1}}\right \rangle _{L_{2}({\Omega })}$ for which $\text {supp} \sigma ^{\mathscr{U}}_{\mu ^{\prime }}$ is contained in ω for some $\omega \in \mathcal {O}_{\Omega }$ with |ω| = |μ| (the ‘regular’ entries), and where B^s contains the remaining (‘singular’) entries.

Thanks to (${s}_{4}^{\mathscr{U}}$), for the regular entries we can estimate

$$\begin{array}{@{}rcl@{}} |\langle \sigma_{\mu^{\prime}}^{\mathscr{U}},\sigma_{\mu}^{\mathscr{V}_{1}}\rangle_{L_{2}({\Omega})}| && \lesssim \|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{L_{1}({\Omega})} 4^{-|\mu^{\prime}|} |\sigma_{\mu}^{\mathscr{V}_{1}}|_{W_{\infty}^{2}(\text{supp}\sigma_{\mu^{\prime}}^{\mathscr{U}})}\\ && \lesssim 4^{-|\mu^{\prime}|} 2^{-|\mu^{\prime}|\frac{n}{2}} \|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{L_{2({\Omega})}} 2^{|\mu|} 2^{|\mu|\frac{n}{2}} \|\sigma_{\mu}^{\mathscr{V}_{1}}\|_{H^{1}({\Omega})}\\ && \eqsim 2^{(|\mu|-|\mu^{\prime}|)(1+\frac{n}{2})} \|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{-1}({\Omega})} \end{array} $$

where we used $\|\sigma _{\mu }^{\mathscr{V}_{1}}\|_{H^{1}({\Omega })} \eqsim 1$, and $\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{L_{2}({\Omega })}^{2} \leq \|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{1}({\Omega })} \|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{-1}({\Omega })}\lesssim 2^{|\mu ^{\prime }|} \|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{L_{2}({\Omega })}\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{-1}({\Omega })}$. An application of Lemma A.1 with ξ = n and ρ = 1 shows that $\||B^{r}|\|\lesssim 2^{-k}$.

Since the wavelets $\sigma _{\mu }^{\mathscr{V}_{1}}$ are piecewise polynomial functions in H¹(Ω), they are contained in $W_{\infty }^{1}({\Omega })$. Using (s₄), for the remaining singular entries we estimate

$$\begin{array}{@{}rcl@{}} |\langle \sigma_{\mu^{\prime}}^{\mathscr{U}},\sigma_{\mu}^{\mathscr{V}_{1}}\rangle_{L_{2}({\Omega})}| &\lesssim& \|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{L_{1}({\Omega})} 2^{-|\mu^{\prime}|} |\sigma_{\mu}^{\mathscr{V}_{1}}|_{W_{\infty}^{1}(\text{supp}\sigma_{\mu^{\prime}}^{\mathscr{U}})}\\ &\eqsim& 2^{(|\mu|-|\mu^{\prime}|)\frac{n}{2}} \|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{-1}({\Omega})} \end{array} $$

again by $\|\sigma _{\mu }^{\mathscr{V}_{1}}\|_{H^{1}({\Omega })} \eqsim 1$, and $\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{L_{2}({\Omega })} \lesssim 2^{|\mu ^{\prime }|}\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{-1}({\Omega })}$ (cf. (4.9)). An application of Lemma A.1 with ξ = n − 1 and ρ = 1/2 shows that $\||B^{s}|\|\lesssim 2^{-k/2}$.

Moving to the second inequality, we split the matrix into B^r + B^s, where B^r contains all its entries $\left \langle \frac {\frac {\partial }{\partial x_{i}}\sigma _{\mu ^{\prime }}^{\mathscr{U}}}{\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{1}({\Omega })}},\sigma _{\mu }^{a}\right \rangle _{L_{2}({\Omega })}$ for which $\text {supp}\sigma ^{\mathscr{U}}_{\mu ^{\prime }}$ is contained in ω ∩Ω for some $\omega \in \mathcal {O}_{\Omega }$ with |ω| = |μ| (the ‘regular’ entries), and where B^s contains the remaining (‘singular’) entries.

Thanks to (s₄), for the regular entries we can estimate

$$\begin{array}{@{}rcl@{}} |\langle \frac{\partial}{\partial x_{i}} {\sigma}_{\mu^{\prime}}^{\mathscr{U}},{\sigma}_{\mu}^{a}\rangle_{L_{2}({\Omega})}|= |\langle {\sigma}_{\mu^{\prime}}^{\mathscr{U}}, \frac{\partial}{\partial x_{i}} {\sigma}_{\mu}^{a}\rangle_{L_{2}({\Omega})}|\lesssim \|{\sigma}_{\mu^{\prime}}^{\mathscr{U}}\|_{L_{1}({\Omega})} 2^{-|\mu^{\prime}|} | {\sigma}_{\mu}^{a}|_{{W}_{\infty}^{2}(\text{supp} {\sigma}_{\mu^{\prime}}^{\mathscr{U}})}\\ \lesssim 4^{-|\mu^{\prime}|} 2^{-|\mu^{\prime}|\frac{n}{2}} \| {\sigma}_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})} 4^{|\mu|} 2^{|\mu|\frac{n}{2}} \| {\sigma}_{\mu}^{a}\|_{L_{2}({\Omega})} \eqsim 2^{(|\mu|-|\mu^{\prime}|)(2+\frac{n}{2})} \| {\sigma}_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})}, \end{array} $$

where we used that $\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{L_{2}({\Omega })} \lesssim 2^{-|\mu ^{\prime }|} \|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{1}({\Omega })}$ (4.9). An application of Lemma A.1 with ξ = n and ρ = 2 shows that $\||B^{r}|\|\lesssim 4^{-k}$.

For the remaining singular entries we estimate

$$\begin{array}{@{}rcl@{}} |\langle \frac{\partial}{\partial x_{i}} \sigma_{\mu^{\prime}}^{\mathscr{U}},\sigma_{\mu}^{a}\rangle_{L_{2}({\Omega})}| &\lesssim& \|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{{W^{1}_{1}}({\Omega})} \|\sigma_{\mu}^{a}\|_{L_{\infty}(\text{supp} \sigma_{\mu^{\prime}}^{\mathscr{U}})}\\ &\lesssim& 2^{(|\mu|-|\mu^{\prime}|)\frac{n}{2}} \|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})} \|\sigma_{\mu}^{a}\|_{L_{2}(\text{supp}\sigma_{\mu^{\prime}}^{\mathscr{U}})}. \end{array} $$

An application of Lemma A.1 with ξ = n − 1 and ρ = 1/2 shows that $\||B^{s}|\|\lesssim 2^{-k/2}$. □

Lemma A.6

For $k \in \mathbb {N}_{0}$ , it holds that

$$\left\| \left| \left\langle \frac{({\Theta}^{\mathscr{U}})^{\prime}}{\|({\Theta}^{\mathscr{U}})^{\prime}\|_{L_{2}(\mathrm{I})}},{\Theta}^{\mathscr{V}_{1}}\right\rangle_{L_{2}(\mathrm{I})}|_{\{(\lambda^{\prime},\lambda) \colon |\lambda^{\prime}|>|\lambda|+k\}}\right| \right\| \lesssim 2^{-k/2}. $$

Proof

We split the matrix into B^r + B^s, where B^r contains all (‘regular’) entries $\left \langle \frac {(\theta _{\lambda ^{\prime }}^{\mathscr{U}})^{\prime }}{\|(\theta _{\lambda ^{\prime }}^{\mathscr{U}})^{\prime }\|_{L_{2}(\mathrm {I})}},\theta _{\lambda }^{\mathscr{V}_{1}}\right \rangle _{L_{2}(\mathrm {I})}$ for which $\text {supp} \theta ^{\mathscr{U}}_{\lambda ^{\prime }}$ is contained in ω ∩ I for some $\omega \in \mathcal {O}_{\mathrm {I}}$ with |ω| = |λ| (so that in particular $\theta ^{\mathscr{U}}_{\lambda ^{\prime }}$ vanishes on ∂I), and where B^s contains the remaining (‘singular’) entries.

For the regular entries, we can estimate

$$\begin{array}{@{}rcl@{}} |\langle (\theta_{\lambda^{\prime}}^{\mathscr{U}})^{\prime},\theta_{\lambda}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I})}| =|\langle \theta_{\lambda^{\prime}}^{\mathscr{U}},(\theta_{\lambda}^{\mathscr{V}_{1}})^{\prime}\rangle_{L_{2}(\mathrm{I})}| \lesssim \|\theta_{\lambda^{\prime}}^{\mathscr{U}}\|_{L_{1}(\mathrm{I})} 2^{-|\lambda^{\prime}|} \|\theta_{\lambda}^{\mathscr{V}_{1}}\|_{W_{\infty}^{2}(\text{supp}\theta_{\lambda^{\prime}}^{\mathscr{U}})}\\ \lesssim 2^{-|\lambda^{\prime}|/2} 2^{-|\lambda^{\prime}|} \|(\theta_{\lambda^{\prime}}^{\mathscr{U}})^{\prime}\|_{L_{2}(\mathrm{I})} 2^{-|\lambda^{\prime}|} 4^{|\lambda|} 2^{|\lambda|/2} \|\theta_{\lambda}^{\mathscr{V}_{1}}\|_{L_{2}(\mathrm{I})} \eqsim 2^{\frac{5}{2}(|\lambda|-|\lambda^{\prime}|)} \|(\theta_{\lambda^{\prime}}^{\mathscr{U}})^{\prime}\|_{L_{2}(\mathrm{I})}, \end{array} $$

where we used (t₄), Poincaré’s inequality, an inverse inequality, and $\|\theta _{\lambda }^{\mathscr{V}_{1}}\|_{L_{2}(\mathrm {I})} \eqsim 1$. An application of Lemma A.1 with ξ = 1 and ρ = 2 shows that $\||B^{r}|\|\lesssim 4^{-k}$.

For the remaining singular entries, we estimate

$$\begin{array}{@{}rcl@{}} |\langle (\theta_{\lambda^{\prime}}^{\mathscr{U}})^{\prime},\theta_{\lambda}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I})}| \leq \|(\theta_{\lambda^{\prime}}^{\mathscr{U}})^{\prime}\|_{L_{1}(\mathrm{I})} \|\theta_{\lambda}^{\mathscr{V}_{1}}\|_{L_{\infty}(\mathrm{I})} \lesssim 2^{-\frac{1}{2}(|\lambda|-|\lambda^{\prime}|)} \|(\theta_{\lambda^{\prime}}^{\mathscr{U}})^{\prime}\|_{L_{2}(\mathrm{I})}. \end{array} $$

An application of Lemma A.1 with ξ = 0 and $\rho =\frac {1}{2}$ shows that $\||B^{s}|\|\lesssim 2^{-k/2}$. □

The following Corollary will be used to bound $\|(\textbf {r}_{1}-{\tilde {\textbf {r}}}_{1})|_{\vee _{\mathscr{U}}}\|$.

Corollary A.7

Let ${\Lambda }^{\mathscr{V}_{1}} \subset \vee _{\mathscr{V}_{1}}$ be a multi-tree. Then for $k \in \mathbb {N}_{0}$ ,

$$\left. \begin{array}{r} \| \langle \frac{\partial}{\partial t} {\Psi}^{\mathscr{U}}, {\Psi}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I}\times{\Omega})}|_{(\vee_{\mathscr{U}}\setminus \vee_{\mathscr{U}}({\Lambda}^{\mathscr{V}_{1}},k)) \times {\Lambda}^{\mathscr{V}_{1}} }\| \\ \| \langle b_{i}\frac{\partial}{\partial x_{i}} {\Psi}^{\mathscr{U}}, {\Psi}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I}\times{\Omega})}|_{(\vee_{\mathscr{U}}\setminus \vee_{\mathscr{U}}({\Lambda}^{\mathscr{V}_{1}},k)) \times {\Lambda}^{\mathscr{V}_{1}} }\|\\ \| \langle c {\Psi}^{\mathscr{U}}, {\Psi}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I}\times{\Omega})}|_{(\vee_{\mathscr{U}}\setminus \vee_{\mathscr{U}}({\Lambda}^{\mathscr{V}_{1}},k)) \times {\Lambda}^{\mathscr{V}_{1}} }\| \end{array} \right\} \lesssim 2^{-k/2}. $$

Proof

(a). From $\|\theta ^{\mathscr{U}}_{\lambda } \otimes \sigma ^{\mathscr{U}}_{\mu }\|_{\mathscr{U}} \geq \|\theta ^{\mathscr{U}}_{\lambda }\|_{H^{1}(\mathrm {I})} \|\sigma ^{\mathscr{U}}_{\mu }\|_{H^{-1}({\Omega })}$, for the first inequality it is sufficient to prove that

$$\| \langle \frac{({\Theta}^{\mathscr{U}})^{\prime}}{\|{\Theta}^{\mathscr{U}}\|_{H^{1}(\mathrm{I})}} \otimes \frac{{\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{-1}({\Omega})}}, {\Theta}^{\mathscr{V}_{1}}\otimes {\Sigma}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I}\times{\Omega})}|_{(\vee_{\mathscr{U}}\setminus \vee_{\mathscr{U}}({\Lambda}^{\mathscr{V}_{1}},k)) \times {\Lambda}^{\mathscr{V}_{1}} }\| \lesssim 2^{-k/2}. $$

From ${\Theta }^{\mathscr{U}} / \|{\Theta }^{\mathscr{U}}\|_{H^{1}(\mathrm {I})}$, ${\Theta }^{\mathscr{V}_{1}}$, ${\Sigma }^{\mathscr{U}} / \|{\Sigma }^{\mathscr{U}}\|_{H^{-1}({\Omega })}$, and ${\Sigma }^{\mathscr{V}_{1}}$ being Riesz bases for H¹(I), L₂(I), H^− 1(Ω), and ${H^{1}_{0}}({\Omega })$, we have

$$\begin{array}{@{}rcl@{}} \left\langle \frac{({\Theta}^{\mathscr{U}})^{\prime}}{\|{\Theta}^{\mathscr{U}}\|_{H^{1}(\mathrm{I})}},{\Theta}^{\mathscr{V}_{1}}\right\rangle_{L_{2}(\mathrm{I})} &\in \mathcal{L}(\ell_{2}(\lhd_{\mathscr{V}_{1}}),\ell_{2}(\lhd_{\mathscr{U}})),\\ \left\langle \frac{{\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{-1}({\Omega})}},{\Sigma}^{\mathscr{V}_{1}}\right\rangle_{L_{2}({\Omega})} &\in \mathcal{L}(\ell_{2}(\Diamond_{\mathscr{V}_{1}}),\ell_{2}(\Diamond_{\mathscr{U}})). \end{array} $$

The proof of the first inequality is completed by applications of Proposition A.4 and Lemmata A.5(first statement)–A.6.

(b). From $\text {span} \frac {\partial }{\partial x_{i}} b_{i} {\Psi }^{\mathscr{V}_{1}}|_{{\Lambda }^{\mathscr{V}_{1}}} \subset \text {span} {\Psi }^{a}|_{\vee _{a}({\Lambda }^{\mathscr{V}_{1}},0)}$ (similar to Lemma 4.16), for $\textbf {c} \in \ell _{2}({\Lambda }^{\mathscr{V}_{1}})$ there exists a $\textbf {d} \in \ell _{2}(\vee _{a}({\Lambda }^{\mathscr{V}_{1}},0))$ such that

$$\begin{array}{@{}rcl@{}} \langle b_{i}\frac{\partial}{\partial x_{i}} {\Psi}^{\mathscr{U}}, {\Psi}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I}\times{\Omega})} \textbf{c} &=&\langle {\Psi}^{\mathscr{U}}, -\textbf{c}^{\top} \frac{\partial}{\partial x_{i}} (b_{i} {\Psi}^{\mathscr{V}_{1}})\rangle_{L_{2}(\mathrm{I}\times{\Omega})} \\ &=&\langle {\Psi}^{\mathscr{U}}, \textbf{d}^{\top} {\Psi}^{a}\rangle_{L_{2}(\mathrm{I}\times{\Omega})} =\langle {\Psi}^{\mathscr{U}}, {\Psi}^{a}\rangle_{L_{2}(\mathrm{I}\times{\Omega})} \textbf{d}, \end{array} $$

where

$$\|\textbf{d}\| \eqsim \|\textbf{d}^{\top} {\Psi}^{a}\|_{L_{2}(\mathrm{I} \times {\Omega})}= \|\textbf{c}^{\top} \frac{\partial}{\partial x_{i}} (b_{i} {\Psi}^{\mathscr{V}_{1}})\|_{L_{2}(\mathrm{I} \times {\Omega})} \lesssim \|\textbf{c}\|. $$

From $\|\theta ^{\mathscr{U}}_{\lambda } \otimes \sigma ^{\mathscr{U}}_{\mu }\|_{\mathscr{U}} \geq \|\theta ^{\mathscr{U}}_{\lambda }\|_{L_{2}(\mathrm {I})} \|\sigma ^{\mathscr{U}}_{\mu }\|_{H^{1}({\Omega })}$, $\vee _{\mathscr{U}}({\Lambda }^{\mathscr{V}_{1}},k)=\vee _{\mathscr{U}}(\vee _{a}$$({\Lambda }^{\mathscr{V}_{1}},0),k)$ it remains to be proven that

$$\begin{array}{@{}rcl@{}} &&\!\!\!\| \langle \frac{{\Theta}^{\mathscr{U}}}{\|{\Theta}^{\mathscr{U}}\|_{L_{2}(\mathrm{I})}} \otimes \frac{\frac{\partial}{\partial x_{i}} {\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{1}({\Omega})}}, {\Theta}^{a}\otimes {\Sigma}^{a}\rangle_{L_{2}(\mathrm{I}\times{\Omega})}|_{(\vee_{\mathscr{U}}\setminus \vee_{\mathscr{U}}(\vee_{a}({\Lambda}^{\mathscr{V}_{1}},0),k)) \times \vee_{a}({\Lambda}^{\mathscr{V}_{1}},0) }\|\\&&\!\!\! \lesssim 2^{-k/2}. \end{array} $$

Indeed, this gives

$$\begin{array}{@{}rcl@{}} &&\left\|\left( \langle b_{i}\frac{\partial}{\partial x_{i}} {\Psi}^{\mathscr{U}}, {\Psi}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I}\times{\Omega})} \textbf{c}\right)\right\|_{\vee_{\mathscr{U}} \setminus \vee_{\mathscr{U}}({\Lambda}^{\mathscr{V}_{1}},k)}\|\\ &&=\|\left( \langle {\Psi}^{\mathscr{U}}, {\Psi}^{a}\rangle_{L_{2}(\mathrm{I}\times{\Omega})} \textbf{d}\right)\left|\right._{\vee_{\mathscr{U}} \setminus \vee_{\mathscr{U}}({\Lambda}^{\mathscr{V}_{1}},k)}\| \lesssim 2^{-k/2} \|\textbf{d}\|\lesssim 2^{-k/2} \|\textbf{c}\|, \end{array} $$

showing the second inequality.

From ${\Theta }^{\mathscr{U}} / \|{\Theta }^{\mathscr{U}}\|_{L_{2}(\mathrm {I})}$, Θ^a, ${\Sigma }^{\mathscr{U}} / \|{\Sigma }^{\mathscr{U}}\|_{H^{1}({\Omega })}$, and Σ^a being Riesz bases for L₂(I), L₂(I), ${H_{0}^{1}}({\Omega })$, and L₂(Ω), we have

$$\begin{array}{@{}rcl@{}} \left\langle \frac{{\Theta}^{\mathscr{U}}}{\|{\Theta}^{\mathscr{U}}\|_{L_{2}(\mathrm{I})}},{\Theta}^{a}\right\rangle_{L_{2}(\mathrm{I})} &\in \mathcal{L}(\ell_{2}(\lhd_{a}),\ell_{2}(\lhd_{\mathscr{U}})),\\ \left\langle \frac{\frac{\partial}{\partial x_{i}}{\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{1}({\Omega})}},{\Sigma}^{a}\right\rangle_{L_{2}({\Omega})} &\in \mathcal{L}(\ell_{2}(\Diamond_{\mathscr{V}_{a}}),\ell_{2}(\Diamond_{\mathscr{U}})). \end{array} $$

The proof of the remaining inequality is completed by applications of Proposition A.4, Lemma A.2 for $({\Theta }^{*},{\Theta }^{\circ })=(\frac {{\Theta }^{\mathscr{U}}}{\|{\Theta }^{\mathscr{U}}\|_{L_{2}(\mathrm {I})}},{\Theta }^{a})$, and the second statement from Lemma A.5.

(c). A subset of the arguments that showed the second inequality gives the third one. □

Lemma A.8

For$k \in \mathbb {N}_{0}$and 1 ≤ i ≤ n, it holdsthat

$$\left\| \left| \left\langle {\Sigma}^{\mathscr{P}},\partial_{i} {\Sigma}^{\mathscr{V}_{1}}\right\rangle_{L_{2}({\Omega})^{n}}|_{\{(\mu^{\prime},\mu)\colon |\mu^{\prime}|>|\mu|+k\}}\right| \right\| \lesssim 2^{-k/2},\\ $$

Proof

We split the matrix into B^r + B^s, where B^r contains all its entries $\left \langle \sigma _{\mu ^{\prime }}^{\mathscr{P}},\right .\!\!$$\left .\partial _{i} \sigma _{\mu }^{\mathscr{V}_{1}}\right \rangle _{L_{2}({\Omega })^{n}}$ for which $\text {supp} \sigma ^{\mathscr{P}}_{\mu ^{\prime }}$ is contained in ω ∩Ω for some $\omega \in \mathcal {O}_{\Omega }$ with |ω| = |μ| (the ‘regular’ entries), and where B^s contains the remaining (‘singular’) entries.

For the regular entries using (s₄) we can estimate

$$\begin{array}{@{}rcl@{}} |\langle \sigma_{\mu^{\prime}}^{\mathscr{P}},\partial_{i} \sigma_{\mu}^{\mathscr{V}_{1}}\rangle_{L_{2}({\Omega})^{n}}| &\lesssim& \|\sigma_{\mu^{\prime}}^{\mathscr{P}}\|_{L_{1}({\Omega})^{n}} 2^{-|\mu^{\prime}|} |\sigma_{\mu}^{\mathscr{V}_{1}}|_{W_{\infty}^{2}(\text{supp}\sigma_{\mu^{\prime}}^{\mathscr{P}})}\\ &\eqsim& 2^{(|\mu|-|\mu^{\prime}|)(1+\frac{n}{2})} \end{array} $$

An application of Lemma A.1 with ξ = n and ρ = 1 shows that $\||B^{r}|\|\lesssim 2^{-k}$.

Since the wavelets $\sigma _{\mu }^{\mathscr{V}_{1}}$ are piecewise polynomial, and functions in H¹(Ω), they are contained in $W_{\infty }^{1}({\Omega })$. For the remaining singular entries we estimate

$$|\langle \sigma_{\mu^{\prime}}^{\mathscr{P}},\partial_{i} \sigma_{\mu}^{\mathscr{V}_{1}}\rangle_{L_{2}({\Omega})^{n}}| \lesssim \|\sigma_{\mu^{\prime}}^{\mathscr{P}}\|_{L_{1}({\Omega})^{n}} |\sigma_{\mu}^{\mathscr{V}_{1}}|_{W_{\infty}^{1}(\text{supp} \sigma_{\mu^{\prime}}^{\mathscr{P}})} \lesssim 2^{(|\mu|-|\mu^{\prime}|)\frac{n}{2}}. $$

An application of Lemma A.1 with ξ = n − 1 and ρ = 1/2 shows that $\||B^{s}|\|\lesssim 2^{-k/2}$. □

The following Corollary will be used to bound $\|(\textbf {r}_{1}-{\tilde {\textbf {r}}}_{1})|_{\vee _{\vec {\mathscr{P}}}}\|$.

Corollary A.9

Let ${\Lambda }^{\mathscr{V}_{1}} \subset \vee _{\mathscr{V}_{1}}$ be a multi-tree. Then for $k \in \mathbb {N}_{0}$ ,

$$\left\| \langle {\Psi}^{\vec{\mathscr{P}}}, \nabla_{x} {\Psi}^{\mathscr{V}_{1}}\rangle_{L_{2}(\mathrm{I}\times{\Omega})^{n}}|_{(\vee_{\vec{\mathscr{P}}} \setminus \vee_{\vec{\mathscr{P}}}({\Lambda}^{\mathscr{V}_{1}},k)) \times {\Lambda}^{\mathscr{V}_{1}} }\right\| \lesssim 2^{-k/2}. $$

Proof

Using Lemma A.2 for $({\Theta }^{*},{\Theta }^{\circ })=({\Theta }^{\mathscr{P}},{\Theta }^{\mathscr{V}_{1}})$, Lemma A.8, $\langle {\Theta }^{\mathscr{P}}, {\Theta }^{\mathscr{V}_{1}}\rangle _{L_{2}(\mathrm {I})} \in \mathcal {L}(\lhd _{\mathscr{V}_{1}},\lhd _{\mathscr{P}})$, and $\langle {\Sigma }^{\mathscr{P}}, \partial _{i} {\Sigma }^{\mathscr{V}_{1}}\rangle _{L_{2}({\Omega })^{n}} \in \mathcal {L}(\Diamond _{\mathscr{V}_{1}},\Diamond _{\mathscr{P}})$, the proof follows from Proposition A.4. □

Lemma A.10

For$k \in \mathbb {N}_{0}$,1 ≤ i ≤ n,it holds that

$$\begin{array}{@{}rcl@{}} \left\| \left| \left\langle \frac{\nabla {\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{1}({\Omega})}}, {\Sigma}^{a} \textbf{e}_{i} \right\rangle_{L_{2}({\Omega})^{n}}|_{\{(\mu^{\prime},\mu)\colon |\mu^{\prime}|>|\mu|+k\}}\right| \right\| &\lesssim& 2^{-k/2},\\ \left\| \langle {\Sigma}^{\mathscr{P}}, {\Sigma}^{a} \rangle_{L_{2}({\Omega})^{n}}|_{\{(\mu^{\prime},\mu)\colon |\mu^{\prime}|>|\mu|+k\}}\right\| &\lesssim& 2^{-k/2}. \end{array} $$

Proof

For proving the first inequality, we split the matrix into B^r + B^s, where B^r contains all its entries $\left \langle \frac {\nabla \sigma _{\mu ^{\prime }}^{\mathscr{U}}}{\|\sigma _{\mu ^{\prime }}^{\mathscr{U}}\|_{H^{1}({\Omega })}}, \sigma _{\mu }^{a} \textbf {e}_{i} \right \rangle _{L_{2}({\Omega })^{n}}$ for which $\text {supp} \sigma ^{\mathscr{U}}_{\mu ^{\prime }}$ is contained in ω ∩Ω for some $\omega \in \mathcal {O}_{\Omega }$ with |ω| = |μ| (the ‘regular’ entries), and where B^s contains the remaining (‘singular’) entries.

For the regular entries, using (s₄) and the first inequality in (4.9), we estimate

$$\begin{array}{@{}rcl@{}} &&\left| \left\langle \frac{\partial_{i} \sigma_{\mu^{\prime}}^{\mathscr{U}}}{\|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})}}, \sigma_{\mu}^{a} \right\rangle_{L_{2}({\Omega})^{n}}\left|\vphantom{\left\langle \frac{\partial_{i} \sigma_{\mu^{\prime}}^{\mathscr{U}}}{\|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})}}, \sigma_{\mu}^{a} \right\rangle_{L_{2}({\Omega})^{n}}} ={\kern-.5pt} \right| \left\langle \frac{\sigma_{\mu^{\prime}}^{\mathscr{U}}}{\|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})}}, \partial_{i} \sigma_{\mu}^{a} \right\rangle_{L_{2}({\Omega})^{n}}\right|\\ &&\lesssim 2^{-|\mu^{\prime}|} \frac{\|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{L_{1}({\Omega})}}{\|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})}} |\sigma^{a}_{\mu}|_{W^{2}(\text{supp} \sigma_{\mu^{\prime}}^{\mathscr{U}})} \lesssim 2^{-|\mu^{\prime}|}2^{-|\mu^{\prime}|n/2} 2^{-|\mu^{\prime}|} 2^{|\mu|n/2} 2^{2|\mu|}\\&&= 2^{(2+n/2)(|\mu^{\prime}|-|\mu|)}. \end{array} $$

An application of Lemma A.1 with ξ = n and ρ = 2 shows that $\|B^{r}\|\lesssim 4^{-k}$.

For the singular entries, we estimate

$$ \left|\left\langle \frac{\partial_{i} \sigma_{\mu^{\prime}}^{\mathscr{U}}}{\|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})}}, \sigma_{\mu}^{a} \right\rangle_{L_{2}({\Omega})^{n}}\right| \lesssim \frac{\|\partial_{i} \sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{L_{1}({\Omega})}}{\|\sigma_{\mu^{\prime}}^{\mathscr{U}}\|_{H^{1}({\Omega})}} \|\sigma^{a}_{\mu}\|_{L_{\infty}({\Omega})} \lesssim 2^{(|\mu^{\prime}|-|\mu|)n/2}. $$

(3)

An application of Lemma A.1 with ξ = n − 1 and ρ = 1/2 shows that $\||B^{s}|\|\lesssim 2^{-k/2}$.

The proof of the second inequality proceeds along the by now well-known steps. Using assumption (s₄) on ${\Sigma }^{\mathscr{P}}$ one shows that $\||B^{r}|\|\lesssim 2^{-k}$, whereas $\||B^{s}|\|\lesssim 2^{-k/2}$. □

The following Corollary will be used to bound $\|\textbf {r}_{3}-{\tilde {\textbf {r}}}_{3}\|$.

Corollary A.11

Let Λ^a ⊂∨_abe a multi-tree. Thenfor$k \in \mathbb {N}_{0}$,

$$\left\| \langle \nabla_{x}{\Psi}^{\mathscr{U}}, {\Psi}^{a} \textbf{e}_{i} \rangle_{L_{2}(\mathrm{I}\times{\Omega})^{n}}|_{(\vee_{\mathscr{U}} \setminus \vee_{\mathscr{U}}({\Lambda}^{a},k)) \times {\Lambda}^{a} }\right\| \lesssim 2^{-k/2}. $$

$$\left\| \langle {\Psi}^{\vec{\mathscr{P}}}, {\Psi}^{a} \textbf{e}_{i} \rangle_{L_{2}(\mathrm{I}\times{\Omega})^{n}}|_{(\vee_{\vec{\mathscr{P}}} \setminus \vee_{\vec{\mathscr{P}}}({\Lambda}^{a},k)) \times {\Lambda}^{a} }\right\| \lesssim 2^{-k/2}. $$

Proof

From $\|\theta ^{\mathscr{U}}_{\lambda } \otimes \sigma ^{\mathscr{U}}_{\mu }\|_{\mathscr{U}} \geq \|\theta ^{\mathscr{U}}_{\lambda }\|_{L_{2}(\mathrm {I})} \|\sigma ^{\mathscr{U}}_{\mu }\|_{H^{1}({\Omega })}$, in order to prove the first result it suffices to show that

$$\left\| \left \langle \frac{{\Theta}^{\mathscr{U}}}{\|{\Theta}^{\mathscr{U}}\|_{L_{2}(\mathrm{I})}} \otimes \frac{\nabla {\Sigma}^{\mathscr{U}}}{\|{\Sigma}^{\mathscr{U}}\|_{H^{1}({\Omega})}}, {\Theta}^{a} \otimes {\Sigma}^{a} \textbf{e}_{i}\right\rangle_{L_{2}(\mathrm{I}\times{\Omega})^{n}}|_{\vee_{\mathscr{U}} \setminus \vee_{\mathscr{U}}({\Lambda}^{a},k) \times {\Lambda}^{a} }\right\| \lesssim 2^{-k/2}. $$

This and the second result follow from applications of Proposition A.4, Lemma A.2, and Lemma A.10. □