Exponential Convergence of hp-FEM for Elliptic Problems in Polyhedra: Mixed Boundary Conditions and Anisotropic Polynomial Degrees

Abstract

We prove exponential rates of convergence of hp-version finite element methods on geometric meshes consisting of hexahedral elements for linear, second-order elliptic boundary value problems in axiparallel polyhedral domains. We extend and generalize our earlier work for homogeneous Dirichlet boundary conditions and uniform isotropic polynomial degrees to mixed Dirichlet–Neumann boundary conditions and to anisotropic, which increase linearly over mesh layers away from edges and vertices. In particular, we construct $H^1$-conforming quasi-interpolation operators with N degrees of freedom and prove exponential consistency bounds $\exp (-b\root 5 \of {N})$ for piecewise analytic functions with singularities at edges, vertices and interfaces of boundary conditions, based on countably normed classes of weighted Sobolev spaces with non-homogeneous weights in the vicinity of Neumann edges.

Appendix: Proof of Theorem 4.3

We outline the major steps of the proof of Theorem 4.3.

1.1 Approximation Results

We first establish auxiliary approximation results.

1.1.1 Univariate Approximation Properties

The following consistency bound holds for the $H^1$-projector $\pi _{p,1}$ in (4.1) on $\widehat{I}=(-1,1)$; see [26, Corollary 3.15].

Lemma 8.1

Let $p\ge 1$, $\widehat{u}\in H^{s+1}(\widehat{I})$ and $0\le s\le p$. Then there holds

$$\begin{aligned} \Vert \widehat{u}-\widehat{\pi }_{p,1} \widehat{u}\Vert ^2_{H^{1}(\widehat{I})} \lesssim \Psi _{p,s} \Vert \widehat{u}^{(s+1)}\Vert _{L^2(\widehat{I})}^2. \end{aligned}$$

(8.1)

Here,

$$\begin{aligned} \Psi _{q,r} := \frac{\Gamma (q+1-r)}{\Gamma (q+1+r)}, \qquad 0\le r\le q, \end{aligned}$$

(8.2)

where $\Gamma$ is the Gamma function satisfying $\Gamma (m+1)=m!$ for any $m\in \mathbb {N}_0$.

The subsequent $H^1$-norm error bound holds for the $L^2$-projection $\widehat{\pi }_{p,0}$ (see also [26, Theorem 3.11] for p-optimal bounds).

Lemma 8.2

Let $p\ge 1$, $\widehat{u}\in H^{s+1}(\widehat{I})$ and $0\le s\le p$. Then there holds

$$\begin{aligned} \Vert \widehat{u}-\widehat{\pi }_{p,0}\widehat{u}\Vert ^2_{H^1(\widehat{I})} \lesssim p^4 \Psi _{p,s}\Vert \widehat{u}^{(s+1)}\Vert ^2_{L^2(\widehat{I})}\;. \end{aligned}$$

(8.3)

Proof

We recall from [25, Lemma 5.1] that

$$\begin{aligned} \Vert (\widehat{\pi }_{p,0} \widehat{u})^{(s)}\Vert _{L^2(\widehat{I})} \lesssim \max \{1,p\}^{2s} \Vert \widehat{u}^{(s)}\Vert _{L^2(\widehat{I})},\qquad p\ge 0,\ s\ge 0. \end{aligned}$$

(8.4)

With (8.4) and for $p\ge 1$, we find that

$$\begin{aligned} \Vert \widehat{u}-\widehat{\pi }_{p,0}\widehat{u}\Vert _{H^1(\widehat{I})}&\le \Vert \widehat{u}-\widehat{\pi }_{p,1} \widehat{u}\Vert _{H^1(\widehat{I})}+\Vert \widehat{\pi }_{p,0}(\widehat{u}-\widehat{\pi }_{p,1}\widehat{u})\Vert _{H^1(\widehat{I})}\\&\lesssim p^2 \Vert \widehat{u}-\widehat{\pi }_{p,1} \widehat{u}\Vert _{H^1(\widehat{I})}. \end{aligned}$$

Referring to (8.1) yields (8.3). $\square$

1.1.2 Approximation Properties of $\widehat{\pi }_{\varvec{p},r}$

We next derive approximation results for the tensor projectors in (4.3). On $\widehat{K}=\widehat{I}^3$, we introduce the tensor-product space

$$\begin{aligned} H^{1}_{\mathrm {mix}}(\widehat{K}):=H^{1}_{\mathrm {mix}}(\widehat{K}^\perp )\otimes H^{1}(\widehat{K}^{\Vert }):= H^{1}(\widehat{I})\otimes H^{1}(\widehat{I})\otimes H^{1}(\widehat{I}). \end{aligned}$$

(8.5)

endowed with the standard (tensor-product) norm $\Vert \cdot \Vert _{H^{1}_{\mathrm {mix}}(\widehat{K})}$. Let $K=K^\perp \otimes K^\perp$ be an axiparallel element, $\varvec{p}_K=(p_K^\perp ,p_K^{\Vert })$ an elemental degree vector and $r_K\in \{0,1\}$ an elemental conformity index in edge-parallel direction. For $u:K \rightarrow \mathbb {R}$, we denote by $\widehat{u}:=u\circ \Phi _K$ the pullback to the reference element $\widehat{K}$. In this setting, the tensor projection $\widehat{\pi }_{\varvec{p}_K,r_K}\widehat{u}=\widehat{\pi }^\perp _{p_K^\perp ,0}\otimes \widehat{\pi }^{\Vert }_{p_K^{\Vert },r_K}\widehat{u}$ defined in (4.3) satisfies the subsequent bounds.

Proposition 8.3

The error $\widehat{\eta }^\perp _0=\widehat{u}- \widehat{\pi }^\perp _{p_K^\perp ,0}\widehat{u}$ in edge-perpendicular direction satisfies

$$\begin{aligned} \Vert \widehat{\eta }^\perp _0 \Vert ^2_{H^{1}_{\mathrm {mix}}(\widehat{K})} \lesssim (p_K^\perp )^8 \Psi _{p_K^\perp ,s^\perp _K} E^{\perp }_{s_K^\perp }(K;u), \end{aligned}$$

(8.6)

for any $0 \le s_K^\perp \le p_K^\perp$, with

$$\begin{aligned} E^\perp _{s}(K;u ) :=\sum _{|\varvec{\alpha }^\perp |=s+1}^{s+2}\ \sum _{\alpha ^\Vert =0,1} (h_K^\perp )^{2|\varvec{\alpha }^\perp |-2} (h_K^{\Vert })^{2\alpha ^{\Vert }-1} \Vert \mathsf {D}^{\varvec{\alpha }^\perp }_{\perp }\mathsf {D}^{\alpha ^\Vert }_\Vert u \Vert ^2_{L^2(K)} \;. \end{aligned}$$

(8.7)

The error $\widehat{\eta }^{\Vert }=\widehat{u} -\widehat{\pi }_{p_K^{\Vert },r_K} \widehat{u}$ in edge-parallel direction satisfies

$$\begin{aligned} \Vert \widehat{\mathsf {D}}^{\varvec{\alpha }^\perp }_\perp \widehat{\mathsf {D}}^{\alpha ^{\Vert }}_{\Vert }\widehat{\eta }^{\Vert }\Vert ^2_{L^2(\widehat{K})}\lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^\perp )^{2|\varvec{\alpha }^\perp |-2} (h_K^{\Vert })^{2s_K^{\Vert }+1}\Vert \mathsf {D}_\perp ^{\varvec{\alpha }^\perp } \mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u\Vert ^2_{L^2(K)}, \end{aligned}$$

(8.8)

for any $r_K=0,1$, $|\varvec{\alpha }^\perp |\ge 0$, $\alpha ^{\Vert }=0,1$, and $0\le s_K^{\Vert }\le p_K^{\Vert }$.

Proof

We have

$$\begin{aligned} \widehat{\eta }^\perp _0 = \widehat{u}-\widehat{\pi }^{(1)}_{p_K^\perp ,0} \otimes \widehat{\pi }^{(2)}_{p_K^\perp ,0} \widehat{u} = (\widehat{u}-\widehat{\pi }^{(1)}_{p_K^\perp ,0} \widehat{u})+\widehat{\pi }^{(1)}_{p_K^\perp ,0} \left( \widehat{u} - \widehat{\pi }^{(2)}_{p_K^\perp ,0} \widehat{u}\right) \;. \end{aligned}$$

Hence, by the triangle inequality and the stability property (8.4) of the univariate $L^2$-projector $\widehat{\pi }^{(1)}_{p_K^\perp ,0}$, we find

$$\begin{aligned} \Vert \widehat{\eta }^\perp _0\Vert ^2_{H^{1}_\mathrm {mix}(\widehat{K})} \lesssim (p_K^\perp )^4 \Big ( \sum _{i=1}^2 \Vert \widehat{u} - \widehat{\pi }^{(i)}_{p_K^\perp ,0}\widehat{u} \Vert _{H^{1}_\mathrm {mix}(\widehat{K})}^2\Big ) \;. \end{aligned}$$

The univariate approximation properties (8.3) in Lemma 8.2 now imply

$$\begin{aligned} \Vert \widehat{\eta }^\perp _0\Vert ^2_{H^{1}_\mathrm {mix}(\widehat{K})} \lesssim (p_K^\perp )^8 \Psi _{p_K^\perp ,s_K^\perp } \Big (&\sum _{0\le \alpha _2^\perp ,\alpha ^{\Vert }\le 1} \Vert \widehat{\mathsf {D}}^{(s_K^\perp +1,\alpha _2^\perp ,\alpha ^{\Vert })} \widehat{u}\Vert ^2_{L^2(\widehat{K})}\Big . \\&+ \Big .\sum _{0\le \alpha ^\perp _1,\alpha ^{{\Vert }} \le 1} \Vert \widehat{\mathsf {D}}^{(\alpha ^\perp _1,s_K^\perp +1,\alpha ^{\Vert })} \widehat{u}\Vert ^2_{L^2(\widehat{K})} \Big )\;, \end{aligned}$$

for any $0\le s_K^\perp \le p_K^\perp$ and where we write $\mathsf {D}^{\varvec{\alpha }}=\mathsf {D}^{(\alpha _1,\alpha _2,\alpha _3)}$ for a multi-index $\varvec{\alpha }=(\alpha _1,\alpha _2,\alpha _3)$. This bound and a scaling argument as in [24, Section 5.1.4] yield the desired bound (8.6) for $\widehat{\eta }^\perp _0$.

The bound for $\widehat{\eta }^{\Vert }$ is an immediate consequence of the consistency bounds (8.1) ($r_K=1$) and (8.3) ($r_K=0$) applied in edge-parallel direction, combined again with a scaling argument as in [24, Section 5.1.4]. $\square$

1.1.3 Weighted Norm Estimates in Plane Domains

In plane domains perpendicular to edges, we shall use estimates in weighted spaces analogous to those in [13, Section 3]. To state them, let $\mathfrak {K}$ be an axiparallel and shape-regular rectangle of diameter $h_\mathfrak {K}$ which is affinely equivalent to the reference square $\widehat{\mathfrak {K}}=\widehat{I}^2$. Let ${\varvec{c}}$ be a corner of $\mathfrak {K}$ and set $r(\varvec{x})=|\varvec{x}-{\varvec{c}}|$. For a weight exponent $\beta \in [0,1)$, we denote by $L^2_\beta (\mathfrak {K})$ the weighted $L^2$-space endowed with the weighted norm $\Vert u\Vert _{L^2_\beta (\mathfrak {K})}:=\Vert r^{\beta }u\Vert _{L^2(\mathfrak {K})}$. For $m= 1,2$, the weighted Sobolev space $H^{m,m}_\beta (\mathfrak {K})$ is defined as the completion of all $C^\infty (\overline{\mathfrak {K}})$-functions with respect to the norm $\Vert u\Vert ^2_{H^{m,m}_\beta (\mathfrak {K})}:=\Vert u\Vert ^2_{H^{m-1}(\mathfrak {K})}+|u|^2_{H^{m,m}_\beta (\mathfrak {K})}$, where $|u|^2_{H^{m,m}_\beta (\mathfrak {K})}:=\sum _{|\varvec{\alpha }|=m}\Vert r^{\beta }\mathsf {D}^{\varvec{\alpha }} u\Vert ^2_{L^2(\mathfrak {K})}$. We denote by $\pi ^2_{p,0}$ the $L^2$-projection onto the tensor-product polynomial space $\mathbb {Q}_p(\mathfrak {K})$ obtained by mapping $\widehat{\pi }^2_{p,0}$ on $\widehat{\mathfrak {K}}$.

Lemma 8.4

Let $\beta \in [0,1)$ be a weight exponent. For $u\in H^{1,1}_\beta (\mathfrak {K})$ and $p\ge 0$, there holds

$$\begin{aligned} \Vert u-\pi ^2_{p,0} u\Vert ^2_{L^2(\mathfrak {K})} \lesssim h_\mathfrak {K}^{2-2\beta } |u|^2_{H^{1,1}_\beta (\mathfrak {K})}. \end{aligned}$$

(8.9)

Similarly, for $u\in H^{2,2}_\beta (\mathfrak {K})$ and $p\ge 1$, there holds

$$\begin{aligned} \Vert u-\pi ^2_{p,0} u\Vert ^2_{L^2(\mathfrak {K})}+h_\mathfrak {K}^2\Vert \nabla (u-\pi ^2_{p,0}u)\Vert ^2_{L^2(\mathfrak {K})} \lesssim p^4 h_\mathfrak {K}^{4-2\beta } |u|^2_{H^{2,2}_\beta (\mathfrak {K})}. \end{aligned}$$

(8.10)

The implied constants depend on the aspect ratio of $\mathfrak {K}$.

Proof

To prove (8.9), we apply the triangle inequality and the stability of the $L^2$-projection $\pi ^2_{p,0}$ to obtain

$$\begin{aligned} \Vert u-\pi ^2_{p,0} u\Vert _{L^2(\mathfrak {K})} \lesssim \Vert u-\pi ^2_{0,0} u\Vert _{L^2(\mathfrak {K})}+\Vert \pi ^2_{p,0}(u-\pi ^2_{0,0}u)\Vert _{L^2(\mathfrak {K})}\lesssim \Vert u-\pi ^2_{0,0}u\Vert _{L^2(\mathfrak {K})}. \end{aligned}$$

The proof of (8.9) for $p=0$ can then be found in [19, Proposition 27].

To show (8.10), upon scaling it is sufficient to consider the reference square $\widehat{\mathfrak {K}}=(-1,1)^2$. We denote by $\widehat{{\mathsf {p}}}^2_{1,0}$ the $L^2$-projection onto the polynomial space $\mathbb {P}_1(\widehat{\mathfrak {K}})$. With the stability bound (8.4), it follows that

$$\begin{aligned} \Vert \widehat{u}-\widehat{\pi }^2_{p,0}\widehat{u}\Vert _{H^1(\widehat{\mathfrak {K}})}^2&\lesssim \Vert \widehat{u}-\widehat{{\mathsf {p}}}^2_{1,0}\Vert ^2_{H^1(\widehat{\mathfrak {K}})} + \Vert \widehat{\pi }^2_{p,0}(\widehat{u} - \widehat{{\mathsf {p}}}^2_{1,0}\widehat{u})\Vert _{H^1(\widehat{\mathfrak {K}})}^2 \\&\lesssim p^4\Vert \widehat{u}-\widehat{{\mathsf {p}}}^2_{1,0}\widehat{u}\Vert _{H^1(\widehat{\mathfrak {K}})}^2. \end{aligned}$$

Hence, it suffices to prove (8.10) for $\widehat{{\mathsf {p}}}^2_{1,0}$: We claim that there is a constant $\widehat{C}>0$ independent of $\widehat{u}$ such that

$$\begin{aligned} \Vert \widehat{u}\Vert _{H^{2,2}_\beta (\widehat{\mathfrak {K}})} \le \widehat{C}\big (|\widehat{u}|_{H^{2,2}_\beta (\widehat{\mathfrak {K}})}+\Vert \widehat{{\mathsf {p}}}^2_{1,0}\widehat{u}\Vert _{L^2(\widehat{\mathfrak {K}})}\big )\;. \end{aligned}$$

(8.11)

The bound (8.11) follows with standard arguments from the Peetre–Tartar lemma (see [10, Lemma A.38]) and the fact that the embedding $H^{2,2}_\beta (\widehat{\mathfrak {K}}) \hookrightarrow H^1(\widehat{\mathfrak {K}})$ is compact (see [13, Lemma 3.4]). Invoking (8.11) for $\widehat{u}-\widehat{{\mathsf {p}}}^2_{1,0}\widehat{u}$ and noting that $|\widehat{{\mathsf {p}}}^2_{1,0}\widehat{u}|_{H^{2,2}_\beta (\mathfrak {K})}=0$, $\widehat{{\mathsf {p}}}^2_{1,0}(\widehat{u}-\widehat{{\mathsf {p}}}^2_{1,0}\widehat{u})=0$, results in $\Vert \widehat{u}-\widehat{{\mathsf {p}}}^2_{1,0}\widehat{u}\Vert _{H^{2,2}_\beta (\widehat{\mathfrak {K}})} \le \widehat{C} |\widehat{u}|_{H^{2,2}_\beta (\widehat{\mathfrak {K}})}$, which finishes the proof. $\square$

1.1.4 Edge-Parallel Interpolation

We construct univariate hp-projectors and establish exponential convergence bounds for univariate geometric refinements on the interval $\omega =(0,1)$ toward $x=0$. These results will be used for the hp-approximations along edges ${\varvec{e}}\in \mathcal E_{{\varvec{c}}}$ toward corners ${\varvec{c}}\in \mathcal C$.

In $\omega$ and for $\sigma \in (0,1)$, we introduce geometric meshes $\mathcal T^{\ell }_\sigma =\{I_j\}_{j=1}^{\ell +1}$ with elements given by $I_{1}=(0,\sigma ^\ell )$ and $I_j=(\sigma ^{\ell +2-j},\sigma ^{\ell +1-j})$ for $2\le j \le \ell +1$, respectively. We introduce the local mesh sizes $h_1:=\sigma ^\ell$ and

$$\begin{aligned} h_j := \sigma ^{\ell +1-j}(1-\sigma ), \qquad 2\le j \le \ell +1 \;. \end{aligned}$$

(8.12)

Then, there is a constant $\kappa >0$ solely depending on $\sigma \in (0,1)$ with

$$\begin{aligned} \kappa ^{-1} h_j \le |x| \le \kappa h_j,\qquad x\in I_j,\ 2\le j \le \ell +1 \;. \end{aligned}$$

(8.13)

On the geometric mesh $\mathcal T^{\ell }_\sigma$, let ${\varvec{p}}^{\Vert }=(p_1^{\Vert },\ldots ,p^{\Vert }_{\ell +1})\in \mathbb {N}^{\ell +1}$ be an (edge-parallel) polynomial degree vector with $p^{\Vert }_j = \max \{1, \lfloor \mathfrak {s}j \rfloor \}$, for $\mathfrak {s}>0$ as in Sect. 3.2.1. We set $|{\varvec{p}}^{\Vert } |=\max _{j=1}^{\ell +1} p^{\Vert }_j$ and introduce the space

$$\begin{aligned} V^{0}(\mathcal T^{\ell }_\sigma ,\varvec{p}^{\Vert }) := \left\{ \, v\in L^2(\omega ) \,:\, v|_{I_j} \in {\mathbb P}_{p_j^{\Vert }}(I_j),\ j = 1,\ldots ,\ell +1 \,\right\} . \end{aligned}$$

(8.14)

For conformity indices $r_j\in \{0,1\}$, we denote by $\pi$ the projection onto $V^{0}(\mathcal T^{\ell }_\sigma ,\varvec{p}^{\Vert })$, given on interval $I_j$ as the (scaled) univariate projector $\pi _{p^{\Vert }_j,r_j}:H^{r_j}(I_j) \rightarrow {\mathbb P}_{p^{\Vert }_j}(I_j)$. For $u\in H^1(\omega )$, we define the approximation errors $\eta := u - \pi u$, and set

$$\begin{aligned} T_j[\eta ]^2 := h_j^{-2}\Vert \eta \Vert ^2_{L^2(I_j)} + \Vert \eta ^\prime \Vert _{L^2(I_j)}^2. \end{aligned}$$

(8.15)

Lemma 8.5

For a weight exponent $\beta >0$, let $u\in H^1(\omega )$ be such that

$$\begin{aligned} \Vert |x|^{-1-\beta +s} u^{(s)}\Vert _{L^2(\omega )} \le C_u ^{s+1} \Gamma (s+1), \qquad s\ge 2. \end{aligned}$$

(8.16)

Then, for any conformity indices $r_j\in \{0,1\}$, there exist $b,C>0$ independent of $\ell \ge 1$ such that $\sum _{j=2}^{\ell +1} T_j [\eta ]^2 \le C \exp (-2 b\ell ).$

Proof

Fix $I_j\in {\mathcal T}_\sigma ^{\ell }$ for $2\le j\le \ell +1$. A straightforward scaling argument yields , where as usual we denote by $\widehat{v}$ the pullback operator from $v|_{I_j}$ to the reference interval $\widehat{I}=(-1,1)$. The bounds in (8.1) and (8.3) yield

for any $1 \le s^{\Vert }_j \le p^{\Vert }_j$, where we exclude $s^{\Vert }_j=0$ in (8.1), (8.3) to ensure that $s \ge 2$ in (8.16). Scaling the right-hand side above back to element $I_j$ results in

(8.17)

Moreover, by the equivalence (8.13),

$$\begin{aligned} \Vert u^{(s^{\Vert }_j+1)}\Vert _{L^2(I_j)}^2 \simeq h_j^{2+2\beta -2(s^{\Vert }_j+1)} \Vert |x|^{-1-\beta +(s^{\Vert }_j+1)} u^{(s^{\Vert }_j+1)}\Vert ^2_{L^2(I_j)}. \end{aligned}$$

(8.18)

By combining (8.17), (8.18) with the regularity assumption (8.16), we find that

(8.19)

for $1 \le s^{\Vert }_j\le p^{\Vert }_j$. An interpolation argument as in [24, Lemma 5.8] shows that the bound (8.19) holds for any real $s^{\Vert }_j\in [1,p^{\Vert }_j]$.

Next, we sum the bound (8.19) over all intervals $2\le j\le \ell +1$. In view of (8.12), we obtain

$$\begin{aligned} \sum _{j=2}^{\ell +1} T_j[\eta ]^2 \lesssim |{\varvec{p}}^{\Vert } |^{4} \left( \sum _{j=2}^{\ell +1} \sigma ^{2(\ell +1-j)\beta } \min _{s^{\Vert }_j\in [1,p^{\Vert }_j]}\left[ C^{2s^{\Vert }_j} \Psi _{p^{\Vert }_j,s^{\Vert }_j} \Gamma (s_j+2)^2\right] \right) . \end{aligned}$$

By [24, Lemma 5.12], the bracket on the right-hand side above is exponentially small. Adjusting the constants to absorb $|\varvec{p}^{\Vert }|^4$ finishes the proof. $\square$

Similarly, we obtain the following result.

Lemma 8.6

For a weight exponent $\beta >0$, let $u\in L^2(\omega )$ be such that

$$\begin{aligned} \Vert |x|^{-\beta +s} u^{(s)}\Vert _{L^2(\omega )} \le C_u^{s+2} \Gamma (s+2), \qquad s\ge 1 \;. \end{aligned}$$

(8.20)

For any conformity indices $r_j\in \{0,1\}$, there exist $b,C > 0$ independent of $\ell \ge 1$ such that $\sum _{j=2}^{\ell +1} \Vert \eta \Vert ^2_{L^2(I_j)} \le C \exp (-2 b\ell ).$

Proof

This follows as in Lemma 8.5 or [25, Proposition 5.5]. $\square$

1.2 Reference Corner–Edge Mesh

We consider the reference corner–edge mesh patch $\widetilde{{\mathcal M}}^{\ell ,{\varvec{c}}{\varvec{e}}}_\sigma$ on $\widetilde{Q}$ for ${\varvec{c}}\in \mathcal C$ and ${\varvec{e}}\in \mathcal E_{{\varvec{c}}}$; cf. Fig. 1 (right). As in [25, Section 7], it is sufficient to focus on the elements in $\widetilde{{\mathcal M}}^{\ell ,{\varvec{c}}{\varvec{e}}}_\sigma$ near the corner–edge pair. To this end, we introduce the submesh $\widetilde{\mathcal K}^{\ell ,{\varvec{c}}{\varvec{e}}}_\sigma \subset \widetilde{{\mathcal M}}^{\ell ,{\varvec{c}}{\varvec{e}}}_\sigma$ given by

$$\begin{aligned} \widetilde{\mathcal K}_\sigma ^{\ell ,{\varvec{c}}{\varvec{e}}} = \bigcup _{j=1}^{\ell +1} \bigcup _{i=1}^{j} \widetilde{\mathfrak L}^{ij}_{{\varvec{c}}{\varvec{e}}}, \end{aligned}$$

(8.21)

where the sets $\widetilde{\mathfrak L}^{ij}_{{\varvec{c}}{\varvec{e}}}$ stand for layers of elements with identical scaling properties with respect to ${\varvec{c}}$ and ${\varvec{e}}$; cf. [24, Section 5.2.4]. The index j indicates the number of the geometric mesh layers in edge-parallel direction along the edge ${\varvec{e}}$, whereas the index i indicates the number of mesh layers in direction perpendicular to ${\varvec{e}}$. In agreement with [25, Section 7.1], we split $\widetilde{\mathcal K}^{\ell ,{\varvec{c}}{\varvec{e}}}_\sigma$ into interior elements away from ${\varvec{c}}$ and ${\varvec{e}}$, boundary layer elements along ${\varvec{e}}$ (but away from ${\varvec{c}}$), and corner elements abutting at ${\varvec{c}}$. That is, we have $\widetilde{\mathcal K}^{\ell ,{\varvec{c}}{\varvec{e}}}_\sigma = \widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}\mathop {\cup }\limits ^{.}\widetilde{\mathfrak T}^\ell _{{\varvec{e}}}\mathop {\cup }\limits ^{.}\widetilde{\mathfrak T}^{\ell }_{{\varvec{c}}}$, with

$$\begin{aligned} \widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}:=\bigcup _{j=2}^{\ell +1} \bigcup _{i=2}^{j} \widetilde{\mathfrak L}^{ij}_{{\varvec{c}}{\varvec{e}}},\qquad \widetilde{\mathfrak T}^\ell _{{\varvec{e}}}:= \bigcup _{j=2}^{\ell +1} \widetilde{\mathfrak L}^{1j}_{{\varvec{c}}{\varvec{e}}},\qquad \widetilde{\mathfrak T}_{\varvec{c}}^\ell :=\widetilde{\mathfrak L}^{11}_{{\varvec{c}}{\varvec{e}}}. \end{aligned}$$

(8.22)

Here, for $2\le i,j\le \ell +1$, interior elements $K\in \widetilde{\mathfrak L}^{ij}_{{\varvec{c}}{\varvec{e}}}$ satisfy

$$\begin{aligned} r_{\varvec{e}}|_K\simeq h_K^\perp \simeq \sigma ^{\ell +1-i}, \qquad r_{\varvec{c}}|_K\simeq h_K^{\Vert } \simeq \sigma ^{\ell +1-j}. \end{aligned}$$

(8.23)

Similarly, boundary layer elements $K\in \widetilde{\mathfrak L}_{{\varvec{c}}{\varvec{e}}}^{1j}$ satisfy

$$\begin{aligned} r_{{\varvec{e}}}|_K\lesssim h_K^\perp \simeq \sigma ^{\ell }, \qquad r_{\varvec{c}}|_K\simeq h_K^{\Vert } \simeq \sigma ^{\ell +1-j},\qquad 2\le j \le \ell +1. \end{aligned}$$

(8.24)

Finally, a corner element in the layer $\widetilde{\mathfrak T}_{\varvec{c}}^\ell =\widetilde{\mathfrak L}^{11}_{{\varvec{c}}{\varvec{e}}}$ is isotropic with $r_{\varvec{e}}|_K\lesssim h_K \simeq \sigma ^\ell$, and $r_{\varvec{c}}|_K\lesssim h_K \simeq \sigma ^\ell$. The sets $\widetilde{\mathfrak L}^{1j}_{{\varvec{c}}{\varvec{e}}}$ and $\widetilde{\mathfrak L}^{11}_{{\varvec{c}}{\varvec{e}}}$ are in fact singletons, and without loss of generality ${K_j}\in \widetilde{\mathfrak L}^{1j}_{{\varvec{c}}{\varvec{e}}}$ can be written in the form

$$\begin{aligned} K_j= K^\perp \times K_j^{\Vert }, \qquad 2\le j \le \ell +1\;, \end{aligned}$$

(8.25)

cf. (3.3), where $K^\perp =(0,\sigma ^\ell )^2$, and the sequence $\{ K_j^{\Vert }\}_{j=2}^{\ell +1}$ forms a one-dimensional geometric mesh ${\mathcal T}_\sigma ^{\ell }$ along the edge ${\varvec{e}}$ as in Sect. 8.1.4. The $\mathfrak {s}$-linearly increasing polynomial degree distributions on $\widetilde{\mathcal K}_\sigma ^{\ell ,{\varvec{c}}{\varvec{e}}}$ in (8.21) are given by

$$\begin{aligned} \forall \, K\in \widetilde{\mathfrak L}^{ij}_{{\varvec{c}}{\varvec{e}}}: \qquad \varvec{p}_K = (p_i^\perp , p_j^\Vert )\simeq (\max \{1,\lfloor \mathfrak {s}i \rfloor \}, \max \{1,\lfloor \mathfrak {s}j\rfloor \} ). \end{aligned}$$

(8.26)

In the sequel, we introduce the domain $\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell :=\big (\cup _{K\in \widetilde{\mathcal K}_\sigma ^{\ell ,{\varvec{c}}{\varvec{e}}}}\overline{K}\big )^\circ$. Analogously to (2.6) and for exponents $\varvec{\beta }=\{\beta _{\varvec{c}},\beta _{\varvec{e}}\}$, we introduce the non-homogeneous reference corner–edge semi-norm on $\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell$:

$$\begin{aligned} \left| u\right| _{\widetilde{N}^{k}_{\varvec{\beta }}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2 := \sum _{|\varvec{\alpha }|=k} \left\| r_{\varvec{c}}^{\max \{\beta _{\varvec{c}}+|\varvec{\alpha }|,0\}}\rho _{{\varvec{c}}{\varvec{e}}}^{\max \{\beta _{\varvec{e}}+|\varvec{\alpha }^\perp |,0\}} \mathsf {D}^{\varvec{\alpha }}u\right\| _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2\;, \end{aligned}$$

(8.27)

for any $k\ge 0$ and where $r_{\varvec{c}}$ and $r_{\varvec{e}}$ are the distances to ${\varvec{c}}$ and ${\varvec{e}}$, respectively, and $\rho _{{\varvec{c}}{\varvec{e}}}=r_{\varvec{e}}/r_{\varvec{c}}$. For $m>k_{\varvec{\beta }}$ as in (2.7), the weighted Sobolev spaces $\widetilde{N}^m_{\varvec{\beta }}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ are defined as in Sect. 2.2 with respect to the norms $\Vert \cdot \Vert ^2_{\widetilde{N}^{m}_{\varvec{\beta }}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}=\sum _{k=0}^m |\cdot |^2_{\widetilde{N}^k_{\varvec{\beta }}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}$. The corresponding analytic reference class $B_{\varvec{\beta }}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ consists of all functions $u:\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell \rightarrow \mathbb {R}$ such that $u\in \widetilde{N}^k_{\varvec{\beta }}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ for $k>k_{\varvec{\beta }}$ and such that there is a constant $d_u>0$ with

$$\begin{aligned} |u|_{\widetilde{N}^{k}_{\varvec{\beta }}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}\le d_u^{k+1} \Gamma (k+1)\qquad \forall \, k>k_{\varvec{\beta }}. \end{aligned}$$

(8.28)

In the following, we restrict ourselves to the classes $B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ and $B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ for exponents $\varvec{b}=\{b_{\varvec{c}},b_{\varvec{e}}\}$ in (0, 1) as in Remark 2.3. In the first case, we have $k_{\varvec{\beta }}\in (1,2)$ and the norms on the right-hand in (8.27) are given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \quad \Vert \mathsf {D}^{\varvec{\alpha }} u\Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2 &{} \quad |\varvec{\alpha }|=0,1,\ |\varvec{\alpha }^\perp |=0,1,\\ \quad \Vert r_{\varvec{c}}^{-1-b_{\varvec{c}}+|\varvec{\alpha }|}\mathsf {D}^{\varvec{\alpha }} u\Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2 &{} \quad |\varvec{\alpha }|\ge 2,\ |\varvec{\alpha }^\perp |= 0,1,\\ \quad \Vert r_{\varvec{c}}^{b_{\varvec{e}}-b_{\varvec{c}}+\alpha ^{\Vert }} r_{\varvec{e}}^{-1-b_{\varvec{e}}+|\varvec{\alpha }^\perp |} \mathsf {D}^{\varvec{\alpha }} u\Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2 &{} \quad |\varvec{\alpha }|\ge 2, \ |\varvec{\alpha }^\perp |\ge 2. \end{array}\right. } \end{aligned}$$

(8.29)

Similarly, for the second analytic class $B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, we have $k_{\varvec{\beta }}\in (0,1)$ and the norms on the right-hand side of (8.27) take the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \quad \Vert u\Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2 &{} \quad |\varvec{\alpha }|=0,\ |\varvec{\alpha }^\perp |=0,\\ \quad \Vert r_{\varvec{c}}^{-b_{\varvec{e}}+\alpha ^{\Vert }} \mathsf {D}^{\varvec{\alpha }} u\Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2 &{} \quad |\varvec{\alpha }|= 1,\ |\varvec{\alpha }^\perp |= 0,\\ \quad \Vert r_{\varvec{c}}^{b_{\varvec{e}}-b_{\varvec{c}}+\alpha ^{\Vert }} r_{\varvec{e}}^{-b_{\varvec{e}}+|\varvec{\alpha }^\perp |} \mathsf {D}^{\varvec{\alpha }} u\Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}^2 &{} \quad |\varvec{\alpha }|\ge 1,\ |\varvec{\alpha }^\perp |\ge 1. \end{array}\right. } \end{aligned}$$

(8.30)

In the axiparallel setting considered in the present paper, when functions $u\in B_{-1-\varvec{b}}(\Omega )$ and $u\in B_{-\varvec{b}}(\Omega )$ as in Theorem 4.3 are localized and scaled to $\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell$, they belong to the reference classes $B_{-1 -\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ and $B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, respectively; cf. [20, Section 3.4].

To prove Theorem 4.3 in the reference corner–edge framework, it is now enough to bound the error contributions as in (3.34), (3.33) over $\widetilde{\mathcal K}_\sigma ^{\ell ,{\varvec{c}}{\varvec{e}}}=\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}\mathop {\cup }\limits ^{.}\widetilde{\mathfrak T}^\ell _{\varvec{e}}\mathop {\cup }\limits ^{.}\widetilde{\mathfrak T}_{\varvec{c}}^\ell$.

Proposition 8.7

For $b_{\varvec{c}},b_{\varvec{e}}\in (0,1)$, let $u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )\cap H^{1+\theta }(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ for some $\theta \in (0,1)$, and let $\pi u=\pi _0^\perp \otimes \pi ^{\Vert } u$ be the base interpolant (4.4) over $\widetilde{\mathcal K}_\sigma ^{\ell ,{\varvec{c}}{\varvec{e}}}$ for any conformity indices $r_K\in \{0,1\}$. For the errors $\eta$, $\eta _0^\perp$, $\eta ^{\Vert }$ in (4.7), we have

$$\begin{aligned} \Upsilon ^\perp _{\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}}[\eta _0^\perp ]^2+\Upsilon ^{\Vert }_{\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}}[\eta ^{\Vert }]^2 +\Upsilon ^\perp _{\widetilde{\mathfrak T}^\ell _{{\varvec{e}}}}[\eta _0^\perp ]^2+ \Upsilon ^{\Vert }_{\widetilde{\mathfrak T}^\ell _{{\varvec{e}}}}[\eta ^{\Vert }]^2+\Upsilon ^{\Vert }_{\widetilde{\mathfrak T}^\ell _{\varvec{c}}}[\eta ]^2 \le C\exp (-2b\ell ), \end{aligned}$$

with $b,C>0$ independent of $\ell$.

Moreover, for $b_{\varvec{c}},b_{\varvec{e}}\in (0,1)$, let $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )\cap H^{\theta }(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ for some $\theta \in (0,1)$, and let $\pi _0 u=\pi _0^\perp \otimes \pi ^{\Vert }_0 u$ be the $L^2$-projection over $\widetilde{\mathcal K}_\sigma ^{\ell ,{\varvec{c}}{\varvec{e}}}$. For the errors $\eta _0$, $\eta ^\perp _0$, $\eta _0^{\Vert }$ in (4.7), we have

$$\begin{aligned} \Vert \eta _0^\perp \Vert ^2_{L^2(\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}})} +\Vert \eta _0^{\Vert }\Vert ^2_{L^2(\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}})} +\Vert \eta _0^\perp \Vert ^2_{L^2(\widetilde{\mathfrak T}^\ell _{{\varvec{e}}})} +\Vert \eta _0^{\Vert }\Vert ^2_{L^2(\widetilde{\mathfrak T}^\ell _{{\varvec{e}}})} + \Vert \eta _0\Vert ^2_{L^2(\widetilde{\mathfrak T}^\ell _{\varvec{c}})} \le C\exp (-2b\ell ), \end{aligned}$$

with $b,C>0$ independent of $\ell$.

The remainder of this section is devoted to the proof of Proposition 8.7.

1.3 Proof of Proposition 8.7

We bound the errors in Proposition 8.7 separately for the set $\widetilde{\mathfrak O}_{{\varvec{c}}{\varvec{e}}}^\ell$ (Propositions 8.9, 8.10), for $\widetilde{\mathfrak T}_{\varvec{e}}^\ell$ (Propositions 8.11, 8.12), and for $\widetilde{\mathfrak T}_{\varvec{c}}^\ell$ (Proposition 8.13).

1.3.1 Convergence on $\widetilde{\mathfrak O}_{{\varvec{c}}{\varvec{e}}}^\ell$

We begin our analysis by recalling essential scaling properties; see [24, Section 5.1.4].

Lemma 8.8

Let $K=(0,h_K^\perp )^2\times (0,h_K^{\Vert })$ be of the form (3.3). Let $v:K\rightarrow \mathbb {R}$, and $\widehat{v}=v\circ \Phi _{K}^{-1}$. Then:

• (i) $\Vert v\Vert ^2_{L^2(K)} \lesssim (h_K^\perp )^2 h_K^{\Vert }\Vert \widehat{v}\Vert ^2_{L^2(\widehat{K})}$.

• (ii) $(h_K^{\Vert })^{-2}\Vert v\Vert ^2_{L^2(K)}+\Vert \mathsf {D}_{\Vert } v\Vert ^2_{L^2(K)}\lesssim (h_K^\perp )^2 (h_K^{\Vert })^{-1}\big (\Vert \widehat{v}\Vert ^2_{L^2(\widehat{K})}+\Vert \widehat{\mathsf {D}}_{\Vert } \widehat{v}\Vert ^2_{L^2(\widehat{K})}\big )$.

• (iii) $(h_K^\perp )^{-2}\Vert v\Vert ^2_{L^2(K)}+\Vert \mathsf {D}_\perp v\Vert ^2_{L^2(K)}\lesssim h_K^{\Vert } \big ( \Vert \widehat{v}\Vert ^2_{L^2(\widehat{K})}+\Vert \widehat{\mathsf {D}}_\perp \widehat{v}\Vert ^2_{L^2(\widehat{K})}\big )$.

We bound $\eta _0^\perp$ over $\widetilde{\mathfrak O}_{{\varvec{c}}{\varvec{e}}}^\ell$ as follows.

Proposition 8.9

Let $u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, respectively $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$. Then there are constants $b,C>0$ independent of $\ell$ such that $\Upsilon ^\perp _{\widetilde{\mathfrak O}_{{\varvec{c}}{\varvec{e}}}^\ell }[\eta _0^{\perp }]^2 \le C \exp (-2 b\ell )$, respectively $\Vert \eta _0^\perp \Vert ^2_{L^2(\widetilde{\mathfrak O}_{{\varvec{c}}{\varvec{e}}}^\ell )}\le C \exp (-2 b\ell )$.

Proof

Let $u \in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$. We consider an element $K\in \widetilde{\mathfrak L}^{ij}_{{\varvec{c}}{\varvec{e}}}$ with $2\le j\le \ell +1$ and $2\le i \le j$; see (8.22). With Lemma 8.8 (observing that $h_K^\perp \lesssim h_K^{\Vert }$), the approximation results for $\widehat{\eta }^\perp _0$ in Proposition 8.3 and (8.26), we conclude that

$$\begin{aligned} N_{K}^\perp [\eta ^\perp _0]^2&\lesssim h_K^{\Vert } \Vert \widehat{\eta }_0^\perp \Vert ^2_{H^1_\mathrm {mix}(\widehat{K})} \lesssim _p \, h_K^{\Vert } \Psi _{p_i^\perp ,s^\perp _i} E^\perp _{s_i^\perp }(K;u), \end{aligned}$$

for $1\le s_i^\perp \le p_i^\perp$, where $E^\perp _{s_i^\perp }(K;u)$ is the expression in (8.7). Notice that here we exclude the choice $s^\perp _i=0$ to ensure that $|\varvec{\alpha }|\ge |\varvec{\alpha }^\perp |\ge 2$ in $E^\perp _{s_i^\perp }(K;u)$. Thanks to the equivalences (8.24), we insert the appropriate weights as in (8.29) and obtain

$$\begin{aligned} \Vert \mathsf {D}^{\varvec{\alpha }^\perp }_{\perp }\mathsf {D}^{\alpha ^\Vert }_\Vert u \Vert ^2_{L^2(K)}&\simeq (h_K^{\Vert })^{2b_{\varvec{c}}-2b_{\varvec{e}}-2\alpha ^{\Vert }} (h_K^\perp )^{2+2b_{\varvec{e}}-2|\varvec{\alpha }^\perp |}\\&\qquad \qquad \times \Vert r_{\varvec{c}}^{b_{\varvec{e}}-b_{\varvec{c}}+\alpha ^{\Vert }} r_{\varvec{e}}^{-1-b_{\varvec{e}}+|\varvec{\alpha }^\perp |} \mathsf {D}^{\varvec{\alpha }^\perp }_{\perp } \mathsf {D}^{\alpha ^\Vert }_\Vert u \Vert ^2_{L^2(K)}\;, \end{aligned}$$

for $2\le s_i^\perp +1\le |\varvec{\alpha }^\perp |\le s_i^\perp +2$ and $\alpha ^{\Vert }=0,1$. Hence, it follows that

$$\begin{aligned} N_{K}^\perp [\eta ^\perp _0]^2\lesssim _p \Psi _{p_i^\perp ,s_i^\perp } (h_K^{\Vert })^{2b_{\varvec{c}}-2b_{\varvec{e}}} (h_K^\perp )^{2 b_{\varvec{e}}}\sum _{k=s_i^\perp +1}^{s_i^\perp +3} \left| u\right| ^2_{\widetilde{N}^k_{-1-\varvec{b}}(K)}. \end{aligned}$$

The analytic regularity (8.28) then implies the existence of $C>0$ such that

$$\begin{aligned} N_{K}^\perp [\eta _0^\perp ]^2 \lesssim _p \, \Psi _{p^\perp _i,s^\perp _i} (h_K^{\Vert })^{2b_{\varvec{c}}-2b_{\varvec{e}}}(h_K^\perp )^{2b_{\varvec{e}}} C^{2s^\perp _i}\Gamma (s_i^\perp +4)^2, \end{aligned}$$

(8.31)

for all $1\le s_i^\perp \le p_i^\perp$. Summing (8.31) over all layers in $\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}$ in (8.22) in combination with (8.23) results in

$$\begin{aligned} \Upsilon ^\perp _{\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}}[\eta _0^\perp ] \lesssim _p \sum _{j=2}^{\ell +1} \sigma ^{2(b_{\varvec{c}}-b_{\varvec{e}})(\ell +1-j)} \Big (\sum _{i=2}^j \sigma ^{2b_{\varvec{e}}(\ell +1-i)}\Psi _{p_i^\perp ,s_i^\perp } C^{2s_i^\perp } \Gamma (s_i^\perp +4)^2\Big ) \;. \end{aligned}$$

By interpolating to real parameters $s_i^\perp \in [1,p_i^\perp ]$ as in [24, Lemma 5.8], this sum is of the same form as $S^\perp$ in the proof of [24, Proposition 5.17], and the assertion now follows from the arguments there and after adjusting the constants to absorb the algebraic loss in $|\varvec{p}|$.

For $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, we proceed similarly and note that

$$\begin{aligned} \Vert \eta ^\perp _0\Vert ^2_{L^2(K)} \lesssim (h_K^\perp )^2 h_K^{\Vert } \Vert \widehat{\eta }_0^\perp \Vert ^2_{H^1_\mathrm {mix}(\widehat{K})} \lesssim _p (h_K^\perp )^2 h_K^{\Vert } \Psi _{p_i^\perp ,s^\perp _i} E^\perp _{s_i^\perp }(K;u), \end{aligned}$$

for $1\le s_i^\perp \le p_i^\perp$. Hence, we obtain

$$\begin{aligned} \Vert \eta ^\perp _0\Vert ^2_{L^2(K)}\lesssim _p \Psi _{p_i^\perp ,s_i^\perp } (h_K^{\Vert })^{2b_{\varvec{c}}-2b_{\varvec{e}}} (h_K^\perp )^{2b_{\varvec{e}}}\sum _{k=s_i^\perp +1}^{s_i^\perp +3} \left| u\right| ^2_{\widetilde{N}^k_{-\varvec{b}}(K)}. \end{aligned}$$

The second bound follows as before. $\square$

Next, we establish the analog of Proposition 8.9 in edge-parallel direction.

Proposition 8.10

Let $u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, respectively $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$. Then there are constants $b,C>0$ independent of $\ell$ such that $\Upsilon ^{\Vert }_{\widetilde{\mathfrak O}_{{\varvec{c}}{\varvec{e}}}^\ell }[\eta ^{\Vert }]^2 \le C \exp (-2 b\ell )$, respectively $\Vert \eta _0^{\Vert }\Vert ^2_{L^2(\widetilde{\mathfrak O}_{{\varvec{c}}{\varvec{e}}}^\ell )} \le C \exp (-2 b\ell )$.

Proof

For $u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, we claim that

$$\begin{aligned} N_{K}^{\Vert }[\eta ^{\Vert }]^2 \lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_{K}^{\Vert })^{2b_{\varvec{c}}} \Vert u\Vert ^2_{\widetilde{N}^{s_K^{\Vert }+2}_{-1-\varvec{b}}(K)}, \end{aligned}$$

(8.32)

for any $K\in \widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}$ and $1\le s_K^{\Vert }\le p_K^{\Vert }$. To prove (8.32), we start by employing Lemma 8.8 and the approximation property for $\widehat{\eta }^{\Vert }$ in Proposition 8.3:

$$\begin{aligned} (h_K^{\Vert })^{-2} \Vert \eta ^{\Vert }\Vert ^2_{L^2(K)}+\Vert \mathsf {D}_{\Vert }\eta ^{\Vert }\Vert ^2_{L^2(K)}&\lesssim (h_K^\perp )^{2}(h_K^{\Vert })^{-1} \sum _{\alpha ^{\Vert }=0,1}\Vert \widehat{\mathsf {D}}_{\Vert }^{\alpha ^{\Vert }} \widehat{\eta }^{\Vert } \Vert ^2_{L^2(\widehat{K})}\\&\lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^{\Vert })^{2s_K^{\Vert }} \Vert \mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u \Vert ^2_{L^2(K)} \end{aligned}$$

for any $1 < s_K^{\Vert }\le p_K^{\Vert }$, where we again exclude the choice $s_K^{\Vert }=0$ so that $|\varvec{\alpha }|\ge 2$. We then insert suitable weights with the aid of (8.23), (8.29) to obtain

$$\begin{aligned} \Vert \mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u\Vert ^2_{L^2(K)}&\simeq (h_K^{\Vert })^{2b_{\varvec{c}}-2s_K^{\Vert }}\Vert r_{\varvec{c}}^{-1-b_{\varvec{c}}+s_K^{\Vert }+1}\mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u\Vert ^2_{L^2(K)}. \end{aligned}$$

Hence,

$$\begin{aligned} (h_K^{\Vert })^{-2} \Vert \eta ^{\Vert }\Vert ^2_{L^2(K)}+\Vert \mathsf {D}_{\Vert }\eta ^{\Vert }\Vert ^2_{L^2(K)} \lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^{\Vert })^{2b_{\varvec{c}}} | u|^2_{\widetilde{N}^{s_K^{\Vert }+1}_{-1-\varvec{b}}(K)} \;. \end{aligned}$$

By proceeding similarly, we find that, for $|\varvec{\alpha }^\perp |=1$,

$$\begin{aligned} \Vert \mathsf {D}^{\varvec{\alpha }^\perp }_{\perp } \eta ^{\Vert }\Vert ^2_{L^2(K)}&\lesssim h_K^{\Vert } \sum _{\alpha ^{\Vert }=0,1}\Vert \widehat{\mathsf {D}}^{\varvec{\alpha }^\perp }_{\perp }\widehat{\mathsf {D}}_{\Vert }^{\alpha ^{\Vert }} \widehat{\eta }^{\Vert } \Vert ^2_{L^2(\widehat{K})} \\&\lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^{\Vert })^{2s_K^{\Vert }+2} \Vert \mathsf {D}_{\perp }^{\varvec{\alpha }^\perp }\mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u \Vert ^2_{L^2(K)} \\&\lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^{\Vert })^{2b_{\varvec{c}}}\Vert r_{\varvec{c}}^{-b_{\varvec{c}}+s_K^{\Vert }+1}\mathsf {D}_\perp ^{\varvec{\alpha }^\perp }\mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u\Vert ^2_{L^2(K)}\\&\lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^{\Vert })^{2b_{\varvec{c}}} | u|^2_{\widetilde{N}^{s_K^{\Vert }+2}_{-1-\varvec{b}}(K)}. \end{aligned}$$

This establishes the bound (8.32).

For $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, we use analogous arguments based on Lemma 8.8, Proposition 8.3 and (8.30). This results in

$$\begin{aligned} \begin{aligned} \Vert \eta ^{\Vert }_0\Vert ^2_{L^2(K)}&\lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^{\Vert })^{2s_K^{\Vert }+2} \Vert \mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u \Vert ^2_{L^2(K)}\\&\lesssim _p \Psi _{p_K^{\Vert },s_K^{\Vert }} (h_K^{\Vert })^{2 b_{\varvec{c}}} \Vert r_{\varvec{c}}^{-b_{\varvec{c}}+s_K^{\Vert }+1}\mathsf {D}_{\Vert }^{s_K^{\Vert }+1} u\Vert ^2_{L^2(K)}\\&\lesssim _p (h_K^{\Vert })^{2 b_{\varvec{c}}} |u|^2_{\widetilde{N}^{s_K^{\Vert }+1}_{-\varvec{b}}(K)}. \end{aligned} \end{aligned}$$

(8.33)

Next, we sum the bounds in (8.32), (8.33) over all layers of $\widetilde{\mathfrak O}^{\ell }_{{\varvec{c}}{\varvec{e}}}$. By noticing (8.23), (8.26) and the analytic regularity (8.28), we conclude that

$$\begin{aligned}&u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell ):&\quad&\Upsilon ^{\Vert }_{\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}}}[\eta ^{\Vert }]^2 \lesssim _p \sum _{j=2}^{\ell +1} \sum _{i=2}^{j} \Psi _{p_j^{\Vert },s_j^{\Vert }} \sigma ^{2(\ell +1-j)b_{\varvec{c}}} C^{2s_j^{\Vert }}\Gamma (s_j^{\Vert }+3)^2,\\&u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell ):&\quad&\Vert \eta _0^{\Vert }\Vert ^2_{L^2(\widetilde{\mathfrak O}^\ell _{{\varvec{c}}{\varvec{e}}})} \lesssim _p \sum _{j=2}^{\ell +1} \sum _{i=2}^{j} \Psi _{p_j^{\Vert },s_j^{\Vert }} \sigma ^{2(\ell +1-j)b_{\varvec{c}}} C^{2s_j^{\Vert }}\Gamma (s_j^{\Vert }+2)^2. \end{aligned}$$

The terms in the sums above are independent of the inner index i. Interpolation to non-integer differentiation orders $s_j^{\Vert }\in [1 , p_j^{\Vert }]$ as in [24, Lemma 5.8], applying [24, Lemma 5.12] and absorbing the algebraic loss in $|\varvec{p}|$ complete the proof. $\square$

1.3.2 Convergence on $\widetilde{\mathfrak T}^{\ell }_{{\varvec{e}}}$

We first consider edge-perpendicular elements.

Proposition 8.11

Let $u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, respectively $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$. Then there are constants $b,C>0$ independent of $\ell$ such that $\Upsilon ^\perp _{\widetilde{\mathfrak T}^\ell _{{\varvec{e}}}}[\eta _0^{\perp }]^2\le C\exp (-2b\ell )$, respectively $\Vert \eta _0^\perp \Vert ^2_{L^2(\widetilde{\mathfrak T}^\ell _{{\varvec{e}}})}\le C\exp (-2b\ell )$.

Proof

Let $K=K_j=K^\perp \times K_j^{\Vert }$ be an element of $\widetilde{\mathfrak T}_{\varvec{e}}^\ell$ as in (8.25). We claim that

$$\begin{aligned} (h_{K}^{\perp })^{-2}\Vert \eta _0^\perp \Vert ^2_{L^2(K)} + \Vert \mathsf {D}_\perp \eta _0^{\perp }\Vert ^2_{L^2(K}&\lesssim \sigma ^{2 \min \{b_{\varvec{c}},b_{\varvec{e}}\} \ell } |u|^2_{\widetilde{N}^2_{-1-\varvec{b}}(K)}\;, \end{aligned}$$

(8.34)

$$\begin{aligned} \Vert \mathsf {D}_{\Vert } \eta _0^\perp \Vert ^2_{L^2(K)}&\lesssim \sigma ^{2 \min \{b_{\varvec{c}},b_{\varvec{e}}\} \ell } |u|^2_{\widetilde{N}^{3}_{-1-\varvec{b}}(K)} \;, \end{aligned}$$

(8.35)

$$\begin{aligned} \Vert \eta _0^\perp \Vert ^2_{L^2(K)}&\lesssim \sigma ^{2\min \{b_{\varvec{c}},b_{\varvec{e}}\}\ell }|u|^2_{\widetilde{N}^1_{-\varvec{b}}(K)}. \end{aligned}$$

(8.36)

To show (8.34), let $s=|\varvec{\alpha }^\perp |=0,1$. Applying the bound (8.10) with $\beta =1-b_{\varvec{e}}$ (noting that $p_K^\perp =\max \{1,\mathfrak {s}\}$ by (8.26)) and from (8.24), (8.29), we see that

$$\begin{aligned} (h_{K}^{\perp })^{2(s-1)} \Vert \mathsf {D}_\perp ^{\varvec{\alpha }^\perp } \eta _0^{\perp }\Vert ^2_{L^2(K)}&\lesssim (h_K^\perp )^{2b_{\varvec{e}}} \Vert r_{\varvec{e}}^{1-b_{\varvec{e}}}\mathsf {D}_\perp ^2 u\Vert ^2_{L^2(K)},\\&\lesssim (h_{K}^{\Vert })^{-2(b_{\varvec{e}}-b_{\varvec{c}})}(h_K^\perp )^{2b_{\varvec{e}}} \Vert r_{\varvec{c}}^{b_{\varvec{e}}-b_{\varvec{c}}} r_{\varvec{e}}^{1-b_{\varvec{e}}} \mathsf {D}^2_\perp u\Vert ^2_{L^2(K)}, \end{aligned}$$

where $|\mathsf {D}^2_\perp v|^2=\sum _{|\varvec{\alpha }^\perp |=2}|\mathsf {D}^{\varvec{\alpha }^\perp }_\perp v|^2$. Thus, combining these estimates and expressing the mesh sizes in terms of $\sigma$, see (8.24), we have

$$\begin{aligned} (h_{K}^{\perp })^{2(s-1)}\Vert \mathsf {D}_\perp ^{\varvec{\alpha }^\perp } \eta _0^{\perp }\Vert ^2_{L^2(K)}&\lesssim \sigma ^{2 b_{\varvec{c}}(\ell +1-j)+ {2}b_{\varvec{e}}(j-1)} |u|^2_{\widetilde{N}^2_{-1-\varvec{b}}(K)} \\&\lesssim \sigma ^{2 \min \{b_{\varvec{c}},b_{\varvec{e}}\} \ell } |u|^2_{\widetilde{N}^2_{-1-\varvec{b}}(K)}, \end{aligned}$$

which yields (8.34). To prove (8.35), we similarly conclude that

$$\begin{aligned} \Vert \mathsf {D}_{\Vert } \eta _0^\perp \Vert ^2_{L^2(K)}&\lesssim (h_{K}^\perp )^{2+2b_{\varvec{e}}} \Vert r_{\varvec{e}}^{1-b_{\varvec{e}}} \mathsf {D}_\perp ^{2} \mathsf {D}_{\Vert } u\Vert _{L^2(K)}^2 \\&\lesssim (h^{\Vert }_{K})^{-2-2b_{\varvec{e}}+2b_{\varvec{c}}} (h_{K}^{\perp })^{2+2 b_{\varvec{e}}} \Vert r_{\varvec{c}}^{b_{\varvec{e}}-b_{\varvec{c}}+1} r_{\varvec{e}}^{1-b_{\varvec{e}}} \mathsf {D}_\perp ^{2} \mathsf {D}_{\Vert } u\Vert _{L^2(K)}^2 \\&\lesssim \sigma ^{2\min \{b_{\varvec{c}},b_{\varvec{e}}\}\ell }|u|^2_{\widetilde{N}^{3}_{-1-\varvec{b}}(K)}\;. \end{aligned}$$

For the bound (8.36), we employ an analogous argument based on (8.9) (with $\beta =1-b_{\varvec{e}}$). Indeed, with (8.24) and (8.30), we conclude that

$$\begin{aligned} \Vert \eta _0^\perp \Vert ^2_{L^2(K)}&\lesssim (h_K^\perp )^{2b_{\varvec{e}}} \Vert r_{\varvec{e}}^{1-b_{\varvec{e}}} \mathsf {D}_\perp u\Vert ^2_{L^2(K)}\\&\lesssim (h_K^{\Vert })^{2b_{\varvec{c}}-2b_{\varvec{e}}}(h_K^\perp )^{2b_{\varvec{e}}} \Vert r_{\varvec{c}}^{b_{\varvec{e}}-b_{\varvec{c}}}r_{\varvec{e}}^{1-b_{\varvec{e}}} \mathsf {D}_\perp u\Vert ^2_{L^2(K)}\\&\lesssim \sigma ^{2\min \{b_{\varvec{c}},b_{\varvec{e}}\}\ell }|u|^2_{\widetilde{N}^1_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}, \end{aligned}$$

which is (8.36).

The assertions now follow by summing the estimates (8.34), (8.35) and (8.36) over all elements $K\in \widetilde{\mathfrak T}_{\varvec{e}}^\ell$ (i.e., over $2 \le j\le \ell +1$) and by suitably adjusting the constants. $\square$

A similar estimate holds for the approximation errors in direction parallel to ${\varvec{e}}$.

Proposition 8.12

Let $u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, respectively $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$. Then there are constants $b,C>0$ independent of $\ell$ such that $\Upsilon ^{\Vert }_{\widetilde{\mathfrak T}^\ell _{{\varvec{e}}}}[\eta ^{\Vert }]^2\le C \exp (-2b\ell )$, respectively $\Vert \eta _0^{\Vert }\Vert ^2_{L^2(\widetilde{\mathfrak T}^\ell _{{\varvec{e}}})}\le C \exp (-2b\ell )$.

Proof

For $u\in B_{-1-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ and $|\varvec{\alpha }^\perp |=1$, we have $u, \mathsf {D}_{\Vert } u, \mathsf {D}^{\varvec{\alpha }^\perp }_\perp u\in L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^{\ell })$ due to (8.28), and there holds

$$\begin{aligned} \Vert r_{\varvec{c}}^{-1-b_{\varvec{c}}+\alpha ^{\Vert }} \mathsf {D}^{\alpha ^{\Vert }}_{\Vert } u \Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}&\le C^{\alpha ^{\Vert }+1}\Gamma (\alpha ^{\Vert }+1),&\qquad&\alpha ^{\Vert } \ge 2,\\ \Vert r_{\varvec{c}}^{-b_{\varvec{c}}+\alpha ^{\Vert }} \mathsf {D}_{\Vert }^{\alpha ^{\Vert }}\mathsf {D}_\perp u \Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}&\le C^{\alpha ^{\Vert }+2}\Gamma (\alpha ^{\Vert }+2),&\qquad&\alpha ^{\Vert } \ge 1. \end{aligned}$$

Similarly, for $u\in B_{-\varvec{b}}(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ it follows with (8.30) that $u\in L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ and

$$\begin{aligned} \Vert r_{\varvec{c}}^{-b_{\varvec{c}}+\alpha ^{\Vert }} \mathsf {D}_{\Vert }^{\alpha ^{\Vert }} u\Vert _{L^2(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )}\le C^{\alpha ^{\Vert }+1} \Gamma (\alpha ^{\Vert }+1),\qquad \alpha ^{\Vert }\ge 1. \end{aligned}$$

In view of (8.24), (8.25), these properties correspond to the one-dimensional analytic regularity assumptions considered in (8.16) and (8.20), respectively. Moreover, due to (8.26), the polynomial degrees $p_K^{\Vert }$ along the edge ${\varvec{e}}$ are $\mathfrak {s}$-linearly increasing away from the corner ${\varvec{c}}$. Hence, Lemma 8.5, respectively Lemma 8.6 along with the tensor-product structure of the elements yield the assertions. $\square$

1.3.3 Convergence on $\widetilde{\mathfrak T}_{\varvec{c}}^\ell$

It remains to show exponential convergence on $\widetilde{\mathfrak T}_{\varvec{c}}^\ell$.

Proposition 8.13

Let $u\in H^{1+\theta }(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$, respectively $u\in H^{\theta }(\widetilde{\Omega }_{{\varvec{c}}{\varvec{e}}}^\ell )$ for some $\theta \in (0,1)$. Then there exist constants $b,C>0$ independent of $\ell$ such that $\Upsilon ^{\Vert }_{\widetilde{\mathfrak T}_{\varvec{c}}^\ell }[\eta ]^2\le C\exp (-2b\ell )$, respectively $\Vert \eta _0\Vert ^2_{L^2(\widetilde{\mathfrak T}_{\varvec{c}}^\ell )}\le C\exp (-2b\ell )$.

Proof

The element $K \in \widetilde{\mathfrak T}^\ell _{\varvec{c}}$ is isotropic with $h_K\simeq \sigma ^\ell$; cf. (8.22). Standard h-version approximation properties then show that $N^{\Vert }_{K}[\eta ]^2 \lesssim h_K^{2\theta } \Vert u\Vert ^2_{H^{1+\theta }(K)}$, respectively $\Vert \eta _0\Vert ^2_{L^2(K)} \lesssim h_K^{2\theta } \Vert u\Vert ^2_{H^{\theta }(K)}$. $\square$