∎

¹¹institutetext: Hiroki Ishizaka ²²institutetext: Graduate School of Science and Engineering, Ehime University, Matsuyama, Japan
²²email: h.ishizaka005@gmail.com ³³institutetext: Kenta Kobayashi⁴⁴institutetext: Graduate School of Business Administration, Hitotsubashi University, Kunitachi, Japan
⁴⁴email: kenta.k@r.hit-u.ac.jp ⁵⁵institutetext: Takuya Tsuchiya ⁶⁶institutetext: Graduate School of Science and Engineering, Ehime University, Matsuyama, Japan
⁶⁶email: tsuchiya@math.sci.ehime-u.ac.jp

Anisotropic interpolation error estimates using a new geometric parameter

Hiroki Ishizaka Kenta Kobayashi Takuya Tsuchiya

(Received: date / Accepted: date)

Abstract

We present precise anisotropic interpolation error estimates for smooth functions using a new geometric parameter and derive inverse inequalities on anisotropic meshes. In our theory, the interpolation error is bounded in terms of the diameter of a simplex and the geometric parameter. Imposing additional assumptions makes it possible to obtain anisotropic error estimates. This paper also includes corrections to an error in Theorem 2 of our previous paper, “General theory of interpolation error estimates on anisotropic meshes” (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191).

Keywords:

Finite element Interpolation error estimates Anisotropic meshes

MSC:

65D05 65N30

^†^†journal: Japan Journal of Industrial and Applied Mathematics

1 Introduction

Analyzing the errors of interpolations on $d$ -simplices is an important subject in numerical analysis. It is particularly crucial for finite element error analysis. Let us briefly outline the problems considered in this paper using the Lagrange interpolation operator.

Let $d\in\{1,2,3\}$ . Let $\widehat{T}\subset\mathbb{R}^{d}$ and $T_{0}\subset\mathbb{R}^{d}$ be a reference element and a simplex, respectively, that are affine equivalent. Let us consider two Lagrange finite elements $\{\widehat{T},\widehat{P}:=\mathcal{P}^{k},\widehat{\Sigma}\}$ and $\{{T}_{0},{P}:=\mathcal{P}^{k},{\Sigma}\}$ with associated normed vector spaces $V(\widehat{T}):=\mathcal{C}(\widehat{T})$ and $V(T_{0}):=\mathcal{C}(T_{0})$ with $k\in\mathbb{N}$ , where $\mathcal{P}^{m}$ is the space of polynomials with degree at most $m\in\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$ . For $\hat{\varphi}\in V(\widehat{T})$ , we use the correspondences

\displaystyle(\varphi_{0}:T_{0}\to\mathbb{R})\to(\hat{\varphi}:={\varphi}_{0}\circ{\Phi}:\widehat{T}\to\mathbb{R}),

where $\Phi$ is an affine mapping. Let $I_{\widehat{T}}^{k}:V(\widehat{T})\to\mathcal{P}^{k}$ and $I_{{T}_{0}}^{k}:V({T}_{0})\to\mathcal{P}^{k}$ be the corresponding Lagrange interpolation operators. Details can be found in Section 2.3.

We first consider the case in which $d=1$ . Let $\Omega:=(0,1)\subset\mathbb{R}$ . For $N\in\mathbb{N}$ , let $\mathbb{T}_{h}=\{0=x_{0}\textless x_{1}\textless\cdots\textless x_{N}\textless x_{N+1}=1\}$ be a mesh of $\overline{\Omega}$ such as

\displaystyle\displaystyle\overline{\Omega}:=\bigcup_{i=1}^{N}T_{0}^{i},\quad\mathop{\rm{int}}T_{0}^{i}\cap\mathop{\rm{int}}T_{0}^{j}=\emptyset\quad\text{for $i\neq j$},

where $T_{0}^{i}:=[x_{i},x_{i+1}]$ for $0\leq i\leq N$ . We denote $h_{i}:=x_{i+1}-x_{i}$ for $0\leq i\leq N$ . If we set $x_{j}:=\frac{j}{N+1}$ for $j=0,1,\ldots,N,N+1$ , the mesh $\mathbb{T}_{h}$ is said to be the uniform mesh. If we set $x_{j}:=g\left(\frac{j}{N+1}\right)$ for $j=1,\ldots,N,N+1$ with a grading function $g$ , the mesh $\mathbb{T}_{h}$ is said to be the graded mesh with respect to $x=0$ ; see BabSur94 . In particular, when $g(y):=y^{\varepsilon}$ ( $\varepsilon\textgreater 0$ ), the mesh is called the radical mesh. To obtain the Lagrange interpolation error estimates, we impose standard assumptions and specify that $\ell,m\in\mathbb{N}_{0}$ and $p,q\in[1,\infty]$ such that

\displaystyle\displaystyle 0\leq m\leq\ell+1:\quad W^{\ell+1,p}(\widehat{T})\hookrightarrow W^{m,q}(\widehat{T}).

(1)

Under these assumptions, the following holds for any $\varphi_{0}\in W^{\ell+1,p}(T_{0}^{i})$ with $\hat{\varphi}=\varphi_{0}\circ\Phi$ :

\displaystyle\displaystyle|\varphi_{0}-{I}_{T_{0}^{i}}^{k}\varphi_{0}|_{W^{m,q}(T_{0}^{i})}

\displaystyle\leq ch_{i}^{\frac{1}{q}-\frac{1}{p}+\ell+1-m}|\varphi_{0}|_{W^{\ell+1,p}(T_{0}^{i})}.

(2)

The proof of this statement is standard; see ErnGue04 . When $p=q$ , it is possible to obtain optimal error estimates even if the scale is different for each element. When $q\textgreater p$ , the order of convergence of the interpolation operator may deteriorate.

We now consider the cases in which $d=2,3$ . Let $\Omega\subset\mathbb{R}^{d}$ be a bounded polyhedral domain. Let $\mathbb{T}_{h}=\{T_{0}\}$ be a simplicial mesh of $\overline{\Omega}$ made up of closed $d$ -simplices, such as

\displaystyle\displaystyle\overline{\Omega}=\bigcup_{T_{0}\in\mathbb{T}_{h}}T_{0}

with $h:=\max_{T_{0}\in\mathbb{T}_{h}}h_{T_{0}}$ , where $h_{T_{0}}:=\mathop{\mathrm{diam}}(T_{0})$ . For simplicity, we assume that $\mathbb{T}_{h}$ is conformal. That is, $\mathbb{T}_{h}$ is a simplicial mesh of $\overline{\Omega}$ without hanging nodes. Let $\widehat{T}\subset\mathbb{R}^{d}$ be the reference element defined in Section 2 and $\Phi$ be the affine mapping defined in Eq. (13). For any $T_{0}\in\mathbb{T}_{h}$ , it holds that $T_{0}=\Phi(\widehat{T})$ . Under the standard assumptions and Eq. (1), the following holds for any $\varphi_{0}\in W^{\ell+1,p}(T_{0})$ with $\hat{\varphi}=\varphi_{0}\circ\Phi$ :

\displaystyle\displaystyle|\varphi_{0}-{I}_{T_{0}}^{k}\varphi_{0}|_{W^{m,q}(T_{0})}

\displaystyle\leq c|T_{0}|^{\frac{1}{q}-\frac{1}{p}}\left(\frac{\alpha_{\max}}{\alpha_{\min}}\right)^{m}\left(\frac{H_{T_{0}}}{h_{T_{0}}}\right)^{m}h_{T_{0}}^{\ell+1-m}|\varphi_{0}|_{W^{\ell+1,p}(T_{0})},

(3)

where $|T_{0}|$ is the measure of $T_{0}$ , the parameters $\alpha_{\max}$ and $\alpha_{\min}$ are defined in Eq. (50), and the parameter $H_{T_{0}}$ is as proposed in a recent paper IshKobTsu ; see Section 2.4 for a definition. The proof of estimate (3) can be found in Section 5. Compared with the one-dimensional case, the quantities $\alpha_{\max}/\alpha_{\min}$ and $H_{T_{0}}/h_{T_{0}}$ negatively affect the order of convergence and do not appear in Eq.(2). The two quantities $\alpha_{\max}/\alpha_{\min}$ and $H_{T_{0}}/h_{T_{0}}$ are considered in Section 7.1. As a mesh condition, the shape-regularity condition is widely used and well known. This condition states that there exists a constant $\gamma\textgreater 0$ such that

\displaystyle\displaystyle\rho_{T_{0}}\geq\gamma h_{T_{0}}\quad\forall\mathbb{T}_{h}\in\{\mathbb{T}_{h}\},\quad\forall T_{0}\in\mathbb{T}_{h},

(4)

where $\rho_{T_{0}}$ is the radius of the inscribed ball of $T_{0}$ . Under this condition, it holds that

\displaystyle\displaystyle|\varphi_{0}-{I}_{T_{0}}^{k}\varphi_{0}|_{W^{m,q}(T_{0})}

\displaystyle\leq c|T_{0}|^{\frac{1}{q}-\frac{1}{p}}h_{T_{0}}^{\ell+1-m}|\varphi_{0}|_{W^{\ell+1,p}(T_{0})};

(5)

see Section 7.1.1. If condition (4) is violated (i.e., the simplex becomes too flat as $h_{T_{0}}\to 0$ ), the quantity

\displaystyle\displaystyle\left(\frac{\alpha_{\max}}{\alpha_{\min}}\right)^{m}\left(\frac{H_{T_{0}}}{h_{T_{0}}}\right)^{m}h_{T_{0}}^{\ell+1-m}

may diverge even when $p=q$ . The effect of the quantity $|T_{0}|^{\frac{1}{q}-\frac{1}{p}}$ on the interpolation error estimates is considered in Section 7.2.

In some cases, it is not necessary for condition (4) to hold to obtain Eq. (5). The shape-regularity condition can be relaxed to the maximum-angle condition, as stated in Eqs. (20) and (21), for both two-dimensional BabAzi76 and three-dimensional cases Kri92 . Anisotropic interpolation theory has also been developed ApeDob92 ; Ape99 ; CheShiZha04 . The idea of Apel et al. is to construct a set of functionals satisfying conditions (54), (55), and (56). The introduction of these functionals makes it possible to remove the quantity $\alpha_{\max}/\alpha_{\min}$ . Under the conditions of the maximum angle and coordinate system, anisotropic interpolation error estimates can then be deduced (e.g., see Ape99 ).

In contrast, this paper proposes anisotropic interpolation error estimates using the new parameter under conditions (54), (55), and (56) and Assumption 1; i.e., we derive the following anisotropic error estimate (Theorem B, in particular, Corollary 1):

	$\displaystyle\|{\varphi}_{0}-I_{{T}_{0}}^{k}{\varphi}_{0}\|_{W^{m,q}({T}_{0})}$
	$\displaystyle\quad\leq c\|T_{0}\|^{\frac{1}{q}-\frac{1}{p}}\left(\frac{H_{T_{0}}}{h_{T_{0}}}\right)^{m}\sum_{\|\gamma\|=\ell-m}\mathscr{H}^{\gamma}\|\partial^{\gamma}{(\varphi_{0}\circ\Phi_{T_{0}})}\|_{W^{m,p}(\Phi_{T_{0}}^{-1}(T_{0}))},$		(6)

where $\Phi_{T_{0}}$ is defined in Eq. (12), $\gamma:=(\gamma_{1},\ldots,\gamma_{d})\in\mathbb{N}_{0}^{d}$ is a multi-index, and $\mathscr{H}^{\gamma}$ is specified in Definition 3. Theorem B applies to interpolations other than the Lagrange interpolation, and the basis for the proof of Theorem B is the scaling argument described in Section 3.

Because the new geometric parameter is used in the interpolation error analysis, the coefficient $c$ used in the error estimation is independent of the geometry of the simplices, and the error estimations obtained may therefore be applied to arbitrary meshes, including very “flat” or anisotropic simplices. Furthermore, we are naturally able to consider the following geometric condition as being sufficient to obtain optimal order estimates (when $p=q$ ): there exists $\gamma_{0}\textgreater 0$ such that

\displaystyle\displaystyle\frac{H_{T_{0}}}{h_{T_{0}}}\leq\gamma_{0}\quad\forall\mathbb{T}_{h}\in\{\mathbb{T}_{h}\},\quad\forall T_{0}\in\mathbb{T}_{h}.

(7)

Condition (7) appears to be simpler than the maximum-angle condition. Furthermore, the quantity $H_{T_{0}}/h_{T_{0}}$ can be easily calculated in the numerical process of finite element methods. Therefore, the new condition may be useful. A recent paper IshKobSuzTsu21 showed that the new condition is satisfied if and only if the maximum-angle condition holds. We expect the new mesh condition to become an alternative to the maximum-angle condition.

Furthermore, under Assumption 1, component-wise inverse inequalities can be deduced as (see Section 7.3):

\displaystyle\displaystyle\|\partial^{\gamma}\varphi_{h}\|_{L^{q}(T)}\leq C^{IVC}|T|^{\frac{1}{q}-\frac{1}{p}}\mathscr{H}^{-\gamma}\|{\varphi}_{h}\|_{L^{p}({T})}.

In a previous paper IshKobTsu , the present authors developed new interpolation error estimations in a general framework and derived Raviart–Thomas interpolations on $d$ -simplices. However, the statement of Theorem 2 in IshKobTsu includes a mistake. That is, under standard assumptions, the quantity $\alpha_{\max}/\alpha_{\min}$ cannot be removed. We need to modify the statement of this theorem to correct this error. The current paper presents Theorems A (see Section 5) and B (see Section 6), which replace Theorem 2 of IshKobTsu . In Section 4, we explain the inaccuracies in the proof of Theorem 2 in IshKobTsu and describe how the results can be recovered using our Theorems A and B. Furthermore, the Babuška and Aziz technique is generally not applicable on anisotropic meshes in the proof of Theorem 3 in IshKobTsu . Details will be discussed in a coming paper Ish21 .

When there is no ambiguity, we use the notation and definitions given in IshKobTsu . Throughout this paper, $c$ denotes a constant independent of $h$ (defined later), unless specified otherwise. These values may change in each context. $\mathbb{R}_{+}$ is the set of positive real numbers.

2 Strategy for constructing anisotropic interpolation theory

In standard interpolation theory, one introduces an affine mapping that connects the reference element to the mesh element. However, on anisotropic meshes, the interpolation errors may be overestimated. Therefore, our strategy is to divide the transformation into three affine mappings.

2.1 Standard positions of simplices

We recall (IshKobTsu, , Section 3). Let us first define a diagonal matrix $\widehat{A}^{(d)}$ as

\displaystyle\displaystyle\widehat{A}^{(d)}:=\mathop{\mathrm{diag}}(\alpha_{1},\ldots,\alpha_{d}),\quad\alpha_{i}\in\mathbb{R}_{+}\quad\forall i.

(8)

2.1.1 Two-dimensional case

Let $\widehat{T}\subset\mathbb{R}^{2}$ be the reference triangle with vertices $\hat{x}_{1}:=(0,0)^{T}$ , $\hat{x}_{2}:=(1,0)^{T}$ , and $\hat{x}_{3}:=(0,1)^{T}$ .

Let $\widetilde{\mathfrak{T}}^{(2)}$ be the family of triangles

\displaystyle\displaystyle\widetilde{T}=\widehat{A}^{(2)}(\widehat{T})

with vertices $\tilde{x}_{1}:=(0,0)^{T}$ , $\tilde{x}_{2}:=(\alpha_{1},0)^{T}$ , and $\tilde{x}_{3}:=(0,\alpha_{2})^{T}$ .

We next define the regular matrices $\widetilde{A}\in\mathbb{R}^{2\times 2}$ by

\displaystyle\displaystyle\widetilde{A}:=\begin{pmatrix}1&s\\ 0&t\\ \end{pmatrix},

(9)

with parameters

\displaystyle\displaystyle s^{2}+t^{2}=1,\quad t\textgreater 0.

For $\widetilde{T}\in\widetilde{\mathfrak{T}}^{(2)}$ , let $\mathfrak{T}^{(2)}$ be the family of triangles

\displaystyle\displaystyle T

\displaystyle=\widetilde{A}(\widetilde{T})

with vertices $x_{1}:=(0,0)^{T},\ x_{2}:=(\alpha_{1},0)^{T},\ x_{3}:=(\alpha_{2}s,\alpha_{2}t)^{T}$ . We then have that $\alpha_{1}=|x_{1}-x_{2}|\textgreater 0$ and $\alpha_{2}=|x_{1}-x_{3}|\textgreater 0$ .

2.1.2 Three-dimensional case

Let $\widehat{T}_{1}$ and $\widehat{T}_{2}$ be reference tetrahedra with the following vertices:

(i): $\widehat{T}_{1}$ has the vertices $\hat{x}_{1}:=(0,0,0)^{T}$ , $\hat{x}_{2}:=(1,0,0)^{T}$ , $\hat{x}_{3}:=(0,1,0)^{T}$ , $\hat{x}_{4}:=(0,0,1)^{T}$ ;
(ii): $\widehat{T}_{2}$ has the vertices $\hat{x}_{1}:=(0,0,0)^{T}$ , $\hat{x}_{2}:=(1,0,0)^{T}$ , $\hat{x}_{3}:=(1,1,0)^{T}$ , $\hat{x}_{4}:=(0,0,1)^{T}$ .

Let $\widetilde{\mathfrak{T}}_{i}^{(3)}$ , $i=1,2$ , be the family of triangles

\displaystyle\displaystyle\widetilde{T}_{i}=\widehat{A}^{(3)}(\widehat{T}_{i}),\quad i=1,2,

with vertices

(i): $\tilde{x}_{1}:=(0,0,0)^{T}$ , $\tilde{x}_{2}:=(\alpha_{1},0,0)^{T}$ , $\tilde{x}_{3}:=(0,\alpha_{2},0)^{T}$ , and $\tilde{x}_{4}:=(0,0,\alpha_{3})^{T}$ ;
(ii): $\tilde{x}_{1}:=(0,0,0)^{T}$ , $\tilde{x}_{2}:=(\alpha_{1},0,0)^{T}$ , $\tilde{x}_{3}:=(\alpha_{1},\alpha_{2},0)^{T}$ , and $\tilde{x}_{4}:=(0,0,\alpha_{3})^{T}$ .

We next define the regular matrices $\widetilde{A}_{1},\widetilde{A}_{2}\in\mathbb{R}^{3\times 3}$ by

\displaystyle\displaystyle\widetilde{A}_{1}:=\begin{pmatrix}1&s_{1}&s_{21}\\ 0&t_{1}&s_{22}\\ 0&0&t_{2}\\ \end{pmatrix},\ \widetilde{A}_{2}:=\begin{pmatrix}1&-s_{1}&s_{21}\\ 0&t_{1}&s_{22}\\ 0&0&t_{2}\\ \end{pmatrix}

(10)

with parameters

\displaystyle\displaystyle\begin{cases}s_{1}^{2}+t_{1}^{2}=1,\ s_{1}\textgreater 0,\ t_{1}\textgreater 0,\ \alpha_{2}s_{1}\leq\alpha_{1}/2,\\ s_{21}^{2}+s_{22}^{2}+t_{2}^{2}=1,\ t_{2}\textgreater 0,\ \alpha_{3}s_{21}\leq\alpha_{1}/2.\end{cases}

For $\widetilde{T}_{i}\in\widetilde{\mathfrak{T}}_{i}^{(3)}$ , $i=1,2$ , let $\mathfrak{T}_{i}^{(3)}$ , $i=1,2$ , be the family of tetrahedra

\displaystyle\displaystyle T_{i}

\displaystyle=\widetilde{A}_{i}(\widetilde{T}_{i}),\quad i=1,2,

with vertices

	$\displaystyle x_{1}:=(0,0,0)^{T},\ x_{2}:=(\alpha_{1},0,0)^{T},\ x_{4}:=(\alpha_{3}s_{21},\alpha_{3}s_{22},\alpha_{3}t_{2})^{T},$
	$\displaystyle\begin{cases}x_{3}:=(\alpha_{2}s_{1},\alpha_{2}t_{1},0)^{T}\quad\text{for case (i)},\\ x_{3}:=(\alpha_{1}-\alpha_{2}s_{1},\alpha_{2}t_{1},0)^{T}\quad\text{for case (ii)}.\end{cases}$

We then have $\alpha_{1}=|x_{1}-x_{2}|\textgreater 0$ , $\alpha_{3}=|x_{1}-x_{4}|\textgreater 0$ , and

\displaystyle\displaystyle\alpha_{2}=\begin{cases}|x_{1}-x_{3}|\textgreater 0\quad\text{for case (i)},\\ |x_{2}-x_{3}|\textgreater 0\quad\text{for case (ii)}.\end{cases}

In the following, we impose conditions for $T\in\mathfrak{T}^{(2)}$ in the two-dimensional case and $T\in\mathfrak{T}_{1}^{(3)}\cup\mathfrak{T}_{2}^{(3)}=:\mathfrak{T}^{(3)}$ in the three-dimensional case.

Condition 1 (Case in which $d=2$ )

Let $T\in\mathfrak{T}^{(2)}$ with the vertices $x_{i}$ ( $i=1,\ldots,3$ ) introduced in Section 2.1.1. We assume that $\overline{x_{2}x_{3}}$ is the longest edge of $T$ ; i.e., $h_{T}:=|x_{2}-x_{3}|$ . Recall that $\alpha_{1}=|x_{1}-x_{2}|$ and $\alpha_{2}=|x_{1}-x_{3}|$ . We then assume that $\alpha_{2}\leq\alpha_{1}$ . Note that $\alpha_{1}=\mathcal{O}(h_{T})$ .

Condition 2 (Case in which $d=3$ )

Let $T\in\mathfrak{T}^{(3)}$ with the vertices $x_{i}$ ( $i=1,\ldots,4$ ) introduced in Section 2.1.2. Let $L_{i}$ ( $1\leq i\leq 6$ ) be the edges of $T$ . We denote by $L_{\min}$ the edge of $T$ that has the minimum length; i.e., $|L_{\min}|=\min_{1\leq i\leq 6}|L_{i}|$ . We set $\alpha_{2}:=|L_{\min}|$ and assume that

\displaystyle\text{the end points of $L_{\min}$ are either $\{x_{1},x_{3}\}$ or $\{x_{2},x_{3}\}$}.

Among the four edges that share an end point with $L_{\min}$ , we take the longest edge $L^{({\min})}_{\max}$ . Let $x_{1}$ and $x_{2}$ be the end points of edge $L^{({\min})}_{\max}$ . Thus, we have that

\displaystyle\displaystyle\alpha_{1}=|L^{(\min)}_{\max}|=|x_{1}-x_{2}|.

Consider cutting $\mathbb{R}^{3}$ with the plane that contains the midpoint of edge $L^{(\min)}_{\max}$ and is perpendicular to the vector $x_{1}-x_{2}$ . We then have two cases:

(Type i): $x_{3}$ and $x_{4}$ belong to the same half-space;
(Type ii): $x_{3}$ and $x_{4}$ belong to different half-spaces.

In each respective case, we set

(Type i): $x_{1}$ and $x_{3}$ as the end points of $L_{\min}$ , that is, $\alpha_{2}=|x_{1}-x_{3}|$ ;
(Type ii): $x_{2}$ and $x_{3}$ as the end points of $L_{\min}$ , that is, $\alpha_{2}=|x_{2}-x_{3}|$ .

Finally, we set $\alpha_{3}=|x_{1}-x_{4}|$ . Note that we implicitly assume that $x_{1}$ and $x_{4}$ belong to the same half-space. In addition, note that $\alpha_{1}=\mathcal{O}(h_{T})$ .

Remark 1

Let $\mathbb{T}_{h}$ be a conformal mesh. We assume that any simplex $T_{0}\in\mathbb{T}_{h}$ is transformed into $T_{1}\in\mathfrak{T}^{(2)}$ such that Condition 1 is satisfied (in the two-dimensional case) or $T_{i}\in\mathfrak{T}_{i}^{(3)}$ , $i=1,2$ , such that Condition 2 is satisfied (in the three-dimensional case) through appropriate rotation, translation, and mirror imaging. Note that none of the lengths of the edges of a simplex or the measure of the simplex is changed by the transformation.

Assumption 1

In anisotropic interpolation error analysis, we may impose the following geometric conditions for the simplex $T$ :

1.

If $d=2$ , there are no additional conditions;
2.

If $d=3$ , there exists a positive constant $M$ , independent of $h_{T}$ , such that $|s_{22}|\leq M\frac{\alpha_{2}t_{1}}{\alpha_{3}}$ . Note that if $s_{22}\neq 0$ , this condition means that the order with respect to $h_{T}$ of $\alpha_{3}$ coincides with the order of $\alpha_{2}$ , whereas if $s_{22}=0$ , the order of $\alpha_{3}$ may be different from that of $\alpha_{2}$ .

2.2 Affine mappings

In our strategy, we adopt the following affine mappings.

Definition 1 (Affine mappings)

Let $\widetilde{T},\widehat{T}\subset\mathbb{R}^{d}$ be the simplices defined in Sections 2.1.1 and 2.1.2. That is,

\displaystyle\displaystyle\widetilde{T}=\widehat{\Phi}(\widehat{T}),\quad{T}=\widetilde{\Phi}(\widetilde{T})\quad\text{with}\quad\tilde{x}:=\widehat{\Phi}(\hat{x}):=\widehat{{A}}^{(d)}\hat{x},\quad x:=\widetilde{\Phi}(\tilde{x}):=\widetilde{{A}}\tilde{x}.

We then define an affine mapping $\Phi_{T}:\widehat{T}\to T$ as

\displaystyle\displaystyle\Phi_{T}:=\widetilde{\Phi}\circ\widehat{\Phi}:\widehat{T}\to T,\ {x}:={\Phi}_{T}(\hat{x}):={{A}}_{T}\hat{x},\quad{A}_{T}:=\widetilde{{A}}\widehat{{A}}^{(d)}.

(11)

Furthermore, let $\Phi_{T_{0}}$ be an affine mapping defined as

\displaystyle\displaystyle\Phi_{T_{0}}:T\ni x\mapsto{A}_{T_{0}}x+b_{T_{0}}\in T_{0},

(12)

where $b_{T_{0}}\in\mathbb{R}^{d}$ and ${A}_{T_{0}}\in O(d)$ is a rotation and mirror imaging matrix. We then define an affine mapping $\Phi:\widehat{T}\to T_{0}$ as

\displaystyle\displaystyle\Phi:={\Phi}_{T_{0}}\circ{\Phi}_{T}:\widehat{T}\to T_{0},\ x^{(0)}:=\Phi(\hat{x})=({\Phi}_{T_{0}}\circ{\Phi}_{T})(\hat{x})={A}\hat{x}+b_{T_{0}},

(13)

where ${A}:={A}_{T_{0}}{A}_{T}$ .

2.3 Finite element generation

We follow the procedure described in (ErnGue04, , Section 1.4.1 and 1.2.1); see also (IshKobTsu, , Section 3.5).

For the reference element $\widehat{T}$ defined in Sections 2.1.1 and 2.1.2, let $\{\widehat{T},\widehat{{P}},\widehat{\Sigma}\}$ be a fixed reference finite element, where $\widehat{{P}}$ is a vector space of functions $\hat{p}:\widehat{T}\to\mathbb{R}^{n}$ for some positive integer $n$ (typically $n=1$ or $n=d$ ) and $\widehat{\Sigma}$ is a set of $n_{0}$ linear forms $\{\hat{\chi}_{1},\ldots,\hat{\chi}_{n_{0}}\}$ such that

\displaystyle\displaystyle\widehat{{P}}\ni\hat{p}\mapsto(\hat{\chi}_{1}(\hat{p}),\ldots,\hat{\chi}_{n_{0}}(\hat{p}))^{T}\in\mathbb{R}^{n_{0}}

is bijective; i.e., $\widehat{\Sigma}$ is a basis for $\mathcal{L}(\widehat{P};\mathbb{R})$ . Further, we denote by $\{\hat{\theta}_{1},\ldots,\hat{\theta}_{n_{0}}\}$ in $\widehat{{P}}$ the local ( $\mathbb{R}^{n}$ -valued) shape functions such that

\displaystyle\displaystyle\hat{\chi}_{i}(\hat{\theta}_{j})=\delta_{ij},\quad 1\leq i,j\leq n_{0}.

Let $V(\widehat{T})$ be a normed vector space of functions $\hat{\varphi}:\widehat{T}\to\mathbb{R}^{n}$ such that $\widehat{P}\subset V(\widehat{T})$ and the linear forms $\{\hat{\chi}_{1},\ldots,\hat{\chi}_{n_{0}}\}$ can be extended to $V(\widehat{T})^{\prime}$ . The local interpolation operator ${I}_{\widehat{T}}$ is then defined by

\displaystyle\displaystyle{I}_{\widehat{T}}:V(\widehat{T})\ni\hat{\varphi}\mapsto\sum_{i=1}^{n_{0}}\hat{\chi}_{i}(\hat{\varphi})\hat{\theta}_{i}\in\widehat{{P}}.

(14)

It is obvious that

	$\displaystyle\hat{\chi}_{i}({I}_{\widehat{T}}\hat{\varphi})=\hat{\chi}_{i}(\hat{\varphi})\quad\forall\hat{\varphi}\in V(\widehat{T}),\quad i=1,\ldots,n_{0},$		(15)
	$\displaystyle I_{\widehat{T}}\hat{p}=\hat{p}\quad\forall\hat{p}\in\widehat{P}.$		(16)

Let ${\Phi}$ be the affine mapping defined in Eq. (13). For $T_{0}={\Phi}(\widehat{T})$ , we first define a Banach space $V(T_{0})$ of $\mathbb{R}^{n}$ -valued functions that is the counterpart of $V(\widehat{T})$ and define a linear bijection mapping by

\displaystyle\displaystyle\psi:=\psi_{\widehat{T}}\circ\psi_{\widetilde{T}}\circ\psi_{T}:V(T_{0})\ni\varphi\mapsto\hat{\varphi}:=\psi(\varphi):=\varphi\circ\Phi\in V(\widehat{T})

with the three linear bijection mappings

	$\displaystyle\psi_{{T}}:V({T}_{0})\ni\varphi_{0}\mapsto{\varphi}:=\psi_{{T}}(\varphi_{0}):=\varphi_{0}\circ{\Phi}_{T_{0}}\in V({T}),$
	$\displaystyle\psi_{\widetilde{T}}:V({T})\ni\varphi\mapsto\tilde{\varphi}:=\psi_{\widetilde{T}}(\varphi):=\varphi\circ\widetilde{\Phi}\in V(\widetilde{T}),$
	$\displaystyle\psi_{\widehat{T}}:V(\widetilde{T})\ni\tilde{\varphi}\mapsto\hat{\varphi}:=\psi_{\widehat{T}}(\tilde{\varphi}):=\tilde{\varphi}\circ\widehat{\Phi}\in V(\widehat{T}).$

The triple $\{\widetilde{T},\widetilde{P},\widetilde{\Sigma}\}$ is defined as

\displaystyle\displaystyle\begin{cases}\displaystyle\widetilde{T}=\widehat{\Phi}(\widehat{T});\\ \displaystyle\widetilde{P}=\{\psi_{\widehat{T}}^{-1}(\hat{p});\ \hat{p}\in\widehat{{P}}\};\\ \displaystyle\widetilde{\Sigma}=\{\{\tilde{\chi}_{i}\}_{1\leq i\leq n_{0}};\ \tilde{\chi}_{i}=\hat{\chi}_{i}(\psi_{\widehat{T}}(\tilde{p})),\forall\tilde{p}\in\widetilde{P},\hat{\chi}_{i}\in\widehat{\Sigma}\}.\end{cases}

The triples $\{{T},{P},{\Sigma}\}$ and $\{{T}_{0},{P}_{0},{\Sigma}_{0}\}$ are similarly defined. These triples are finite elements and the local shape functions are $\tilde{\theta}_{i}=\psi_{\widehat{T}}^{-1}(\hat{\theta}_{i})$ , $\theta_{i}=\psi_{\widetilde{T}}^{-1}(\tilde{\theta}_{i})$ , and $\theta_{0,i}:=\psi_{T}^{-1}(\theta_{i})$ for $1\leq i\leq n_{0}$ , and the associated local interpolation operators are respectively defined by

$\displaystyle\displaystyle{I}_{\widetilde{T}}:V(\widetilde{T})\ni\tilde{\varphi}\mapsto{I}_{\widetilde{T}}\tilde{\varphi}$	$\displaystyle:=\sum_{i=1}^{n_{0}}\tilde{\chi}_{i}(\tilde{\varphi})\tilde{\theta}_{i}\in\widetilde{P},$	(17)
$\displaystyle{I}_{T}:V(T)\ni\varphi\mapsto{I}_{T}\varphi$	$\displaystyle:=\sum_{i=1}^{n_{0}}\chi_{i}(\varphi)\theta_{i}\in{P},$	(18)
$\displaystyle{I}_{T_{0}}:V(T_{0})\ni\varphi_{0}\mapsto{I}_{T_{0}}\varphi_{0}$	$\displaystyle:=\sum_{i=1}^{n_{0}}\chi_{0,i}(\varphi_{0})\theta_{i}\in{P}_{0}.$	(19)

Proposition 1

The diagrams

commute.

Proof

See, for example, (ErnGue04, , Proposition 1.62). ∎

2.4 New parameters

In a previous paper IshKobTsu , we proposed two geometric parameters,

Definition 2

The parameter $H_{T}$ is defined as

\displaystyle\displaystyle H_{T}:=\frac{\prod_{i=1}^{d}\alpha_{i}}{|T|}h_{T},

and the parameter $H_{T_{0}}$ is defined as

\displaystyle\displaystyle H_{T_{0}}:=\frac{h_{T_{0}}^{2}}{|T_{0}|}\min_{1\leq i\leq 3}|L_{i}|\quad\text{if $d=2$},\quad H_{T_{0}}:=\frac{h_{T_{0}}^{2}}{|T_{0}|}\min_{1\leq i,j\leq 6,i\neq j}|L_{i}||L_{j}|\quad\text{if $d=3$}

where $L_{i}$ denotes the edges of the simplex $T_{0}\subset\mathbb{R}^{d}$ .

The following lemma shows the equivalence between $H_{T}$ and $H_{T_{0}}$ .

Lemma 1

It holds that

\displaystyle\displaystyle\frac{1}{2}H_{T_{0}}\textless H_{T}\textless 2H_{T_{0}}.

Furthermore, in the two-dimensional case, $H_{T_{0}}$ is equivalent to the circumradius $R_{2}$ of $T_{0}$ .

Proof

The proof can be found in (IshKobTsu, , Lemma 3). ∎

Remark 2

We set

\displaystyle\displaystyle H(h):=\max_{T_{0}\in\mathbb{T}_{h}}H_{T_{0}}.

As we stated in the Introduction, if the maximum-angle condition is violated, the parameter $H(h)$ may diverge as $h\to 0$ on anisotropic meshes. Therefore, imposing the maximum-angle condition for mesh partitions guarantees the convergence of finite element methods Apeet21 . Reference BabAzi76 studied cases in which the finite element solution may not converge to the exact solution.

We now state the following theorem concerning the new condition.

Theorem 1

Condition (7) holds if and only if there exist $0\textless\gamma_{1},\gamma_{2}\textless\pi$ such that

\displaystyle\displaystyle d=2:\quad\theta_{T_{0},\max}\leq\gamma_{1}\quad\forall\mathbb{T}_{h}\in\{\mathbb{T}_{h}\},\quad\forall T_{0}\in\mathbb{T}_{h},

(20)

where $\theta_{T_{0},\max}$ is the maximum angle of $T_{0}$ , and

\displaystyle\displaystyle d=3:\quad\theta_{T_{0},\max}\leq\gamma_{2},\quad\psi_{T_{0},\max}\leq\gamma_{2}\quad\forall\mathbb{T}_{h}\in\{\mathbb{T}_{h}\},\quad\forall T_{0}\in\mathbb{T}_{h},

(21)

where $\theta_{T_{0},\max}$ is the maximum angle of all triangular faces of the tetrahedron $T_{0}$ and $\psi_{T_{0},\max}$ is the maximum dihedral angle of $T_{0}$ . Conditions (20) and (21) together constitute the maximum-angle condition.

Proof

In the case of $d=2$ , we use the previous result presented in Kri91 ; i.e., there exists a constant $\gamma_{3}\textgreater 0$ such that

\displaystyle\displaystyle\frac{R_{2}}{h_{T_{0}}}\leq\gamma_{3}\quad\forall\mathbb{T}_{h}\in\{\mathbb{T}_{h}\},\quad\forall T_{0}\in\mathbb{T}_{h},

if and only if condition (20) is satisfied. Combining this result with $H_{T_{0}}$ being equivalent to the circumradius $R_{2}$ of $T_{0}$ , we have the desired conclusion. In the case of $d=3$ , the proof can be found in a recent paper IshKobSuzTsu21 . ∎

Lemma 2

It holds that


$\displaystyle\displaystyle\\|\widehat{{A}}^{(d)}\\|_{2}$	$\displaystyle\leq h_{T},\quad\\|\widehat{{A}}^{(d)}\\|_{2}\\|(\widehat{{A}}^{(d)})^{-1}\\|_{2}=\frac{\max\{\alpha_{1},\cdots,\alpha_{d}\}}{\min\{\alpha_{1},\cdots,\alpha_{d}\}},$	(22a)
$\displaystyle\\|\widetilde{{A}}\\|_{2}$	$\displaystyle\leq\begin{cases}\sqrt{2}\quad\text{if $d=2$},\\ 2\quad\text{if $d=3$},\end{cases}\quad\\|\widetilde{{A}}\\|_{2}\\|\widetilde{{A}}^{-1}\\|_{2}\leq\begin{cases}\frac{\alpha_{1}\alpha_{2}}{\|T\|}=\frac{H_{T}}{h_{T}}\quad\text{if $d=2$},\\ \frac{2}{3}\frac{\alpha_{1}\alpha_{2}\alpha_{3}}{\|T\|}=\frac{2}{3}\frac{H_{T}}{h_{T}}\quad\text{if $d=3$},\end{cases}$	(22b)
$\displaystyle\\|A_{T_{0}}\\|_{2}$	$\displaystyle=1,\quad\\|A_{T_{0}}^{-1}\\|_{2}=1.$	(22c)

Furthermore, we have

\displaystyle\displaystyle|\det({A}_{T})|=|\det(\widetilde{{A}})||\det(\widehat{{A}}^{(d)})|=d!|T|,\quad|\det({A}_{T_{0}})|=1.

(23)

Proof

The proof of (22b) can be found in (IshKobTsu, , (4.4), (4.5), (4.6), and (4.7)). The inequality (22a) is easily proved. Because $A_{T_{0}}\in O(d)$ , one easily finds that $A_{T_{0}}^{-1}\in O(d)$ and recovers Eq. (22c). The proof of equality (23) is standard. ∎

For matrix $A\in\mathbb{R}^{d\times d}$ , we denote by $[{A}]_{ij}$ the $(i,j)$ -component of $A$ . We set $\|A\|_{\max}:=\max_{1\leq i,j\leq d}|[{A}]_{ij}|$ . Furthermore, we use the inequality

\displaystyle\displaystyle\|{A}\|_{\max}\leq\|{A}\|_{2}.

(24)

3 Scaling argument

This section gives estimates related to a scaling argument corresponding to (ErnGue04, , Lemma 1.101). The estimates play major roles in our analysis. Furthermore, we use the following inequality (see (ErnGue04, , Exercise 1.20)). Let $0\textless r\leq s$ and $a_{i}\geq 0$ , $i=1,2,\ldots,n$ ( $n\in\mathbb{N}$ ), be real numbers. Then, we have that

\displaystyle\displaystyle\left(\sum_{i=1}^{n}a_{i}^{s}\right)^{1/s}\leq\left(\sum_{i=1}^{n}a_{i}^{r}\right)^{1/r}.

(25)

Lemma 3

Let $s\geq 0$ and $1\leq p\leq\infty$ . There exist positive constants $c_{1}$ and $c_{2}$ such that, for all $T_{0}\in\mathbb{T}_{h}$ and $\varphi_{0}\in W^{s,p}(T_{0})$ ,

\displaystyle\displaystyle c_{1}|\varphi_{0}|_{W^{s,p}(T_{0})}\leq|\varphi|_{W^{s,p}(T)}

\displaystyle\leq c_{2}|\varphi_{0}|_{W^{s,p}(T_{0})}

(26)

with $\varphi=\varphi_{0}\circ{\Phi}_{T_{0}}$ .

Proof

The following inequalities can be found in (ErnGue04, , Lemma 1.101). There exists a positive constant $c$ such that, for all $T_{0}\in\mathbb{T}_{h}$ and $\varphi_{0}\in W^{s,p}(T_{0})$ ,

	$\displaystyle\displaystyle\|\varphi\|_{W^{s,p}(T)}$	$\displaystyle\leq c\\|{A}_{T_{0}}\\|_{2}^{s}\|\det({A}_{T_{0}})\|^{-\frac{1}{p}}\|\varphi_{0}\|_{W^{s,p}(T_{0})},$		(27)
	$\displaystyle\|\varphi_{0}\|_{W^{s,p}(T_{0})}$	$\displaystyle\leq c\\|{A}_{T_{0}}^{-1}\\|_{2}^{s}\|\det({A}_{T_{0}})\|^{\frac{1}{p}}\|\varphi\|_{W^{s,p}(T)}$		(28)

with $\varphi=\varphi_{0}\circ{\Phi}_{T_{0}}$ . Using Eqs. (27) and (28) together with Eqs. (22c) and (23) yields Eq. (26). ∎

Lemma 4

Let $m\in\mathbb{N}_{0}$ and $p\in[0,\infty)$ . Let $\beta:=(\beta_{1},\ldots,\beta_{d})\in\mathbb{N}_{0}^{d}$ be a multi-index with $|\beta|=m$ . Then, for any $\hat{\varphi}\in W^{m,p}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ and ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ , it holds that

\displaystyle\displaystyle|{\varphi}|_{W^{m,p}({T})}

\displaystyle\leq C_{1}^{SA}|\det(A_{T})|^{\frac{1}{p}}\|\widetilde{A}^{-1}\|^{m}_{2}\left(\sum_{|\beta|=m}(\alpha^{-\beta})^{p}\|\partial^{\beta}\hat{\varphi}\|^{p}_{L^{p}(\widehat{T})}\right)^{1/p},

(29)

where $C_{1}^{SA}$ is a constant that is independent of $T$ and $\widetilde{T}$ . When $p=\infty$ , for any $\hat{\varphi}\in W^{m,\infty}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ and ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ , it holds that

\displaystyle\displaystyle|{\varphi}|_{W^{m,\infty}({T})}

\displaystyle\leq C_{1}^{SA,\infty}\|\widetilde{A}^{-1}\|^{m}_{2}\max_{|\beta|=m}\left(\alpha^{-\beta}\|\partial^{\beta}\hat{\varphi}\|_{L^{\infty}(\widehat{T})}\right),

(30)

where $C_{1}^{SA,\infty}$ is a constant that is independent of $T$ and $\widetilde{T}$ .

Proof

Let $p\in[1,\infty)$ . Because the space $\mathcal{C}^{m}(\widehat{T})$ is dense in the space ${W}^{m,p}(\widehat{T})$ , we show that Eq. (29) holds for $\hat{\varphi}\in\mathcal{C}^{m}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ and ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ . From $\hat{x}_{j}=\alpha_{j}^{-1}\tilde{x}_{j}$ , we have that, for any multi-index $\beta$ ,

\displaystyle\displaystyle\partial^{\beta}\tilde{\varphi}

\displaystyle=\alpha_{1}^{-\beta_{1}}\cdots\alpha_{d}^{-\beta_{d}}\partial^{\beta}\hat{\varphi}=\alpha^{-\beta}\partial^{\beta}\hat{\varphi}.

(31)

Through a change of variable, we obtain

\displaystyle\displaystyle|\tilde{\varphi}|_{W^{m,p}(\widetilde{T})}^{p}

\displaystyle=\sum_{|\beta|=m}\|\partial^{\beta}\tilde{\varphi}\|^{p}_{L^{p}(\widetilde{T})}=|\det(\widehat{A}^{(d)})|\sum_{|\beta|=m}(\alpha^{-\beta})^{p}\|\partial^{\beta}\hat{\varphi}\|^{p}_{L^{p}(\widehat{T})}.

(32)

From the standard estimate in (ErnGue04, , Lemma 1.101), we have

\displaystyle\displaystyle|{\varphi}|_{W^{m,p}({T})}

\displaystyle\leq C_{1}^{SA}|\det(\widetilde{A})|^{\frac{1}{p}}\|\widetilde{A}^{-1}\|^{m}_{2}|\tilde{\varphi}|_{W^{m,p}(\widetilde{T})}.

(33)

Inequality (29) follows from Eqs. (32) and (33) with Eq. (23).

We consider the case in which $p=\infty$ . A function $\hat{\varphi}\in W^{m,\infty}(\widehat{T})$ belongs to the space $W^{m,p}(\widehat{T})$ for any $p\in[1,\infty)$ . Therefore, it holds that $\tilde{\varphi}\in W^{m,p}(\widetilde{T})$ for any $p\in[1,\infty)$ and, from Eq. (25), we obtain

$\displaystyle\displaystyle\\|\partial^{\gamma}\tilde{\varphi}\\|_{L^{p}(\widetilde{T})}$	$\displaystyle\leq\|\tilde{\varphi}\|_{W^{\|\gamma\|,p}(\widetilde{T})}$
	$\displaystyle=\|\det(\widehat{A}^{(d)})\|^{\frac{1}{p}}\left(\sum_{\|\beta\|=\|\gamma\|}(\alpha^{-\beta})^{p}\\|\partial^{\beta}\hat{\varphi}\\|^{p}_{L^{p}(\widehat{T})}\right)^{1/p}$
	$\displaystyle\leq\left(\sup_{1\leq p}\|\det(\widehat{A}^{(d)})\|^{\frac{1}{p}}\right)\sum_{\|\beta\|=\|\gamma\|}\alpha^{-\beta}\\|\partial^{\beta}\hat{\varphi}\\|_{L^{p}(\widehat{T})}$
	$\displaystyle\leq c\left(\sup_{1\leq p}\|\det(\widehat{A}^{(d)})\|^{\frac{1}{p}}\right)\sum_{\|\beta\|=\|\gamma\|}\alpha^{-\beta}\\|\partial^{\beta}\hat{\varphi}\\|_{L^{\infty}(\widehat{T})}\textless\infty$	(34)

for the multi-index $\gamma\in\mathbb{N}_{0}^{d}$ with $|\gamma|\leq m$ . This implies that the function $\partial^{\gamma}\tilde{\varphi}$ is in the space $L^{\infty}(\widetilde{T})$ for each $|\gamma|\leq m$ . Therefore, we have that $\tilde{\varphi}\in W^{m,\infty}(\widetilde{T})$ . Taking the limit $p\to\infty$ in Eq. (34) and using $\lim_{p\to\infty}\|\cdot\|_{L^{p}(\widetilde{T})}=\|\cdot\|_{L^{\infty}(\widetilde{T})}$ , we have

\displaystyle\displaystyle|\tilde{\varphi}|_{W^{m,\infty}(\widetilde{T})}\leq c\max_{|\beta|=m}\left(\alpha^{-\beta}\|\partial^{\beta}\hat{\varphi}\|_{L^{\infty}(\widehat{T})}\right).

(35)

From the standard estimate in (ErnGue04, , Lemma 1.101), we have

\displaystyle\displaystyle|{\varphi}|_{W^{m,\infty}({T})}

\displaystyle\leq c\|\widetilde{A}^{-1}\|^{m}_{2}|\tilde{\varphi}|_{W^{m,\infty}(\widetilde{T})}.

(36)

Inequality (30) follows from Eqs. (35) and (36). ∎

We now introduce the following new notation.

Definition 3

We define a parameter $\mathscr{H}_{i}$ , $i=1,\ldots,d$ , as

\displaystyle\displaystyle\begin{cases}\mathscr{H}_{1}:=\alpha_{1},\quad\mathscr{H}_{2}:=\alpha_{2}t\quad\text{if $d=2$},\\ \mathscr{H}_{1}:=\alpha_{1},\quad\mathscr{H}_{2}:=\alpha_{2}t_{1},\quad\mathscr{H}_{3}:=\alpha_{3}t_{2}\quad\text{if $d=3$}.\end{cases}

For a multi-index $\beta=(\beta_{1},\ldots,\beta_{d})\in\mathbb{N}_{0}^{d}$ , we use the notation

\displaystyle\displaystyle\mathscr{H}^{\beta}:=\mathscr{H}_{1}^{\beta_{1}}\cdots\mathscr{H}_{d}^{\beta_{d}},\quad\mathscr{H}^{-\beta}:=\mathscr{H}_{1}^{-\beta_{1}}\cdots\mathscr{H}_{d}^{-\beta_{d}}.

We also define $\alpha^{\beta}:=\alpha_{1}^{\beta_{1}}\cdots\alpha_{d}^{\beta_{d}}$ and $\alpha^{-\beta}:=\alpha_{1}^{-\beta_{1}}\cdots\alpha_{d}^{-\beta_{d}}$ .

Definition 4

We define vectors $r_{n}\in\mathbb{R}^{d}$ , $n=1,\ldots,d$ , as follows. If $d=2$ ,

\displaystyle\displaystyle r_{1}:=(1,0)^{T},\quad r_{2}:=(s,t)^{T},

and if $d=3$ ,

	$\displaystyle r_{1}:=(1,0,0)^{T},\quad r_{3}:=(s_{21},s_{22},t_{2})^{T},$
	$\displaystyle\begin{cases}r_{2}:=(s_{1},t_{1},0)^{T}\quad\text{for case (i)},\\ r_{2}:=(-s_{1},t_{1},0)^{T}\quad\text{for case (ii)}.\end{cases}$

Furthermore, we define a directional derivative as

\displaystyle\displaystyle\frac{\partial}{\partial{r_{i}}^{(0)}}:=(A_{T_{0}}r_{i})\cdot\nabla_{x^{(0)}}=\sum_{j_{0}=1}^{d}(A_{T_{0}}r_{i})_{j_{0}}\frac{\partial}{\partial x_{j_{0}}^{(0)}},\quad i\in\{1:d\},

where ${A}_{T_{0}}\in O(d)$ is the orthogonal matrix defined in Eq. (12). For a multi-index $\beta=(\beta_{1},\ldots,\beta_{d})\in\mathbb{N}_{0}^{d}$ , we use the notation

\displaystyle\displaystyle\partial^{\beta}_{r^{(0)}}:=\frac{\partial^{|\beta|}}{(\partial r_{1}^{(0)})^{\beta_{1}}\ldots(\partial r_{d}^{(0)})^{\beta_{d}}}.

Note 1

Recall that

	$\displaystyle\|s\|\leq 1,\ \alpha_{2}\leq\alpha_{1}\quad\text{if $d=2$},$
	$\displaystyle\|s_{1}\|\leq 1,\ \|s_{21}\|\leq 1,\ \alpha_{2}\leq\alpha_{3}\leq\alpha_{1}\quad\text{if $d=3$}.$

When $d=3$ , if Assumption 1 is imposed, there exists a positive constant $M$ , independent of $h_{T}$ , such that $|s_{22}|\leq M\frac{\alpha_{2}t_{1}}{\alpha_{3}}$ . Thus, if $d=2$ , we have

\displaystyle\alpha_{1}|[\widetilde{A}]_{j1}|\leq\mathscr{H}_{j},\quad\alpha_{2}|[\widetilde{A}]_{j2}|\leq\mathscr{H}_{j},\quad j=1,2,

and if $d=3$ , for $\widetilde{A}\in\{\widetilde{A}_{1},\widetilde{A}_{2}\}$ and $j=1,2,3$ , we have

\displaystyle\alpha_{1}|[\widetilde{A}]_{j1}|\leq\mathscr{H}_{j},\quad\alpha_{2}|[\widetilde{A}]_{j2}|\leq\mathscr{H}_{j},\quad\alpha_{3}|[\widetilde{A}]_{j3}|\leq\max\{1,M\}\mathscr{H}_{j},\quad j=1,2,3.

Note 2

We use the following calculations in Lemma 5. For any multi-indices $\beta$ and $\gamma$ , we have

	$\displaystyle\displaystyle\partial^{\beta+\gamma}_{\hat{x}}$	$\displaystyle=\frac{\partial^{\|\beta\|+\|\gamma\|}}{\partial\hat{x}_{1}^{\beta_{1}}\cdots\partial\hat{x}_{d}^{\beta_{d}}\partial\hat{x}_{1}^{\gamma_{1}}\cdots\partial\hat{x}_{d}^{\gamma_{d}}}$
		$\displaystyle=\underbrace{\sum_{i_{1}^{(1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{i_{1}^{(1)}1}\cdots\sum_{i_{\beta_{1}}^{(1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{i_{\beta_{1}}^{(1)}1}}_{\beta_{1}\text{times}}\cdots\underbrace{\sum_{i_{1}^{(d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{i_{1}^{(d)}d}\cdots\sum_{i_{\beta_{d}}^{(d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{i_{\beta_{d}}^{(d)}d}}_{\beta_{d}\text{times}}$
		$\displaystyle\underbrace{\sum_{j_{1}^{(1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{j_{1}^{(1)}1}\cdots\sum_{j_{\gamma_{1}}^{(1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{j_{\gamma_{1}}^{(1)}1}}_{\gamma_{1}\text{times}}\cdots\underbrace{\sum_{j_{1}^{(d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{j_{1}^{(d)}d}\cdots\sum_{j_{\gamma_{d}}^{(d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{j_{\gamma_{d}}^{(d)}d}}_{\gamma_{d}\text{times}}$
		$\displaystyle\underbrace{\frac{\partial^{\beta_{1}}}{\partial{x}_{i_{1}^{1)}}\cdots\partial{x}_{i_{\beta_{1}}^{(1)}}}}_{\beta_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\beta_{d}}}{\partial{x}_{i_{1}^{(d)}}\cdots\partial{x}_{i_{\beta_{d}}^{(d)}}}}_{\beta_{d}\text{times}}\underbrace{\frac{\partial^{\gamma_{1}}}{\partial{x}_{j_{1}^{(1)}}\cdots\partial{x}_{j_{\gamma_{1}}^{(1)}}}}_{\gamma_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\gamma_{d}}}{\partial{x}_{j_{1}^{(d)}}\cdots\partial{x}_{j_{\gamma_{d}}^{(d)}}}}_{\gamma_{d}\text{times}}.$

Let $\hat{\varphi}\in\mathcal{C}^{\ell}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ and ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ . Then, for $1\leq i\leq d$ ,

\displaystyle\displaystyle\left|\frac{\partial\hat{\varphi}}{\partial\hat{x}_{i}}\right|

\displaystyle\leq\sum_{i_{1}^{(1)}=1}^{d}\alpha_{i}\left|[\widetilde{A}]_{i_{1}^{(1)}i}\right|\left|\frac{\partial\varphi}{\partial x_{i_{1}^{(1)}}}\right|\leq\begin{cases}\alpha_{i}\|\widetilde{A}\|_{\max}\sum_{i_{1}^{(1)}=1}^{d}\left|\frac{\partial\varphi}{\partial x_{i_{1}^{(1)}}}\right|\quad\text{or},\\ c\sum_{i_{1}^{(1)}=1}^{d}\mathscr{H}_{i_{1}^{(1)}}\left|\frac{\partial\varphi}{\partial x_{i_{1}^{(1)}}}\right|,\end{cases}

and for $1\leq i,j\leq d$ ,

	$\displaystyle\displaystyle\left\|\frac{\partial^{2}\hat{\varphi}}{\partial\hat{x}_{i}\partial\hat{x}_{j}}\right\|$	$\displaystyle=\left\|\sum_{i_{1}^{(1)},j_{1}^{(1)}=1}^{d}\alpha_{i}\alpha_{j}[\widetilde{A}]_{i_{1}^{(1)}i}[\widetilde{A}]_{j_{1}^{(1)}j}\frac{\partial^{2}\varphi}{\partial x_{i_{1}^{(1)}}\partial x_{j_{1}^{(1)}}}\right\|$
		$\displaystyle\leq\begin{cases}\alpha_{i}\alpha_{j}\\|\widetilde{A}\\|_{\max}^{2}\sum_{i_{1}^{(1)},j_{1}^{(1)}=1}^{d}\left\|\frac{\partial^{2}\varphi}{\partial x_{i_{1}^{(1)}}\partial x_{j_{1}^{(1)}}}\right\|\quad\text{or},\\ \alpha_{j}\sum_{j_{1}^{(1)}=1}^{d}\|[\widetilde{A}]_{j_{1}^{(1)}j}\|\left\|\sum_{i_{1}^{(1)}=1}^{d}\alpha_{i}[\widetilde{A}]_{i_{1}^{(1)}i}\frac{\partial^{2}\varphi}{\partial x_{i_{1}^{(1)}}\partial x_{j_{1}^{(1)}}}\right\|\\ \quad\leq c\alpha_{j}\\|\widetilde{A}\\|_{\max}\sum_{j_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(1)}=1}^{d}\mathscr{H}_{i_{1}^{(1)}}\left\|\frac{\partial^{2}\varphi}{\partial x_{i_{1}^{(1)}}\partial x_{j_{1}^{(1)}}}\right\|\quad\text{or},\\ c\sum_{i_{1}^{(1)}=1}^{d}\sum_{j_{1}^{(1)}=1}^{d}\mathscr{H}_{i_{1}^{(1)}}\mathscr{H}_{j_{1}^{(1)}}\left\|\frac{\partial^{2}\varphi}{\partial x_{i_{1}^{(1)}}\partial x_{j_{1}^{(1)}}}\right\|.\end{cases}$

Lemma 5

Suppose that Assumption 1 is imposed. Let $m\in\mathbb{N}_{0}$ , $\ell\in\mathbb{N}_{0}$ with $\ell\geq m$ and $p\in[0,\infty]$ . Let $\beta:=(\beta_{1},\ldots,\beta_{d})\in\mathbb{N}_{0}^{d}$ and $\gamma:=(\gamma_{1},\ldots,\gamma_{d})\in\mathbb{N}_{0}^{d}$ be multi-indices with $|\beta|=m$ and $|\gamma|=\ell-m$ . Then, for any $\hat{\varphi}\in W^{\ell,p}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ and ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ , it holds that

\displaystyle\displaystyle\|\partial^{\beta}\partial^{\gamma}\hat{\varphi}\|_{L^{p}(\widehat{T})}

\displaystyle\leq C_{2}^{SA}|\det({A}_{T})|^{-\frac{1}{p}}\|\widetilde{A}\|^{m}_{2}\alpha^{\beta}\sum_{|\epsilon|=|\gamma|}\mathscr{H}^{\varepsilon}|\partial^{\epsilon}\varphi|_{W^{m,p}(T)},

(37)

where $C_{2}^{SA}$ is a constant that is independent of $T$ and $\widetilde{T}$ . Here, for $p=\infty$ and any positive real $x$ , $x^{-\frac{1}{p}}=1$ .

Proof

Let $\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{d})\in\mathbb{N}_{0}^{d}$ and $\delta=(\delta_{1},\ldots,\delta_{d})\in\mathbb{N}_{0}^{d}$ be multi-indies with $|\varepsilon|=|\gamma|$ and $|\delta|=|\beta|$ . Let $p\in[1,\infty)$ . Because the space $\mathcal{C}^{\ell}(\widehat{T})$ is dense in the space ${W}^{\ell,p}(\widehat{T})$ , we show that Eq. (37) holds for $\hat{\varphi}\in\mathcal{C}^{\ell}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ and ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ . Through a simple calculation, we obtain

	$\displaystyle\displaystyle\|\partial^{\beta+\gamma}\hat{\varphi}\|$	$\displaystyle=\left\|\frac{\partial^{\ell}\hat{\varphi}}{\partial\hat{x}_{1}^{\beta_{1}}\cdots\partial\hat{x}_{d}^{\beta_{d}}\partial\hat{x}_{1}^{\gamma_{1}}\cdots\partial\hat{x}_{d}^{\gamma_{d}}}\right\|$
		$\displaystyle\leq c\alpha^{\beta}\\|\widetilde{A}\\|_{\max}^{\|\beta\|}\underbrace{\sum_{i_{1}^{(1)}=1}^{d}\cdots\sum_{i_{\beta_{1}}^{(1)}=1}^{d}}_{\beta_{1}\text{times}}\cdots\underbrace{\sum_{i_{1}^{(d)}=1}^{d}\cdots\sum_{i_{\beta_{d}}^{(d)}=1}^{d}}_{\beta_{d}\text{times}}\underbrace{\sum_{j_{1}^{(1)}=1}^{d}\cdots\sum_{j_{\gamma_{1}}^{(1)}=1}^{d}}_{\gamma_{1}\text{times}}\cdots\underbrace{\sum_{j_{1}^{(d)}=1}^{d}\cdots\sum_{j_{\gamma_{d}}^{(d)}=1}^{d}}_{\gamma_{d}\text{times}}$
		$\displaystyle\underbrace{\mathscr{H}_{j_{1}^{(1)}}\cdots\mathscr{H}_{j_{\varepsilon_{1}}^{(1)}}}_{\gamma_{1}\text{times}}\cdots\underbrace{\mathscr{H}_{j_{1}^{(d)}}\cdots\mathscr{H}_{j_{\varepsilon_{d}}^{(d)}}}_{\gamma_{d}\text{times}}$
		$\displaystyle\Biggl{\|}\underbrace{\frac{\partial^{\beta_{1}}}{\partial{x}_{i_{1}^{1)}}\cdots\partial{x}_{i_{\beta_{1}}^{(1)}}}}_{\beta_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\beta_{d}}}{\partial{x}_{i_{1}^{(d)}}\cdots\partial{x}_{i_{\beta_{d}}^{(d)}}}}_{\beta_{d}\text{times}}\underbrace{\frac{\partial^{\gamma_{1}}}{\partial{x}_{j_{1}^{(1)}}\cdots\partial{x}_{j_{\gamma_{1}}^{(1)}}}}_{\gamma_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\gamma_{d}}}{\partial{x}_{j_{1}^{(d)}}\cdots\partial{x}_{j_{\gamma_{d}}^{(d)}}}}_{\gamma_{d}\text{times}}\varphi\Biggr{\|}$
		$\displaystyle\leq c\alpha^{\beta}\\|\widetilde{A}\\|_{\max}^{\|\beta\|}\sum_{\|\delta\|=\|\beta\|}\sum_{\|\varepsilon\|=\|\gamma\|}\mathscr{H}^{\varepsilon}\|{\partial^{\delta}\partial^{\varepsilon}\varphi}\|.$

Using Eq. (24), we then have that

	$\displaystyle\displaystyle\int_{\widehat{T}}\|\partial^{\beta}\partial^{\gamma}\hat{\varphi}\|^{p}d\hat{x}$	$\displaystyle\leq c\\|\widetilde{A}\\|_{2}^{mp}\alpha^{\beta p}\sum_{\|\delta\|=\|\beta\|}\sum_{\|\epsilon\|=\|\gamma\|}\mathscr{H}^{\varepsilon p}\int_{\widehat{T}}\|{\partial^{\delta}\partial^{\varepsilon}\varphi}\|^{p}d\hat{x}$
		$\displaystyle=c\|\det(A_{T})\|^{-1}\\|\widetilde{A}\\|_{2}^{mp}\alpha^{\beta p}\sum_{\|\delta\|=\|\beta\|}\sum_{\|\epsilon\|=\|\gamma\|}\mathscr{H}^{\varepsilon p}\int_{{T}}\|{\partial^{\delta}\partial^{\varepsilon}\varphi}\|^{p}d{x}.$

Therefore, using (25), we obtain

\displaystyle\displaystyle\|\partial^{\beta}\partial^{\gamma}\hat{\varphi}\|_{L^{p}(\widehat{T})}

\displaystyle\leq c|\det(A_{T})|^{-\frac{1}{p}}\|\widetilde{A}\|_{2}^{m}\alpha^{\beta}\sum_{|\epsilon|=|\gamma|}\mathscr{H}^{\varepsilon}|\partial^{\epsilon}\varphi|_{W^{m,p}(T)},

which recovers Eq. (37).

We consider the case in which $p=\infty$ . A function $\varphi\in W^{\ell,\infty}(T)$ belongs to the space $W^{\ell,p}(T)$ for any $p\in[1,\infty)$ . Therefore, it holds that $\hat{\varphi}\in W^{\ell,p}(\widehat{T})$ for any $p\in[1,\infty)$ , and thus,

	$\displaystyle\displaystyle\\|\partial^{\beta}\partial^{\gamma}\hat{\varphi}\\|_{L^{p}(\widehat{T})}$	$\displaystyle\leq c\|\det(A_{T})\|^{-\frac{1}{p}}\\|\widetilde{A}\\|_{2}^{m}\alpha^{\beta}\sum_{\|\epsilon\|=\|\gamma\|}\mathscr{H}^{\varepsilon}\|\partial^{\epsilon}\varphi\|_{W^{m,p}(T)}$
		$\displaystyle\leq c\\|\widetilde{A}\\|_{2}^{m}\alpha^{\beta}\sum_{\|\epsilon\|=\|\gamma\|}\mathscr{H}^{\varepsilon}\|\partial^{\epsilon}\varphi\|_{W^{m,\infty}(T)}\textless\infty.$		(38)

This implies that the function $\partial^{\beta}\partial^{\gamma}\hat{\varphi}$ is in the space $L^{\infty}(\widehat{T})$ . Inequality (37) for $p=\infty$ is obtained by taking the limit $p\to\infty$ in Eq. (38) on the basis that $\lim_{p\to\infty}\|\cdot\|_{L^{p}(\widehat{T})}=\|\cdot\|_{L^{\infty}(\widehat{T})}$ . ∎

Remark 3

In inequality (37), it is possible to obtain the estimates in $T_{0}$ by specifically determining the matrix $A_{T_{0}}$ .

Let $\ell=2$ , $m=1$ , and $p=q=2$ . Recall that

\displaystyle\displaystyle\Phi_{T_{0}}:T\ni x\mapsto x^{(0)}:={A}_{T_{0}}x+b_{T_{0}}\in T_{0}.

For ${\varphi}\in\mathcal{C}^{2}({T})$ with $\varphi_{0}=\varphi\circ\Phi_{T_{0}}^{-1}$ and $1\leq i,j\leq d$ , we have

\displaystyle\displaystyle\left|\frac{\partial^{2}{\varphi}}{\partial{x}_{i}{\partial{x}_{j}}}({x})\right|

\displaystyle=\left|\sum_{i_{1}^{(1)},j_{1}^{(1)}=1}^{2}[A_{T_{0}}]_{i_{1}^{(1)}i}[A_{T_{0}}]_{j_{1}^{(1)}j}\frac{\partial^{2}\varphi_{0}}{\partial x_{i_{1}^{(1)}}^{(0)}\partial x_{j_{1}^{(1)}}^{(0)}}(x)\right|.

Let $d=2$ . We define the matrix $A_{T_{0}}$ as

\displaystyle\displaystyle A_{T_{0}}:=\begin{pmatrix}\cos\frac{\pi}{2}&-\sin\frac{\pi}{2}\\ \sin\frac{\pi}{2}&\cos\frac{\pi}{2}\end{pmatrix}.

Because $\|A_{T_{0}}\|_{\max}=1$ , we have

\displaystyle\displaystyle\left|\frac{\partial^{2}{\varphi}}{\partial{x}_{i}{\partial{x}_{j}}}({x})\right|

\displaystyle\leq\left|\frac{\partial^{2}\varphi_{0}}{\partial x_{i+1}^{(0)}\partial x_{j+1}^{(0)}}(x)\right|,

where the indices $i$ , $i+1$ and $j$ , $j+1$ must be evaluated modulo 2. Because $|\det(A_{T_{0}})|=1$ , it holds that

\displaystyle\displaystyle\left\|\frac{\partial^{2}{\varphi}}{\partial{x}_{i}{\partial{x}_{j}}}\right\|_{L^{2}(T)}\leq\left\|\frac{\partial^{2}\varphi_{0}}{\partial x_{i_{i+1}}^{(0)}\partial x_{j+1}^{(0)}}\right\|_{L^{2}(T_{0})}.

We then have

\displaystyle\displaystyle\sum_{j=1}^{2}\mathscr{H}_{j}\left|\frac{\partial{\varphi}}{{\partial{x}_{j}}}\right|_{H^{1}(T)}

\displaystyle\leq\sum_{j=1}^{2}\mathscr{H}_{j}\left|\frac{\partial\varphi_{0}}{\partial x_{j+1}^{(0)}}\right|_{H^{1}(T_{0})},

where the indices $j$ , $j+1$ must be evaluated modulo 2.

We define the matrix $A_{T_{0}}$ as

\displaystyle\displaystyle A_{T_{0}}:=\begin{pmatrix}\cos\frac{\pi}{4}&-\sin\frac{\pi}{4}\\ \sin\frac{\pi}{4}&\cos\frac{\pi}{4}\end{pmatrix}.

We then have

\displaystyle\displaystyle\left|\frac{\partial^{2}{\varphi}}{\partial{x}_{i}{\partial{x}_{j}}}({x})\right|

\displaystyle\leq\frac{1}{\sqrt{2}}\sum_{i_{1}^{(1)},j_{1}^{(1)}=1}^{2}\left|\frac{\partial^{2}\varphi_{0}}{\partial x_{i_{1}^{(1)}}^{(0)}\partial x_{j_{1}^{(1)}}^{(0)}}(x)\right|,

which leads to

\displaystyle\displaystyle\left\|\frac{\partial^{2}{\varphi}}{\partial{x}_{i}{\partial{x}_{j}}}\right\|_{L^{2}(T)}^{2}\leq c\sum_{i_{1}^{(1)},j_{1}^{(1)}=1}^{2}\left\|\frac{\partial^{2}\varphi_{0}}{\partial x_{i_{1}^{(1)}}^{(0)}\partial x_{j_{1}^{(1)}}^{(0)}}\right\|_{L^{2}(T_{0})}^{2}\leq c|\varphi_{0}|^{2}_{H^{2}(T_{0})}.

Using (25), we than have that

\displaystyle\displaystyle\sum_{j=1}^{2}\mathscr{H}_{j}\left|\frac{\partial{\varphi}}{{\partial{x}_{j}}}\right|_{H^{1}(T)}

\displaystyle\leq\sum_{j=1}^{2}\mathscr{H}_{j}|\varphi_{0}|_{H^{2}(T_{0})}\leq ch_{T_{0}}|\varphi_{0}|_{H^{2}(T_{0})}.

In this case, anisotropic interpolation error estimates cannot be obtained.

Note 3

We use the following calculations in Lemma 6. For any multi-indices $\beta$ and $\gamma$ , we have

	$\displaystyle\displaystyle\partial^{\beta+\gamma}_{\hat{x}}$	$\displaystyle=\frac{\partial^{\|\beta\|+\|\gamma\|}}{\partial\hat{x}_{1}^{\beta_{1}}\cdots\partial\hat{x}_{d}^{\beta_{d}}\partial\hat{x}_{1}^{\gamma_{1}}\cdots\partial\hat{x}_{d}^{\gamma_{d}}}$
		$\displaystyle=\underbrace{\sum_{i_{1}^{(1)},i_{1}^{(0,1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{i_{1}^{(1)}1}[A_{T_{0}}]_{i_{1}^{(0,1)}i_{1}^{(1)}}\cdots\sum_{i_{\beta_{1}}^{(1)},i_{\beta_{1}}^{(0,1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{i_{\beta_{1}}^{(1)}1}[A_{T_{0}}]_{i_{\beta_{1}}^{(0,1)}i_{\beta_{1}}^{(1)}}}_{\beta_{1}\text{times}}\cdots$
		$\displaystyle\underbrace{\sum_{i_{1}^{(d)},i_{1}^{(0,d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{i_{1}^{(d)}d}[A_{T_{0}}]_{i_{1}^{(0,d)}i_{1}^{(d)}}\cdots\sum_{i_{\beta_{d}}^{(d)},i_{\beta_{d}}^{(0,d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{i_{\beta_{d}}^{(d)}d}[A_{T_{0}}]_{i_{\beta_{d}}^{(0,d)}i_{\beta_{d}}^{(d)}}}_{\beta_{d}\text{times}}$
		$\displaystyle\underbrace{\sum_{j_{1}^{(1)},j_{1}^{(0,1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{j_{1}^{(1)}1}[A_{T_{0}}]_{j_{1}^{(0,1)}j_{1}^{(1)}}\cdots\sum_{j_{\gamma_{1}}^{(1)},j_{\gamma_{1}}^{(0,1)}=1}^{d}\alpha_{1}[\widetilde{A}]_{j_{\gamma_{1}}^{(1)}1}[A_{T_{0}}]_{j_{\gamma_{1}}^{(0,1)}j_{\gamma_{1}}^{(1)}}}_{\gamma_{1}\text{times}}\cdots$
		$\displaystyle\underbrace{\sum_{j_{1}^{(d)},j_{1}^{(0,d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{j_{1}^{(d)}d}[A_{T_{0}}]_{j_{1}^{(0,d)}j_{1}^{(d)}}\cdots\sum_{j_{\gamma_{d}}^{(d)},j_{\gamma_{d}}^{(0,d)}=1}^{d}\alpha_{d}[\widetilde{A}]_{j_{\gamma_{d}}^{(d)}d}[A_{T_{0}}]_{j_{\gamma_{d}}^{(0,d)}j_{\gamma_{d}}^{(d)}}}_{\gamma_{d}\text{times}}$
		$\displaystyle\underbrace{\frac{\partial^{\beta_{1}}}{\partial{x}_{i_{1}^{(0,1)}}^{(0)}\cdots\partial{x}_{i_{\beta_{1}}^{(0,1)}}^{(0)}}}_{\beta_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\beta_{d}}}{\partial{x}_{i_{1}^{(0,d)}}^{(0)}\cdots\partial{x}_{i_{\beta_{d}}^{(0,d)}}^{(0)}}}_{\beta_{d}\text{times}}\underbrace{\frac{\partial^{\gamma_{1}}}{\partial{x}_{j_{1}^{(0,1)}}^{(0)}\cdots\partial{x}_{j_{\gamma_{1}}^{(0,1)}}^{(0)}}}_{\gamma_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\gamma_{d}}}{\partial{x}_{j_{1}^{(0,d)}}^{(0)}\cdots\partial{x}_{j_{\gamma_{d}}^{(0,d)}}^{(0)}}}_{\gamma_{d}\text{times}}.$

Let $\hat{\varphi}\in\mathcal{C}^{\ell}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ , ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ and $\varphi_{0}=\varphi\circ\Phi_{T_{0}}^{-1}$ . Then, for $1\leq i\leq d$ ,

	$\displaystyle\displaystyle\left\|\frac{\partial\hat{\varphi}}{\partial\hat{x}_{i}}\right\|$	$\displaystyle=\left\|\sum_{i_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(0,1)}=1}^{d}\alpha_{i}[\widetilde{A}]_{i_{1}^{(1)}i}[A_{T_{0}}]_{i_{1}^{(0,1)}i_{1}^{(1)}}\frac{\partial\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}}\right\|$
		$\displaystyle=\alpha_{i}\left\|\sum_{i_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(0,1)}=1}^{d}[A_{T_{0}}]_{i_{1}^{(0,1)}i_{1}^{(1)}}(r_{i})_{i_{1}^{(1)}}\frac{\partial\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}}\right\|=\alpha_{i}\left\|\frac{\partial\varphi_{0}}{\partial r_{i}^{(0)}}\right\|$
		$\displaystyle\leq\alpha_{i}\\|\widetilde{A}\\|_{\max}\\|A_{T_{0}}\\|_{\max}\sum_{i_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(0,1)}=1}^{d}\left\|\frac{\partial\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}}\right\|,$

and for $1\leq i,j\leq d$ ,

	$\displaystyle\displaystyle\left\|\frac{\partial^{2}\hat{\varphi}}{\partial\hat{x}_{i}\partial\hat{x}_{j}}\right\|$	$\displaystyle=\Biggl{\|}\sum_{i_{1}^{(1)},j_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(0,1)},j_{1}^{(0,1)}=1}^{d}\alpha_{i}\alpha_{j}[\widetilde{A}]_{i_{1}^{(1)}i}[\widetilde{A}]_{j_{1}^{(1)}j}$
		$\displaystyle\quad\quad[A_{T_{0}}]_{i_{1}^{(0,1)}i_{1}^{(1)}}[A_{T_{0}}]_{j_{1}^{(0,1)}j_{1}^{(1)}}\frac{\partial^{2}\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}\partial x_{j_{1}^{(0,1)}}^{(0)}}\Biggr{\|}=\alpha_{i}\alpha_{j}\left\|\frac{\partial^{2}\varphi_{0}}{\partial r_{i}^{(0)}\partial r_{j}^{(0)}}\right\|$
		$\displaystyle\leq\alpha_{i}\alpha_{j}\sum_{j_{1}^{(1)}=1}^{d}\|[\widetilde{A}]_{j_{1}^{(1)}j}\|\Biggl{\|}\sum_{j_{1}^{(0,1)}=1}^{d}[A_{T_{0}}]_{j_{1}^{(0,1)}j_{1}^{(1)}}\frac{\partial^{2}\varphi_{0}}{\partial r_{i}^{(0)}\partial x_{j_{1}^{(0,1)}}^{(0)}}\Biggr{\|}$
		$\displaystyle\leq\alpha_{i}\alpha_{j}\\|\widetilde{A}\\|_{\max}\\|{A}_{T_{0}}\\|_{\max}\sum_{j_{1}^{(0,1)}=1}^{d}\Biggl{\|}\frac{\partial^{2}\varphi_{0}}{\partial r_{i}^{(0)}\partial x_{j_{1}^{(0,1)}}^{(0)}}\Biggr{\|}$
		$\displaystyle\leq\alpha_{i}\alpha_{j}\\|\widetilde{A}\\|_{\max}^{2}\\|{A}_{T_{0}}\\|_{\max}^{2}\sum_{i_{1}^{(0,1)},j_{1}^{(0,1)}=1}^{d}\left\|\frac{\partial^{2}\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}\partial x_{j_{1}^{(0,1)}}^{(0)}}\right\|.$

If Assumption 1 is not imposed, the estimates corresponding to Lemma 5 are as follows.

Lemma 6

Let $m\in\mathbb{N}_{0}$ , $\ell\in\mathbb{N}_{0}$ with $\ell\geq m$ , and $p\in[0,\infty]$ . Let $\beta:=(\beta_{1},\ldots,\beta_{d})\in\mathbb{N}_{0}^{d}$ and $\gamma:=(\gamma_{1},\ldots,\gamma_{d})\in\mathbb{N}_{0}^{d}$ be multi-indices with $|\beta|=m$ and $|\gamma|=\ell-m$ . Then, for any $\hat{\varphi}\in W^{\ell,p}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ , ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ , and $\varphi_{0}=\varphi\circ\Phi_{T_{0}}^{-1}$ , it holds that

\displaystyle\displaystyle\|\partial^{\beta}\partial^{\gamma}\hat{\varphi}\|_{L^{p}(\widehat{T})}

\displaystyle\leq C_{3}^{SA}|\det({A}_{T})|^{-\frac{1}{p}}\|\widetilde{{A}}\|_{2}^{m}\alpha^{\beta}\sum_{|\epsilon|=|\gamma|}{\alpha}^{\varepsilon}|\partial_{r^{(0)}}^{\epsilon}\varphi_{0}|_{W^{m,p}(T_{0})},

(39)

where $C_{3}^{SA}$ is a constant that is independent of $T_{0}$ and $\widetilde{T}$ . Here, for $p=\infty$ and any positive real $x$ , $x^{-\frac{1}{p}}=1$ .

Proof

We follow the proof of Lemma 5. Let $p\in[1,\infty)$ . Because the space $\mathcal{C}^{\ell}(\widehat{T})$ is dense in the space ${W}^{\ell,p}(\widehat{T})$ , we show that Eq. (39) holds for $\hat{\varphi}\in\mathcal{C}^{\ell}(\widehat{T})$ with $\tilde{\varphi}=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ , ${\varphi}=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ , and $\varphi_{0}=\varphi\circ\Phi_{T_{0}}^{-1}$ . For $1\leq i,k\leq d$ ,

\displaystyle\displaystyle\left|\partial^{\beta+\gamma}\hat{\varphi}\right|

\displaystyle\leq c\alpha^{\beta}\|\widetilde{A}\|_{\max}^{|\beta|}\|{A}_{T_{0}}\|_{\max}^{|\beta|}\sum_{|\delta|=|\beta|}\sum_{|\varepsilon|=|\gamma|}\alpha^{\varepsilon}\left|\partial^{\delta}\partial_{r^{(0)}}^{\varepsilon}\varphi_{0}\right|.

Using Eqs. (22c) and (24), we obtain Eq. (39) for $p\in[1,\infty]$ by an argument analogous to that used for Lemma 5. ∎

4 Remarks on anisotropic interpolation analysis

We use the following Bramble–Hilbert-type lemma on anisotropic meshes proposed in (Ape99, , Lemma 2.1).

Lemma 7

Let $D\subset\mathbb{R}^{d}$ with $d\in\{2,3\}$ be a connected open set that is star-shaped with respect to a ball $B$ . Let $\gamma$ be a multi-index with $m:=|\gamma|$ and $\varphi\in L^{1}(D)$ be a function with $\partial^{\gamma}\varphi\in W^{\ell-m,p}(D)$ , where $\ell\in\mathbb{N}$ , $m\in\mathbb{N}_{0}$ , $0\leq m\leq\ell$ , and $p\in[1,\infty]$ . Then, it holds that

\displaystyle\displaystyle\|\partial^{\gamma}(\varphi-Q^{(\ell)}\varphi)\|_{W^{\ell-m,p}(D)}\leq C^{BH}|\partial^{\gamma}\varphi|_{W^{\ell-m,p}(D)},

(40)

where $C^{BH}$ depends only on $d$ , $\ell$ , $\mathop{\mathrm{diam}}D$ , and $\mathop{\mathrm{diam}}B$ , and $Q^{(\ell)}\varphi$ is defined as

\displaystyle\displaystyle(Q^{(\ell)}\varphi)(x):=\sum_{|\delta|\leq\ell-1}\int_{B}\eta(y)(\partial^{\delta}\varphi)(y)\frac{(x-y)^{\delta}}{\delta!}dy\in\mathcal{P}^{\ell-1},

(41)

where $\eta\in\mathcal{C}_{0}^{\infty}(B)$ is a given function with $\int_{B}\eta dx=1$ .

As explained in the Introduction, there exist some mistakes in the proof of Theorem 2 of IshKobTsu , and the statement is not valid in its original form. To clarify the following description, we explain the errors in the proof. Let $\widehat{T}\subset\mathbb{R}^{2}$ be the reference element defined in Section 2.1.1. We set $k=m=1$ , $\ell=2$ , and $p=2$ . For $\hat{\varphi}\in H^{2}(\widehat{T})$ , we set $\tilde{\varphi}:=\hat{\varphi}\circ\widehat{\Phi}^{-1}$ and ${\varphi}:=\tilde{\varphi}\circ\widetilde{\Phi}^{-1}$ . Inequality (29) yields

\displaystyle\displaystyle|{\varphi}-I_{T}\varphi|_{H^{1}({T})}

\displaystyle\leq c|\det(A_{T})|^{\frac{1}{2}}\|\widetilde{A}^{-1}\|_{2}\left(\sum_{i=1}^{2}\alpha^{-2}_{i}\|\partial_{\hat{x}_{i}}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\|^{2}_{L^{2}(\widehat{T})}\right)^{\frac{1}{2}}.

(42)

The coefficient $\alpha_{i}^{-2}$ appears on the right-hand side of Eq. (42). In (IshKobTsu, , Theorem 2), we wrongly claimed that $\alpha_{i}^{-2}$ could be canceled out. In fact, a further assumption is required for this. Using Eq. (41) and the triangle inequality, we have

\displaystyle\displaystyle\|\partial_{\hat{x}_{i}}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\|^{2}_{L^{2}(\widehat{T})}

\displaystyle\leq 2\|\partial_{\hat{x}_{i}}(\hat{\varphi}-Q^{(2)}\hat{\varphi})\|^{2}_{L^{2}(\widehat{T})}+2\|\partial_{\hat{x}_{i}}(Q^{(2)}\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\|^{2}_{L^{2}(\widehat{T})}.

We use inequality (40) to obtain the target inequality (IshKobTsu, , Theorem 2). To this end, we have to show that

\displaystyle\displaystyle\|\partial_{\hat{x}_{i}}(Q^{(2)}\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\|_{L^{2}(\widehat{T})}

\displaystyle\leq c\|\partial_{\hat{x}_{i}}(\hat{\varphi}-Q^{(2)}\hat{\varphi})\|_{H^{1}(\widehat{T})}.

(43)

However, this is unlikely to hold because Eqs. (14) and (16) yield

	$\displaystyle\displaystyle\\|\partial_{\hat{x}_{i}}(Q^{(2)}\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\\|_{L^{2}(\widehat{T})}$	$\displaystyle=\\|\partial_{\hat{x}_{i}}(I_{\widehat{T}}(Q^{(2)}\hat{\varphi})-I_{\widehat{T}}\hat{\varphi})\\|_{L^{2}(\widehat{T})}$
		$\displaystyle\leq c\\|Q^{(2)}\hat{\varphi}-\hat{\varphi}\\|_{H^{2}(\widehat{T})}\leq c\|\hat{\varphi}\|_{H^{2}(\widehat{T})}.$

Using the classical scaling argument (see (ErnGue04, , Lemma 1.101)), we have

\displaystyle\displaystyle|\hat{\varphi}|_{H^{2}(\widehat{T})}\leq c|\det(A_{T})|^{-\frac{1}{2}}\|\widetilde{A}\|_{2}|\varphi|_{H^{2}(T)},

which does not include the quantity $\alpha_{i}$ . Therefore, the quantity $\alpha_{i}^{-1}$ in Eq. (42) remains. Thus, the proof of (IshKobTsu, , Theorem 2) is incorrect.

To overcome this problem, we use some results from previous studies ApeDob92 ; Ape99 . That is, we assume that there exists a linear functional $\mathscr{F}_{1}$ such that

	$\displaystyle\mathscr{F}_{1}\in H^{1}(\widehat{T})^{\prime},$
	$\displaystyle\mathscr{F}_{1}(\partial_{\hat{x}_{i}}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi}))=0\quad i=1,2,\quad\forall\hat{\varphi}\in\mathcal{C}({\widehat{T}}):\ \partial_{\hat{x}_{i}}\hat{\varphi}\in H^{1}(\widehat{T}),$
	$\displaystyle\hat{\eta}\in\mathcal{P}^{1},\quad\mathscr{F}_{1}(\partial_{\hat{x}_{i}}\hat{\eta})=0\quad i=1,2,\quad\Rightarrow\quad\partial_{\hat{x}_{i}}\hat{\eta}=0.$

Because the polynomial spaces are finite-dimensional, all norms are equivalent; i.e., because $|\mathscr{F}_{1}(\partial_{\hat{x}_{i}}(\hat{\eta}-I_{\widehat{T}}\hat{\varphi}))|$ ( $i=1,2$ ) is a norm on $\mathcal{P}^{0}$ , we have that, for $i=1,2$ ,

	$\displaystyle\displaystyle\\|\partial_{\hat{x}_{i}}(\hat{\eta}-I_{\widehat{T}}\hat{\varphi})\\|_{L^{2}(\widehat{T})}$	$\displaystyle\leq c\|\mathscr{F}_{1}(\partial_{\hat{x}_{i}}(\hat{\eta}-I_{\widehat{T}}\hat{\varphi}))\|=c\|\mathscr{F}_{1}(\partial_{\hat{x}_{i}}(\hat{\eta}-\hat{\varphi}))\|$
		$\displaystyle\leq c\\|\partial_{\hat{x}_{i}}(\hat{\eta}-\hat{\varphi})\\|_{H^{1}(\widehat{T})}.$

Setting $\hat{\eta}:=Q^{(2)}\hat{\varphi}$ , we obtain Eq. (43). Using inequality (40) yields

\displaystyle\displaystyle\|\partial_{\hat{x}_{i}}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\|^{2}_{L^{2}(\widehat{T})}

\displaystyle\leq c|\partial_{\hat{x}_{i}}\hat{\varphi}|^{2}_{H^{1}(\widehat{T})},

and so inequality (42) together with Eq. (25) can be written as

\displaystyle\displaystyle|{\varphi}-I_{T}\varphi|_{H^{1}({T})}

\displaystyle\leq c|\det(A_{T})|^{\frac{1}{2}}\|\widetilde{A}^{-1}\|_{2}\sum_{i,j=1}^{2}\alpha^{-1}_{i}\|\partial_{\hat{x}_{i}}\partial_{\hat{x}_{j}}\hat{\varphi}\|_{L^{2}(\widehat{T})}.

(44)

Inequality (37) yields

\displaystyle\displaystyle\|\partial_{\hat{x}_{i}}\partial_{\hat{x}_{j}}\hat{\varphi}\|_{L^{2}(\widehat{T})}

\displaystyle\leq c|\det({A}_{T})|^{-\frac{1}{2}}\|\widetilde{A}\|_{2}\alpha_{i}\sum_{n=1}^{2}\mathscr{H}_{n}\left|\frac{\partial\varphi}{\partial x_{n}}\right|_{H^{1}(T)}.

(45)

Therefore, the quantity $\alpha_{i}^{-1}$ in Eq. (44) and the quantity $\alpha_{i}$ in Eq. (45) cancel out.

5 Classical interpolation error estimates

The following embedding results hold.

Theorem

Let $d\geq 2$ , $s\textgreater 0$ , and $p\in[1,\infty]$ . Let $D\subset\mathbb{R}^{d}$ be a bounded open subset of $\mathbb{R}^{d}$ . If $D$ is a Lipschitz set, we have that

\displaystyle\displaystyle W^{s,p}(D)\hookrightarrow\begin{cases}L^{q}(D)\quad\text{$\forall q\in[p,\frac{pd}{d-sp}]$ if $sp\textless d$},\\ L^{q}(D)\quad\text{$\forall q\in[p,\infty)$, if $sp=d$},\\ L^{\infty}(D)\cap\mathcal{C}^{0,\xi}(\overline{D})\quad\text{$\xi=1-\frac{d}{sp}$ if $sp\textgreater d$}.\end{cases}

(46)

Furthermore,

\displaystyle\displaystyle W^{s,p}(D)\hookrightarrow L^{\infty}(D)\cap\mathcal{C}^{0}(\overline{D})\quad\text{(case $s=d$ and $p=1$)}.

(47)

Proof

See, for example, (ErnGue04, , Corollary B.43, Theorem B.40), (ErnGue21a, , Theorem 2.31), and the references therein. ∎

Remark 4

Let $s\textgreater 0$ and $p\in[1,\infty]$ be such that

\displaystyle\displaystyle s\textgreater\frac{d}{p}\quad\text{if $p\textgreater 1$},\quad s\geq d\quad\text{if $p=1$}.

Then, it holds that $W^{s,p}(D)\hookrightarrow\mathcal{C}^{0}(\overline{D})$ .

Using the new geometric parameter $H_{T_{0}}$ , it is possible to deduce the classical interpolation error estimates; e.g., see (ErnGue04, , Theorem 1.103) and (ErnGue21a, , Theorem 11.13).

Theorem A

Let $1\leq p\leq\infty$ and assume that there exists a nonnegative integer $k$ such that

\displaystyle\displaystyle\mathcal{P}^{k}\subset\widehat{{P}}\subset W^{k+1,p}(\widehat{T})\subset V(\widehat{T}).

Let $\ell$ ( $0\leq\ell\leq k$ ) be such that $W^{\ell+1,p}(\widehat{T})\subset V(\widehat{T})$ with the continuous embedding. Furthermore, assume that $\ell,m\in\mathbb{N}\cup\{0\}$ and $p,q\in[1,\infty]$ such that $0\leq m\leq\ell+1$ and

\displaystyle\displaystyle W^{\ell+1,p}(\widehat{T})\hookrightarrow W^{m,q}(\widehat{T}).

(48)

Then, for any $\varphi_{0}\in W^{\ell+1,p}(T_{0})$ , it holds that

\displaystyle\displaystyle|\varphi_{0}-{I}_{T_{0}}\varphi_{0}|_{W^{m,q}(T_{0})}

\displaystyle\leq C_{*}^{I}|T_{0}|^{\frac{1}{q}-\frac{1}{p}}\left(\frac{\alpha_{\max}}{\alpha_{\min}}\right)^{m}\left(\frac{H_{T_{0}}}{h_{T_{0}}}\right)^{m}h_{T_{0}}^{\ell+1-m}|\varphi_{0}|_{W^{\ell+1,p}(T_{0})},

(49)

where $C_{*}^{I}$ is a positive constant that is independent of $h_{T}$ and $H_{T}$ , and the parameters $\alpha_{\max}$ and $\alpha_{\min}$ are defined as

\displaystyle\displaystyle\alpha_{\max}:=\max\{\alpha_{1},\ldots,\alpha_{d}\},\quad\alpha_{\min}:=\min\{\alpha_{1},\ldots,\alpha_{d}\}.

(50)

Proof

Let $\hat{\varphi}\in W^{\ell+1,p}(\widehat{T})$ . Because $0\leq\ell\leq k$ , $\mathcal{P}^{\ell}\subset\mathcal{P}^{k}\subset\widehat{{P}}$ . Therefore, for any $\hat{\eta}\in\mathcal{P}^{\ell}$ , we have $I_{\widehat{T}}\hat{\eta}=\hat{\eta}$ . Using Eqs. (16) and (48), we obtain

	$\displaystyle\displaystyle\|\tilde{\varphi}-I_{\widehat{T}}\hat{\varphi}\|_{W^{m,q}(\widehat{T})}$	$\displaystyle\leq\|\tilde{\varphi}-\hat{\eta}\|_{W^{m,q}(\widehat{T})}+\|I_{\widehat{T}}(\hat{\eta}-\hat{\varphi})\|_{W^{m,q}(\widehat{T})}$
		$\displaystyle\leq c\\|\tilde{\varphi}-\hat{\eta}\\|_{W^{\ell+1,p}(\widehat{T})},$

where we have used the stability of the interpolation operator $I_{\widehat{T}}$ ; i.e.,

\displaystyle\displaystyle|I_{\widehat{T}}(\hat{\eta}-\hat{\varphi})|_{W^{m,q}(\widehat{T})}

\displaystyle\leq\sum_{i=1}^{n_{0}}|\hat{\chi}_{i}(\hat{\eta}-\hat{\varphi})||\hat{\theta}_{i}|_{W^{m,q}(\widehat{T})}\leq c\|\hat{\eta}-\tilde{\varphi}\|_{W^{\ell+1,p}(\widehat{T})}.

Using the classic Bramble–Hilbert-type lemma (e.g., (BreSco08, , Lemma 4.3.8)), we obtain

\displaystyle\displaystyle|\tilde{\varphi}-I_{\widehat{T}}\hat{\varphi}|_{W^{m,q}(\widehat{T})}

\displaystyle\leq c\inf_{\hat{\eta}\in\mathcal{P}^{\ell}}\|\hat{\eta}-\tilde{\varphi}\|_{W^{\ell+1,p}(\widehat{T})}\leq c|\hat{\varphi}|_{W^{\ell+1,p}(\widehat{T})}.

(51)

Inequalities (26), (29), (25), and (51) yield

	$\displaystyle\|\varphi_{0}-{I}_{T_{0}}\varphi_{0}\|_{W^{m,q}({T}_{0})}\leq c\|\varphi-{I}_{T}\varphi\|_{W^{m,q}({T})}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{\frac{1}{q}}\\|\widetilde{A}^{-1}\\|_{2}^{m}\left(\sum_{\|\beta\|=m}(\alpha^{-\beta})^{q}\\|\partial^{\beta}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\\|^{q}_{L^{q}(\widehat{T})}\right)^{1/q}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{\frac{1}{q}}\\|\widetilde{A}^{-1}\\|_{2}^{m}\max\{\alpha_{1}^{-1},\ldots,\alpha_{d}^{-1}\}^{\|\beta\|}\|\tilde{\varphi}-I_{\widehat{T}}\hat{\varphi}\|_{W^{m,q}(\widehat{T})}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{\frac{1}{q}}\\|\widetilde{A}^{-1}\\|_{2}^{m}\alpha_{\min}^{-\|\beta\|}\|\hat{\varphi}\|_{W^{\ell+1,p}(\widehat{T})}.$		(52)

Using inequalities (25) and (39) together with Eq. (22c), we have

	$\displaystyle\|\hat{\varphi}\|_{W^{\ell+1,p}(\widehat{T})}$
	$\displaystyle\ \leq\sum_{\|\gamma\|=\ell+1-m}\sum_{\|\beta\|=m}\\|\partial^{\beta}\partial^{\gamma}\hat{\varphi}\\|_{L^{p}(\widehat{T})}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{-\frac{1}{p}}\\|\widetilde{{A}}\\|_{2}^{m}\sum_{\|\gamma\|=\ell+1-m}\sum_{\|\beta\|=m}\alpha^{\beta}\sum_{\|\epsilon\|=\|\gamma\|}{\alpha}^{\varepsilon}\|\partial_{r^{(0)}}^{\epsilon}\varphi_{0}\|_{W^{m,p}(T_{0})}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{-\frac{1}{p}}\\|\widetilde{{A}}\\|_{2}^{m}\max\{\alpha_{1},\ldots,\alpha_{d}\}^{\|\beta\|}h_{T_{0}}^{\ell+1-m}\|\varphi_{0}\|_{W^{\ell+1,p}(T_{0})}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{-\frac{1}{p}}\\|\widetilde{{A}}\\|_{2}^{m}\alpha_{\max}^{\|\beta\|}h_{T_{0}}^{\ell+1-m}\|\varphi_{0}\|_{W^{\ell+1,p}(T_{0})}.$		(53)

From Eqs. (52) and (53) together with Eq. (23), we have the desired estimate (49). ∎

6 Anisotropic interpolation error estimates

6.1 Main theorem

Theorem A can be applied to standard isotropic elements as well as some classes of anisotropic elements. If we are concerned with anisotropic elements, it is desirable to remove the quantity $\alpha_{\max}/\alpha_{\min}$ from estimate (49). To this end, we employ the approach described in Ape99 and consider the case of a finite element with $V(\widehat{T}):=\mathcal{C}({\widehat{T}})$ and $\widehat{{P}}:=\mathcal{P}^{k}(\widehat{T})$ (Theorem B). However, one needs stronger assumptions to obtain the optimal estimate. When using finite elements that do not satisfy the assumptions of Theorem B (e.g., $\mathcal{P}^{1}$ -bubble finite element), we have to use Theorem A. In these cases, it may not be possible to obtain optimal order estimates if the shape-regularity condition is violated.

Theorem B

Let $\{\widehat{T},\widehat{{P}},\widehat{\Sigma}\}$ be a finite element with the normed vector space $V(\widehat{T}):=\mathcal{C}({\widehat{T}})$ and $\widehat{{P}}:=\mathcal{P}^{k}(\widehat{T})$ with $k\geq 1$ . Let $I_{\widehat{T}}:V({\widehat{T}})\to\widehat{{P}}$ be a linear operator. Fix $\ell\in\mathbb{N}$ , $m\in\mathbb{N}_{0}$ , and $p,q\in[1,\infty]$ such that $0\leq m\leq\ell\leq k+1$ , $\ell-m\geq 1$ , and assume that the embeddings (46) and (47) with $s:=\ell-m$ hold. Let $\beta$ be a multi-index with $|\beta|=m$ . We set $j:=\dim(\partial^{\beta}\mathcal{P}^{k})$ . Assume that there exist linear functionals $\mathscr{F}_{i}$ , $i=1,\ldots,j$ , such that

	$\displaystyle\mathscr{F}_{i}\in W^{\ell-m,p}(\widehat{T})^{\prime},\quad\forall i=1,\ldots,j,$		(54)
	$\displaystyle\mathscr{F}_{i}(\partial^{\beta}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi}))=0\quad\forall i=1,\ldots,j,\quad\forall\hat{\varphi}\in\mathcal{C}({\widehat{T}}):\ \partial^{\beta}\hat{\varphi}\in W^{\ell-m,p}(\widehat{T}),$		(55)
	$\displaystyle\hat{\eta}\in\mathcal{P}^{k},\quad\mathscr{F}_{i}(\partial^{\beta}\hat{\eta})=0\quad\forall i=1,\ldots,j\quad\Rightarrow\quad\partial^{\beta}\hat{\eta}=0.$		(56)

For any $\hat{\varphi}\in W^{\ell,p}(\widehat{T})\cap\mathcal{C}({\widehat{T}})$ , we set ${\varphi}_{0}:=\hat{\varphi}\circ{\Phi}^{-1}$ . If Assumption 1 is imposed, it holds that

	$\displaystyle\|{\varphi}_{0}-I_{{T}_{0}}{\varphi}_{0}\|_{W^{m,q}({T}_{0})}$
	$\displaystyle\quad\leq C_{1}^{TB}\|T_{0}\|^{\frac{1}{q}-\frac{1}{p}}\left(\frac{H_{T_{0}}}{h_{T_{0}}}\right)^{m}\sum_{\|\gamma\|=\ell-m}\mathscr{H}^{\gamma}\|\partial^{\gamma}{(\varphi_{0}\circ\Phi_{T_{0}})}\|_{W^{m,p}(\Phi_{T_{0}}^{-1}(T_{0}))},$		(57)

where $C_{1}^{TB}$ is a positive constant that is independent of $h_{T_{0}}$ and $H_{T_{0}}$ . Furthermore, if Assumption 1 is not imposed, it holds that

	$\displaystyle\|{\varphi}_{0}-I_{{T}_{0}}{\varphi}_{0}\|_{W^{m,q}({T}_{0})}$
	$\displaystyle\quad\leq C_{2}^{TB}\|T_{0}\|^{\frac{1}{q}-\frac{1}{p}}\left(\frac{H_{T_{0}}}{h_{T_{0}}}\right)^{m}\sum_{\|\gamma\|=\ell-m}{\alpha}^{\gamma}\|\partial_{r^{(0)}}^{\gamma}{\varphi_{0}}\|_{W^{m,p}(T_{0})},$		(58)

where $C_{2}^{TB}$ is a positive constant that is independent of $h_{T_{0}}$ and $H_{T_{0}}$ .

Proof

The introduction of the functionals $\mathscr{F}_{i}$ follows from Ape99 . In fact, under the same assumptions as made in Theorem B, we have (see (Ape99, , Lemma 2.2))

\displaystyle\displaystyle\|\partial^{\beta}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\|_{L^{q}(\widehat{T})}

\displaystyle\leq C^{B}|\partial^{\beta}\hat{\varphi}|_{W^{\ell-m,p}(\widehat{T})},

(59)

where $|\beta|=m$ , $\hat{\varphi}\in\mathcal{C}({\widehat{T}})$ , and $\partial^{\beta}\hat{\varphi}\in W^{\ell-m,p}(\widehat{T})$ .

Inequalities (26), (29), (25), and (59) yield

	$\displaystyle\|\varphi_{0}-{I}_{T_{0}}\varphi_{0}\|_{W^{m,q}({T}_{0})}\leq c\|\varphi-{I}_{T}\varphi\|_{W^{m,q}({T})}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{\frac{1}{q}}\\|\widetilde{A}^{-1}\\|_{2}^{m}\left(\sum_{\|\beta\|=m}(\alpha^{-\beta})^{q}\\|\partial^{\beta}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\\|^{q}_{L^{q}(\widehat{T})}\right)^{1/q}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{\frac{1}{q}}\\|\widetilde{A}^{-1}\\|_{2}^{m}\sum_{\|\beta\|=m}(\alpha^{-\beta})\\|\partial^{\beta}(\hat{\varphi}-I_{\widehat{T}}\hat{\varphi})\\|_{L^{q}(\widehat{T})}$
	$\displaystyle\ \leq c\|\det({A}_{T})\|^{\frac{1}{q}}\\|\widetilde{A}^{-1}\\|_{2}^{m}\sum_{\|\beta\|=m}(\alpha^{-\beta})\|\partial^{\beta}\hat{\varphi}\|_{W^{\ell-m,p}(\widehat{T})}.$		(60)

If Assumption 1 is imposed, then using inequalities (25) and (37) leads to

	$\displaystyle\sum_{\|\beta\|=m}(\alpha^{-\beta})\|\partial^{\beta}\hat{\varphi}\|_{W^{\ell-m,p}(\widehat{T})}$
	$\displaystyle\quad\leq\sum_{\|\gamma\|=\ell-m}\sum_{\|\beta\|=m}(\alpha^{-\beta})\\|\partial^{\beta+\gamma}\hat{\varphi}\\|_{L^{p}(\widehat{T})}$
	$\displaystyle\quad\leq c\|\det({A}_{T})\|^{-\frac{1}{p}}\\|\widetilde{A}\\|^{m}_{2}\sum_{\|\gamma\|=\ell-m}\sum_{\|\beta\|=m}(\alpha^{-\beta})\alpha^{\beta}\sum_{\|\epsilon\|=\|\gamma\|}\mathscr{H}^{\varepsilon}\|\partial^{\epsilon}\varphi\|_{W^{m,p}(T)}$
	$\displaystyle\quad\leq c\|\det({A}_{T})\|^{-\frac{1}{p}}\\|\widetilde{A}\\|^{m}_{2}\sum_{\|\epsilon\|=\ell-m}\mathscr{H}^{\varepsilon}\|\partial^{\epsilon}\varphi\|_{W^{m,p}(T)}.$		(61)

From Eqs. (60) and (61) together with Eqs. (22) and (23), we have the desired estimate (57) using $T=\Phi_{T_{0}}^{-1}(T_{0})$ and $\varphi=\varphi_{0}\circ\Phi_{T_{0}}$ .

If Assumption 1 is not imposed, then an analogous argument using inequality (39) instead of (37) yields estimate (58). ∎

Example 1

Specific finite elements satisfying conditions (54), (55), and (56) are given in ApeDob92 and Ape99 ; see also Section 6.2.

Remark 5

Finite elements that do not satisfy conditions (54), (55), and (56) can be found in (ApeDob92, , Table 3); e.g., the $\mathcal{P}^{1}$ -bubble finite element and the $\mathcal{P}^{3}$ Hermite finite element. In these cases, Theorem A can be applied.

6.2 Examples satisfying conditions (54), (55), and (56) in Theorem B

Corollary 1

Let $\{\widehat{T},\widehat{{P}},\widehat{\Sigma}\}$ be the Lagrange finite element with $V(\widehat{T}):=\mathcal{C}({\widehat{T}})$ and $\widehat{{P}}:=\mathcal{P}^{k}(\widehat{T})$ for $k\geq 1$ . Let $I_{T}:V({T})\to{{P}}$ be the corresponding local Lagrange interpolation operator. Let $m\in\mathbb{N}_{0}$ , $\ell\in\mathbb{N}$ , and $p\in\mathbb{R}$ be such that $0\leq m\leq\ell\leq k+1$ and

	$\displaystyle d=2:\ \begin{cases}p\in(2,\infty]\quad\text{if $m=0$, $\ell=1$},\\ p\in[1,\infty]\quad\text{if $m=0$, $\ell\geq 2$ or $m\geq 1$, $\ell-m\geq 1$},\end{cases}$
	$\displaystyle d=3:\ \begin{cases}p\in\left(\frac{3}{\ell},\infty\right]\quad\text{if $m=0$, $\ell=1,2$},\\ p\in(2,\infty]\quad\text{if $m\geq 1$, $\ell-m=1$},\\ p\in[1,\infty]\quad\text{if $m=0$, $\ell\geq 3$ or $m\geq 1$, $\ell-m\geq 2$}.\end{cases}$

We set $q\in[1,\infty]$ such that $W^{\ell-m,p}(\widehat{T})\hookrightarrow L^{q}(\widehat{T})$ . Then, for all $\hat{\varphi}\in W^{\ell,p}(\widehat{T})$ with ${\varphi}_{0}:=\hat{\varphi}\circ{\Phi}^{-1}$ , we recover Eq. (57) if Assumption 1 is imposed, and Eq. (58) holds.

Furthermore, for any $\hat{\varphi}\in\mathcal{C}(\widehat{T})$ with ${\varphi}_{0}:=\hat{\varphi}\circ{\Phi}^{-1}$ , it holds that

\displaystyle\displaystyle\|{\varphi}_{0}-I_{{T}_{0}}{\varphi}_{0}\|_{L^{\infty}({T}_{0})}\leq c\|{\varphi}_{0}\|_{L^{\infty}(T_{0})}.

Proof

The existence of functionals satisfying Eqs. (54), (55), and (56) is shown in the proof of (Ape99, , Lemma 2.4) for $d=2$ and in the proof of (Ape99, , Lemma 2.6) for $d=3$ . Inequality (59) then holds. This implies that estimates (57) and (58) hold. ∎

Setting $V(T):=\mathcal{C}({{T}})$ , we define the nodal Crouzeix–Raviart interpolation operators as

\displaystyle\displaystyle I_{T}^{CR,S}:V(T)\ni\varphi\mapsto I_{T}^{CR,S}\varphi:=\sum_{i=1}^{d+1}\varphi(x_{F_{i}})\theta_{i}\in\mathcal{P}^{1}.

Corollary 2

Let $\{\widehat{T},\widehat{{P}},\widehat{\Sigma}\}$ be the Crouzeix–Raviart finite element with $V(\widehat{T}):=\mathcal{C}({\widehat{T}})$ and $\widehat{{P}}:=\mathcal{P}^{1}(\widehat{T})$ . Set $I_{T};=I_{T}^{CR,S}$ . Let $m\in\mathbb{N}_{0}$ , $\ell\in\mathbb{N}$ , and $p\in\mathbb{R}$ be such that

	$\displaystyle d=2:\ \begin{cases}p\in(2,\infty]\quad\text{if $m=0$, $\ell=1$},\\ p\in[1,\infty]\quad\text{if $m=0$, $\ell=2$ or $m=1$, $\ell=2$},\end{cases}$
	$\displaystyle d=3:\ \begin{cases}p\in\left(\frac{3}{\ell},\infty\right]\quad\text{if $m=0$, $\ell=1,2$},\\ \displaystyle p\in(2,\infty]\quad\text{if $m=1$, $\ell=2$}.\end{cases}$

Set $q\in[1,\infty]$ such that $W^{\ell-m,p}(\widehat{T})\hookrightarrow L^{q}(\widehat{T})$ . Then, for all $\hat{\varphi}\in W^{\ell,p}(\widehat{T})$ with ${\varphi}_{0}:=\hat{\varphi}\circ{\Phi}^{-1}$ , we recover Eq. (57) if Assumption 1 is imposed, and Eq. (58) holds.

Furthermore, for any $\hat{\varphi}\in\mathcal{C}(\widehat{T})$ with ${\varphi}_{0}:=\hat{\varphi}\circ{\Phi}^{-1}$ , it holds that

\displaystyle\displaystyle\|{\varphi}_{0}-I_{{T}_{0}}{\varphi}_{0}\|_{L^{\infty}({T}_{0})}\leq c\|{\varphi}_{0}\|_{L^{\infty}(T_{0})}.

Proof

For $k=1$ , we only introduce functionals $\mathscr{F}_{i}$ satisfying Eqs. (54), (55), and (56) in Theorem B for each $\ell$ and $m$ .

Let $m=0$ . From the Sobolev embedding theorem, we have $W^{\ell,p}(\widehat{T})\subset\mathcal{C}^{0}(\widehat{T})$ with $1\textless p\leq\infty$ , $d\textless\ell p$ or $p=1$ , $d\leq\ell$ . Under this condition, we use

\displaystyle\displaystyle\mathscr{F}_{i}(\hat{\varphi}):=\hat{\varphi}(\hat{x}_{\widehat{F}_{i}}),\quad\hat{\varphi}\in W^{\ell,p}(\widehat{T}),\quad i=1,\ldots,d+1.

Let $d=2$ and $m=1$ ( $\ell=2$ ). We set $\beta=(1,0)$ . Then, we have that $j=\dim(\partial^{\beta}\mathcal{P}^{1})=1$ . We consider a functional

\displaystyle\displaystyle\mathscr{F}_{1}(\hat{\varphi}):=\int_{0}^{\frac{1}{2}}\hat{\varphi}(\hat{x}_{1},1/2)d\hat{x}_{1},\quad\hat{\varphi}\in W^{2,p}(\widehat{T}).

By an analogous argument, we can set a functional for the case $\beta=(0,1)$ .

Let $d=3$ and $m=1$ ( $\ell=2$ ). We first consider Type (i) in Section 2.1.2. That is, the reference element is $\widehat{T}=\mathop{\mathrm{conv}}\{0,e_{1},e_{2},e_{3}\}$ . Here, $e_{1},\ldots,e_{3}\in\mathbb{R}^{3}$ form the canonical basis. We set $\beta=(1,0,0)$ and consider the functional

\displaystyle\displaystyle\mathscr{F}_{1}(\hat{\varphi}):=\int_{0}^{\frac{1}{3}}\hat{\varphi}(\hat{x}_{1},1/3,1/3)d\hat{x}_{1},\quad\hat{\varphi}\in W^{2,p}(\widehat{T}).

We now consider Type (ii) in Section 2.1.2. That is, the reference element is $\widehat{T}=\mathop{\mathrm{conv}}\{0,e_{1},e_{1}+e_{2},e_{3}\}$ . We set $\beta=(1,0,0)$ and consider the functional

\displaystyle\displaystyle\Phi_{1}(\hat{\varphi}):=\int_{\frac{1}{3}}^{\frac{2}{3}}\hat{\varphi}(\hat{x}_{1},1/3,1/3)d\hat{x}_{1},\quad\hat{\varphi}\in W^{2,p}(\widehat{T}).

By an analogous argument, we can set functionals for cases $\beta=(0,1,0),(0,0,1)$ .

When $m=\ell=0$ and $p=\infty$ , we can easily check that

\displaystyle\displaystyle\|\hat{\varphi}-I_{\widehat{T}}^{CR,S}\hat{\varphi}\|_{L^{\infty}(\widehat{T})}

\displaystyle\leq c\|\hat{\varphi}\|_{L^{\infty}(\widehat{T})}.

∎

7 Concluding remarks

As our concluding remarks, we identify several topics related to the results described in this paper.

7.1 Good elements or not for $d=2,3$ ?

In this subsection, we consider good elements on meshes. Here, we define “good elements” on meshes as those for which there exists some $\gamma_{0}\textgreater 0$ satisfying Eq. (7). We treat a “Right-angled triangle,” “Blade,” and “Dagger” for $d=2$ , and a “Spire,” “Spear,” “Spindle,” “Spike,” “Splinter,” and ”Sliver” for $d=3$ , as introduced in Cheetal00 . We present the quantities $\alpha_{\max}/\alpha_{\min}$ and $H_{T_{0}}/h_{T_{0}}$ for these elements.

7.1.1 Isotropic mesh

We consider the following condition. There exists a constant $\gamma_{1}\textgreater 0$ such that, for any $\mathbb{T}_{h}\in\{\mathbb{T}_{h}\}$ and any simplex $T_{0}\in\mathbb{T}_{h}$ , we have

\displaystyle\displaystyle|T_{0}|\geq\gamma_{1}h_{T_{0}}^{d}.

(62)

Condition (62) is equivalent to the shape-regularity condition; see (BraKorKri08, , Theorem 1).

If geometric condition (62) is satisfied, it holds that

\displaystyle\displaystyle\frac{H_{T_{0}}}{h_{T_{0}}}\leq\frac{h_{T_{0}}^{d}}{|T_{0}|}\leq\frac{1}{\gamma_{1}},\quad\frac{\alpha_{\max}}{\alpha_{\min}}\leq c\frac{h_{T_{0}}}{\alpha_{2}}\leq c\frac{h_{T_{0}}^{d}}{|T_{0}|}\leq\frac{c}{\gamma_{1}}.

If $p=q$ in Theorem A, one can obtain the optimal order $h_{T_{0}}^{\ell+1-m}$ . In this case, elements satisfying geometric condition (62) are “good.”

7.1.2 Anisotropic mesh: two-dimensional case

Let $S_{0}\subset\mathbb{R}^{2}$ be a triangle. Let $0\textless s\ll 1$ , $s\in\mathbb{R}$ , and $\varepsilon,\delta,\gamma\in\mathbb{R}$ . A dagger has one short edge and a blade has no short edge.

Example 2 (Right-angled triangle)

Let $S_{0}\subset\mathbb{R}^{2}$ be the simplex with vertices $x_{1}:=(0,0)^{T}$ , $x_{2}:=(s,0)^{T}$ , and $x_{3}:=(0,s^{\varepsilon})^{T}$ with $1\textless\varepsilon$ . Then, we have that $\alpha_{1}=s$ and $\alpha_{2}=s^{\varepsilon}$ ; i.e.,

\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}

\displaystyle\leq s^{1-\varepsilon}\to\infty\quad\text{as $s\to 0$},\quad\frac{H_{S_{0}}}{h_{S_{0}}}=2.

In this case, the element $S_{0}$ is “good.”

Example 3 (Dagger)

Let $S_{0}\subset\mathbb{R}^{2}$ be the simplex with vertices $x_{1}:=(0,0)^{T}$ , $x_{2}:=(s,0)^{T}$ , and $x_{3}:=(s^{\delta},s^{\varepsilon})^{T}$ with $1\textless\varepsilon\textless\delta$ . Then, we have that $\alpha_{1}=\sqrt{(s-s^{\delta})^{2}+s^{2\varepsilon}}$ and $\alpha_{2}=\sqrt{s^{2\delta}+s^{2\varepsilon}}$ ; i.e.,

	$\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}$	$\displaystyle=\frac{\sqrt{(s-s^{\delta})^{2}+s^{2\varepsilon}}}{\sqrt{s^{2\delta}+s^{2\varepsilon}}}\leq cs^{1-\varepsilon}\to\infty\quad\text{as $s\to 0$},$
	$\displaystyle\frac{H_{S_{0}}}{h_{S_{0}}}$	$\displaystyle=\frac{\sqrt{(s-s^{\delta})^{2}+s^{2\varepsilon}}\sqrt{s^{2\delta}+s^{2\varepsilon}}}{\frac{1}{2}s^{1+\varepsilon}}\leq c.$

In this case, the element $S_{0}$ is “good.”

Remark 6

In the above examples, $\alpha_{2}\approx\alpha_{2}t=\mathscr{H}_{2}$ holds. That is, the good element $S_{0}\subset\mathbb{R}^{2}$ satisfies conditions such as $\alpha_{2}\approx\alpha_{2}t=\mathscr{H}_{2}$ .

Example 4 (Blade)

Let $S_{0}\subset\mathbb{R}^{2}$ be the simplex with vertices $x_{1}:=(0,0)^{T}$ , $x_{2}:=(2s,0)^{T}$ , and $x_{3}:=(s,s^{\varepsilon})^{T}$ with $1\textless\varepsilon$ . Then, we have that $\alpha_{1}=\alpha_{2}=\sqrt{s^{2}+s^{2\varepsilon}}$ ; i.e.,

\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}=1,\quad\frac{H_{S_{0}}}{h_{S_{0}}}=\frac{s^{2}+s^{2\varepsilon}}{s^{1+\varepsilon}}\to\infty\quad\text{as $s\to 0$}.

In this case, the element $S_{0}$ is “not good.”

Example 5 (Dagger)

Let $S_{0}\subset\mathbb{R}^{2}$ be the simplex with vertices $x_{1}:=(0,0)^{T}$ , $x_{2}:=(s,0)^{T}$ , and $x_{3}:=(s^{\delta},s^{\varepsilon})^{T}$ with $1\textless\delta\textless\varepsilon$ . Then, we have that $\alpha_{1}=\sqrt{(s-s^{\delta})^{2}+s^{2\varepsilon}}$ and $\alpha_{2}=\sqrt{s^{2\delta}+s^{2\varepsilon}}$ ; i.e.,

	$\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}$	$\displaystyle=\frac{\sqrt{(s-s^{\delta})^{2}+s^{2\varepsilon}}}{\sqrt{s^{2\delta}+s^{2\varepsilon}}}\leq cs^{1-\delta}\to\infty\quad\text{as $s\to 0$},$
	$\displaystyle\frac{H_{S_{0}}}{h_{S_{0}}}$	$\displaystyle=\frac{\sqrt{(s-s^{\delta})^{2}+s^{2\varepsilon}}\sqrt{s^{2\delta}+s^{2\varepsilon}}}{\frac{1}{2}s^{1+\varepsilon}}\leq cs^{\delta-\varepsilon}\to\infty\quad\text{as $s\to 0$}.$

In this case, the element $S_{0}$ is “not good.”

Refer to caption — Figure 1: Example 2: Right-angled triangle

7.1.3 Anisotropic mesh: three-dimensional case

Example 6

Let $T_{0}\subset\mathbb{R}^{3}$ be a tetrahedron. Let $S_{0}$ be the base of $T_{0}$ ; i.e., $S_{0}=\triangle x_{1}x_{2}x_{3}$ . Recall that

\displaystyle\displaystyle\frac{H_{T_{0}}}{h_{T_{0}}}=\frac{\alpha_{1}\alpha_{2}\alpha_{3}}{|T_{0}|}=\frac{\alpha_{1}\alpha_{2}}{\frac{1}{2}\alpha_{1}\alpha_{2}t_{1}}\frac{\alpha_{3}}{\frac{1}{3}\alpha_{3}t_{2}}\leq\frac{H_{S_{0}}}{h_{S_{0}}}\frac{\alpha_{3}}{\frac{1}{3}\mathscr{H}_{3}}.

(63)

If the triangle $S_{0}$ is “not good,” such as in Examples 4 and 5, the quantity in Eq. (63) may diverge. In the following, we consider the case in which the triangle $S_{0}$ is “good.”

Assume that there exists a positive constant $M$ such that $\frac{H_{S_{0}}}{h_{S_{0}}}\leq M$ . For simplicity, we set $x_{1}:=(0,0,0)^{T}$ , $x_{2}:=(2s,0,0)^{T}$ , and $x_{3}:=(2s-\sqrt{4s^{2}-s^{2\gamma}},s^{\gamma},0)^{T}$ with $1\textless\gamma$ . Then,

\displaystyle\displaystyle\alpha_{1}=2s,\ \alpha_{2}=\sqrt{\frac{4s^{2\gamma}}{2+\sqrt{4-s^{2\gamma-2}}}},

and because $\alpha_{\max}\approx cs$ ,

\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}

\displaystyle\leq\frac{cs}{\alpha_{2}}\leq cs^{1-\gamma}\to\infty\quad\text{as $s\to 0$}.

If we set $x_{4}:=(s,0,s^{\varepsilon})^{T}$ with $1\textless\varepsilon$ , the triangle $\triangle x_{1}x_{2}x_{4}$ is the blade (Example 4). Then,

\displaystyle\displaystyle\alpha_{3}=\sqrt{s^{2}+s^{2\varepsilon}}.

Thus, we have

\displaystyle\displaystyle\frac{H_{T_{0}}}{h_{T_{0}}}\leq c\frac{s^{2s+\gamma}}{s^{1+\gamma+\varepsilon}}\leq cs^{1-\varepsilon}\to\infty\quad\text{as $s\to 0$}.

In this case, the element $T_{0}$ is “not good.”

If we set $x_{4}:=(s^{\delta},0,s^{\varepsilon})^{T}$ with $1\textless\delta\textless\varepsilon\textless\gamma$ , the triangle $\triangle x_{1}x_{2}x_{4}$ is the dagger (Example 5). Then,

\displaystyle\displaystyle\alpha_{3}=\sqrt{s^{2\delta}+s^{2\varepsilon}}.

Thus, we have

\displaystyle\displaystyle\frac{H_{T_{0}}}{h_{T_{0}}}\leq c\frac{s^{1+\gamma+\delta}}{s^{1+\gamma+\varepsilon}}\leq cs^{\delta-\varepsilon}\to\infty\quad\text{as $s\to 0$}.

In this case, the element $T_{0}$ is “not good.”

If we set $x_{4}:=(s^{\delta},0,s^{\varepsilon})^{T}$ with $1\textless\varepsilon\textless\delta\textless\gamma$ , the triangle $\triangle x_{1}x_{2}x_{4}$ is the dagger (Example 3). Then,

\displaystyle\displaystyle\alpha_{3}=\sqrt{s^{2\delta}+s^{2\varepsilon}}.

Thus, we have

\displaystyle\displaystyle\frac{H_{T_{0}}}{h_{T_{0}}}\leq c\frac{s^{1+\gamma+\varepsilon}}{s^{1+\gamma+\varepsilon}}\leq c.

In this case, the element $T_{0}$ is “good” and $\alpha_{3}\approx\alpha_{3}t_{2}=\mathscr{H}_{3}$ holds.

Example 7

In Cheetal00 , the spire has a cycle of three daggers among its four triangles. The splinter has four daggers. The spear and spike have two daggers and two blades as triangles. The spindle has four blades as triangles.

Remark 7

The above examples reveal that the good element $T_{0}\subset\mathbb{R}^{3}$ satisfies conditions such as $\alpha_{2}\approx\alpha_{2}t_{1}=\mathscr{H}_{2}$ and $\alpha_{3}\approx\alpha_{3}t_{2}=\mathscr{H}_{3}$ .

Example 8

Using an element $T_{0}$ called a Sliver, we compare the three quantities $\frac{h_{T_{0}}^{3}}{|{T_{0}}|}$ , $\frac{H_{T_{0}}}{h_{T_{0}}}$ , and $\frac{R_{3}}{h_{T_{0}}}$ , where the parameter $R_{3}$ denotes the circumradius of $T_{0}$ .

Let $T_{0}\subset\mathbb{R}^{3}$ be the simplex with vertices $x_{1}:=(s^{\varepsilon_{2}},0,0)^{T}$ , $x_{2}:=(-s^{\varepsilon_{2}},0,0)^{T}$ , $x_{3}:=(0,-s,s^{\varepsilon_{1}})^{T}$ , and $x_{4}:=(0,s,s^{\varepsilon_{1}})^{T}$ ( $\varepsilon_{1},\varepsilon_{2}\textgreater 1$ ), where $s:=\frac{1}{N}$ , $N\in\mathbb{N}$ . Let $L_{i}$ ( $1\leq i\leq 6$ ) be the edges of $T_{0}$ with $\alpha_{\min}=L_{1}\leq L_{2}\leq\cdots\leq L_{6}=h_{T_{0}}$ . Recall that $\alpha_{\max}\approx h_{T_{0}}$ and

\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}\leq c\frac{L_{6}}{L_{1}},\quad\frac{H_{T_{0}}}{h_{T_{0}}}=\frac{L_{1}L_{2}}{|T_{0}|}h_{T_{0}}.

Table 1:

h_{T_{0}}^{3}/{|T_{0}|}

H_{T_{0}}/h_{T_{0}}

, and

R_{3}/h_{T_{0}}

(

\varepsilon_{1}=1.5

\varepsilon_{2}=1.0

)

$N$	$s$	$L_{6}/L_{1}$	$h_{T_{0}}^{3}/{\|T_{0}\|}$	$H_{T_{0}}/h_{T_{0}}$	$R_{3}/h_{T_{0}}$
32	3.1250e-02	1.4033	6.7882e+01	3.4471e+01	5.0195e-01
64	1.5625e-02	1.4087	9.6000e+01	4.8375e+01	5.0098e-01
128	7.8125e-03	1.4115	1.3576e+02	6.8147e+01	5.0049e-01

Table 2:

h_{T_{0}}^{3}/{|T_{0}|}

H_{T_{0}}/h_{T_{0}}

, and

R_{3}/h_{T_{0}}

(

\varepsilon_{1}=1.0

\varepsilon_{2}=1.5

)

$N$	$s$	$L_{6}/L_{1}$	$h_{T_{0}}^{3}/{\|T_{0}\|}$	$H_{T_{0}}/h_{T_{0}}$	$R_{3}/h_{T_{0}}$
32	3.1250e-02	5.6569	6.7882e+01	8.5513	5.0006e-01
64	1.5625e-02	8.0000	9.6000e+01	8.5184	5.0002e-01
128	7.8125e-03	1.1314e+01	1.3576e+02	8.5018	5.0000e-01

Table 3:

h_{T_{0}}^{3}/{|T_{0}|}

H_{T_{0}}/h_{T_{0}}

, and

R_{3}/h_{T_{0}}

(

\varepsilon_{1}=1.5

\varepsilon_{2}=1.5

)

$N$	$s$	$L_{6}/L_{1}$	$h_{T_{0}}^{3}/{\|T_{0}\|}$	$H_{T_{0}}/h_{T_{0}}$	$R_{3}/h_{T_{0}}$
32	3.1250e-02	5.6569	3.8400e+02	3.4986e+01	1.4170
64	1.5625e-02	8.0000	7.6800e+02	4.8744e+01	2.0010
128	7.8125e-03	1.1314e+01	1.5360e+03	6.8411e+01	2.8288

In Table 1, the angle between $\triangle x_{1}x_{2}x_{3}$ and $\triangle x_{1}x_{2}x_{4}$ tends to $\pi$ as $s\to 0$ , and the simplex $T_{0}$ is “not good.” In Table 2, the angle between $\triangle x_{1}x_{3}x_{4}$ and $\triangle x_{2}x_{3}x_{4}$ tends to $0$ as $s\to 0$ , and the simplex $T_{0}$ is “good.” In Table 3, from the numerical results, the simplex $T_{0}$ is “not good.”

7.2 Effect of the quantity $|T_{0}|^{\frac{1}{q}-\frac{1}{p}}$ on the interpolation error estimates for $d=2,3$

We now consider the effect of the factor $|T_{0}|^{\frac{1}{q}-\frac{1}{p}}$ .

7.2.1 Case in which $q\textgreater p$

When $q\textgreater p$ , the factor may affect the convergence order. In particular, the interpolation error estimate may diverge on anisotropic mesh partitions.

Let $T_{0}\subset\mathbb{R}^{2}$ be the triangle with vertices $x_{1}:=(0,0)^{T}$ , $x_{2}:=(s,0)^{T}$ , $x_{3}:=(0,s^{\varepsilon})^{T}$ for $0\textless s\ll 1$ , $\varepsilon\geq 1$ , $s\in\mathbb{R}$ , and $\varepsilon\in\mathbb{R}$ ; see Figure 2. Then,

\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}=s^{1-\varepsilon},\quad|T_{0}|=\frac{1}{2}s^{1+\varepsilon}.

Let $k=1$ , $\ell=2$ , $m=1$ , $q=2$ , and $p\in(1,2)$ . Then, $W^{1,p}({T}_{0})\hookrightarrow L^{2}({T}_{0})$ and Theorem B lead to

\displaystyle\displaystyle|{\varphi}_{0}-I_{{T}_{0}}{\varphi}_{0}|_{H^{1}({T}_{0})}\leq cs^{-(1+\varepsilon)\frac{2-p}{2p}}\left(s\left|\frac{\partial\varphi_{0}}{\partial x_{1}}\right|_{W^{1,p}({T}_{0})}+s^{\varepsilon}\left|\frac{\partial\varphi_{0}}{\partial x_{2}}\right|_{W^{1,p}({T}_{0})}\right).

When $\varepsilon=1$ (i.e., an isotropic element), we obtain

\displaystyle\displaystyle|{\varphi}_{0}-I_{{T}_{0}}{\varphi}_{0}|_{H^{1}({T}_{0})}\leq ch_{T_{0}}^{\frac{2(p-1)}{p}}|\varphi_{0}|_{W^{2,p}(T_{0})},\quad\frac{2(p-1)}{p}\textgreater 0.

However, when $\varepsilon\textgreater 1$ (i.e., an anisotropic element), the estimate may diverge as $s\to 0$ . Therefore, if $q\textgreater p$ , the convergence order of the interpolation operator may deteriorate.

7.2.2 Case in which $q\textless p$

We consider Theorem B. Let $I_{T_{0}}^{L}:\mathcal{C}^{0}(T_{0})\to\mathcal{P}^{k}$ ( $k\in\mathbb{N}$ ) be the local Lagrange interpolation operator. Let ${\varphi}_{0}\in W^{\ell,\infty}(T_{0})$ be such that $\ell\in\mathbb{N}$ , $2\leq\ell\leq k+1$ . Then, for any $m\in\{0,\ldots,\ell-1\}$ and $q\in[1,\infty]$ , it holds that

\displaystyle\displaystyle|{\varphi}_{0}-I_{{T}_{0}}^{L}{\varphi}_{0}|_{W^{m,q}({T}_{0})}\leq c|T_{0}|^{\frac{1}{q}}\left(\frac{H_{T_{0}}}{h_{T_{0}}}\right)^{m}\sum_{|\gamma|=\ell-m}\mathscr{H}^{\gamma}|\partial^{\gamma}{(\varphi_{0}\circ\Phi_{T_{0}})}|_{W^{m,\infty}(\Phi_{T_{0}}^{-1}(T_{0}))}.

(64)

Therefore, the convergence order is improved by $|T_{0}|^{\frac{1}{q}}$ .

We can perform some numerical tests to confirm this. Let $k=1$ and

\displaystyle\displaystyle\varphi_{0}(x,y,z):=x^{2}+\frac{1}{4}y^{2}+z^{2}.

(I)

Let $T_{0}\subset\mathbb{R}^{3}$ be the simplex with vertices $x_{1}:=(0,0,0)^{T}$ , $x_{2}:=(s,0,0)^{T}$ , $x_{3}:=(0,s^{\varepsilon},0)^{T}$ , and $x_{4}:=(0,0,s^{\delta})^{T}$ ( $1\textless\delta\leq\varepsilon$ ), and $0\textless s\ll 1$ , $s\in\mathbb{R}$ . Then, we have that $\alpha_{1}=\sqrt{s^{2}+s^{2\varepsilon}}$ , $\alpha_{2}=s^{\varepsilon}$ , and $\alpha_{3}:=\sqrt{s^{2\varepsilon}+s^{2\delta}}$ ; i.e.,

\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}\leq cs^{1-\varepsilon},\quad\frac{H_{T_{0}}}{h_{T_{0}}}\leq c.

From Eq. (64) with $m=1$ , $\ell=2$ , and $q=2$ , because $|T_{0}|\approx s^{1+\varepsilon+\delta}$ , we have the estimate

\displaystyle|{\varphi}_{0}-I_{{T}_{0}}^{L}{\varphi}_{0}|_{H^{1}({T}_{0})}\leq ch_{T_{0}}^{\frac{3+\varepsilon+\delta}{2}}.

Computational results for $\varepsilon=3.0$ and $\delta=2.0$ are presented in Table 4.

Table 4: Error of the local interpolation operator (

\varepsilon=3.0,\delta=2.0

)

$N$	$s$	$Err_{s}^{3.0}(H^{1})$	$r$
64	1.5625e-02	2.4336e-08
128	7.8125e-03	1.5209e-09	4.00
256	3.9062e-03	9.5053e-11	4.00

(II)

Let $T_{0}\subset\mathbb{R}^{3}$ be the simplex with vertices $x_{1}:=(0,0,0)^{T}$ , $x_{2}:=(s,0,0)^{T}$ , $x_{3}:=(s/2,s^{\varepsilon},0)^{T}$ , and $x_{4}:=(0,0,s)^{T}$ ( $1\textless\varepsilon\leq 6$ ) and $0\textless s\ll 1$ , $s\in\mathbb{R}$ . Then, we have that $\alpha_{1}=s$ , $\alpha_{2}=\sqrt{s^{2}/4+s^{2\varepsilon}}$ , and $\alpha_{3}:=s$ ; i.e.,

\displaystyle\displaystyle\frac{\alpha_{\max}}{\alpha_{\min}}=\frac{s}{\sqrt{s^{2}/4+s^{2\varepsilon}}}\leq c,\quad\frac{H_{T_{0}}}{h_{T_{0}}}\leq cs^{1-\varepsilon}.

From Eq. (64) with $m=1$ , $\ell=2$ , and $q=2$ , because $|T_{0}|\approx s^{2+\varepsilon}$ , we have the estimate

\displaystyle|{\varphi}_{0}-I_{{T}_{0}}^{L}{\varphi}_{0}|_{H^{1}({T}_{0})}\leq ch_{T_{0}}^{3-\frac{\varepsilon}{2}}.

Computational results for $\varepsilon=3.0,6.0$ are presented in Table 5.

Table 5: Error of the local interpolation operator (

\varepsilon=3.0,6.0

)

$N$	$s$	$Err_{s}^{3.0}(H^{1})$	$r$	$Err_{s}^{6.0}(H^{1})$	$r$
64	1.5625e-02	1.9934e-04		1.0206e-01
128	7.8125e-03	7.0477e-05	1.50	1.0206e-01	0
256	3.9062e-03	2.4917e-05	1.50	1.0206e-01	0

7.3 Inverse inequalities

This section presents some limited results for the inverse inequalities. The results are only stated; the proofs can be found in Ish22 .

Lemma 8

Let $\widehat{P}:=\mathcal{P}^{k}$ with $k\in\mathbb{N}$ . If Assumption 1 is imposed, there exist positive constants $C_{i}^{IV,d}$ , $i=1,\ldots,d$ , independent of $h_{T}$ and ${T}$ , such that, for all ${\varphi}_{h}\in{P}=\{\hat{\varphi}_{h}\circ{\Phi}_{T}^{-1};\ \hat{\varphi}_{h}\in\widehat{P}\}$ ,

\displaystyle\displaystyle\left\|\frac{\partial\varphi_{h}}{\partial x_{i}}\right\|_{L^{q}(T)}

\displaystyle\leq C_{i}^{IV,d}|T|^{\frac{1}{q}-\frac{1}{p}}\frac{1}{\mathscr{H}_{i}}\|\varphi_{h}\|_{L^{p}(T)}.\quad i=1,\ldots,d.

(65)

Remark 8

If Assumption 1 is not imposed, estimate (65) for $i=3$ is

\displaystyle\displaystyle\left\|\frac{\partial\varphi_{h}}{\partial x_{3}}\right\|_{L^{q}(T)}

\displaystyle\leq C_{3}^{IV,3}|T|^{\frac{1}{q}-\frac{1}{p}}\frac{H_{T}}{h_{T}}\frac{1}{\mathscr{H}_{2}}\|\varphi_{h}\|_{L^{p}(T)}.

(66)

This may not be sharp. We leave further arguments for future work.

Theorem D

Let $\widehat{P}:=\mathcal{P}^{k}$ with $k\in\mathbb{N}_{0}$ . Let $\gamma=(\gamma_{1},\ldots,\gamma_{d})\in\mathbb{N}_{0}^{d}$ be a multi-index such that $0\leq|\gamma|\leq k$ . If Assumption 1 is imposed, there exists a positive constant $C^{IVC}$ , independent of $h_{T}$ and ${T}$ , such that, for all ${\varphi}_{h}\in{P}=\{\hat{\varphi}_{h}\circ{\Phi}_{T}^{-1};\ \hat{\varphi}_{h}\in\widehat{P}\}$ ,

\displaystyle\displaystyle\|\partial^{\gamma}\varphi_{h}\|_{L^{q}(T)}\leq C^{IVC}|T|^{\frac{1}{q}-\frac{1}{p}}\mathscr{H}^{-\gamma}\|{\varphi}_{h}\|_{L^{p}({T})}.

(67)

Acknowledgements.

We would like to thank the anonymous referees and the editor of the journal for the valuable comments.

References

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$\displaystyle\displaystyle\\|\widehat{{A}}^{(d)}\\|_{2}$	$\displaystyle\leq h_{T},\quad\\|\widehat{{A}}^{(d)}\\|_{2}\\|(\widehat{{A}}^{(d)})^{-1}\\|_{2}=\frac{\max\{\alpha_{1},\cdots,\alpha_{d}\}}{\min\{\alpha_{1},\cdots,\alpha_{d}\}},$	(22a)
$\displaystyle\\|\widetilde{{A}}\\|_{2}$	$\displaystyle\leq\begin{cases}\sqrt{2}\quad\text{if $d=2$},\\ 2\quad\text{if $d=3$},\end{cases}\quad\\|\widetilde{{A}}\\|_{2}\\|\widetilde{{A}}^{-1}\\|_{2}\leq\begin{cases}\frac{\alpha_{1}\alpha_{2}}{\|T\|}=\frac{H_{T}}{h_{T}}\quad\text{if $d=2$},\\ \frac{2}{3}\frac{\alpha_{1}\alpha_{2}\alpha_{3}}{\|T\|}=\frac{2}{3}\frac{H_{T}}{h_{T}}\quad\text{if $d=3$},\end{cases}$	(22b)
$\displaystyle\\|A_{T_{0}}\\|_{2}$	$\displaystyle=1,\quad\\|A_{T_{0}}^{-1}\\|_{2}=1.$	(22c)

	$\displaystyle\displaystyle\|\varphi\|_{W^{s,p}(T)}$	$\displaystyle\leq c\\|{A}_{T_{0}}\\|_{2}^{s}\|\det({A}_{T_{0}})\|^{-\frac{1}{p}}\|\varphi_{0}\|_{W^{s,p}(T_{0})},$		(27)
	$\displaystyle\|\varphi_{0}\|_{W^{s,p}(T_{0})}$	$\displaystyle\leq c\\|{A}_{T_{0}}^{-1}\\|_{2}^{s}\|\det({A}_{T_{0}})\|^{\frac{1}{p}}\|\varphi\|_{W^{s,p}(T)}$		(28)

$\displaystyle\displaystyle\\|\partial^{\gamma}\tilde{\varphi}\\|_{L^{p}(\widetilde{T})}$	$\displaystyle\leq\|\tilde{\varphi}\|_{W^{\|\gamma\|,p}(\widetilde{T})}$
	$\displaystyle=\|\det(\widehat{A}^{(d)})\|^{\frac{1}{p}}\left(\sum_{\|\beta\|=\|\gamma\|}(\alpha^{-\beta})^{p}\\|\partial^{\beta}\hat{\varphi}\\|^{p}_{L^{p}(\widehat{T})}\right)^{1/p}$
	$\displaystyle\leq\left(\sup_{1\leq p}\|\det(\widehat{A}^{(d)})\|^{\frac{1}{p}}\right)\sum_{\|\beta\|=\|\gamma\|}\alpha^{-\beta}\\|\partial^{\beta}\hat{\varphi}\\|_{L^{p}(\widehat{T})}$
	$\displaystyle\leq c\left(\sup_{1\leq p}\|\det(\widehat{A}^{(d)})\|^{\frac{1}{p}}\right)\sum_{\|\beta\|=\|\gamma\|}\alpha^{-\beta}\\|\partial^{\beta}\hat{\varphi}\\|_{L^{\infty}(\widehat{T})}\textless\infty$	(34)

	$\displaystyle\displaystyle\|\partial^{\beta+\gamma}\hat{\varphi}\|$	$\displaystyle=\left\|\frac{\partial^{\ell}\hat{\varphi}}{\partial\hat{x}_{1}^{\beta_{1}}\cdots\partial\hat{x}_{d}^{\beta_{d}}\partial\hat{x}_{1}^{\gamma_{1}}\cdots\partial\hat{x}_{d}^{\gamma_{d}}}\right\|$
		$\displaystyle\leq c\alpha^{\beta}\\|\widetilde{A}\\|_{\max}^{\|\beta\|}\underbrace{\sum_{i_{1}^{(1)}=1}^{d}\cdots\sum_{i_{\beta_{1}}^{(1)}=1}^{d}}_{\beta_{1}\text{times}}\cdots\underbrace{\sum_{i_{1}^{(d)}=1}^{d}\cdots\sum_{i_{\beta_{d}}^{(d)}=1}^{d}}_{\beta_{d}\text{times}}\underbrace{\sum_{j_{1}^{(1)}=1}^{d}\cdots\sum_{j_{\gamma_{1}}^{(1)}=1}^{d}}_{\gamma_{1}\text{times}}\cdots\underbrace{\sum_{j_{1}^{(d)}=1}^{d}\cdots\sum_{j_{\gamma_{d}}^{(d)}=1}^{d}}_{\gamma_{d}\text{times}}$
		$\displaystyle\underbrace{\mathscr{H}_{j_{1}^{(1)}}\cdots\mathscr{H}_{j_{\varepsilon_{1}}^{(1)}}}_{\gamma_{1}\text{times}}\cdots\underbrace{\mathscr{H}_{j_{1}^{(d)}}\cdots\mathscr{H}_{j_{\varepsilon_{d}}^{(d)}}}_{\gamma_{d}\text{times}}$
		$\displaystyle\Biggl{\|}\underbrace{\frac{\partial^{\beta_{1}}}{\partial{x}_{i_{1}^{1)}}\cdots\partial{x}_{i_{\beta_{1}}^{(1)}}}}_{\beta_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\beta_{d}}}{\partial{x}_{i_{1}^{(d)}}\cdots\partial{x}_{i_{\beta_{d}}^{(d)}}}}_{\beta_{d}\text{times}}\underbrace{\frac{\partial^{\gamma_{1}}}{\partial{x}_{j_{1}^{(1)}}\cdots\partial{x}_{j_{\gamma_{1}}^{(1)}}}}_{\gamma_{1}\text{times}}\cdots\underbrace{\frac{\partial^{\gamma_{d}}}{\partial{x}_{j_{1}^{(d)}}\cdots\partial{x}_{j_{\gamma_{d}}^{(d)}}}}_{\gamma_{d}\text{times}}\varphi\Biggr{\|}$
		$\displaystyle\leq c\alpha^{\beta}\\|\widetilde{A}\\|_{\max}^{\|\beta\|}\sum_{\|\delta\|=\|\beta\|}\sum_{\|\varepsilon\|=\|\gamma\|}\mathscr{H}^{\varepsilon}\|{\partial^{\delta}\partial^{\varepsilon}\varphi}\|.$

	$\displaystyle\displaystyle\left\|\frac{\partial\hat{\varphi}}{\partial\hat{x}_{i}}\right\|$	$\displaystyle=\left\|\sum_{i_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(0,1)}=1}^{d}\alpha_{i}[\widetilde{A}]_{i_{1}^{(1)}i}[A_{T_{0}}]_{i_{1}^{(0,1)}i_{1}^{(1)}}\frac{\partial\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}}\right\|$
		$\displaystyle=\alpha_{i}\left\|\sum_{i_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(0,1)}=1}^{d}[A_{T_{0}}]_{i_{1}^{(0,1)}i_{1}^{(1)}}(r_{i})_{i_{1}^{(1)}}\frac{\partial\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}}\right\|=\alpha_{i}\left\|\frac{\partial\varphi_{0}}{\partial r_{i}^{(0)}}\right\|$
		$\displaystyle\leq\alpha_{i}\\|\widetilde{A}\\|_{\max}\\|A_{T_{0}}\\|_{\max}\sum_{i_{1}^{(1)}=1}^{d}\sum_{i_{1}^{(0,1)}=1}^{d}\left\|\frac{\partial\varphi_{0}}{\partial x_{i_{1}^{(0,1)}}^{(0)}}\right\|,$

Anisotropic interpolation error estimates using a new geometric parameter

Abstract

Keywords:

MSC:

1 Introduction

2 Strategy for constructing anisotropic interpolation theory

2.1 Standard positions of simplices

2.1.1 Two-dimensional case

2.1.2 Three-dimensional case

Condition 1 (Case in which d=2d=2)

Condition 2 (Case in which d=3d=3)

Remark 1

Assumption 1

2.2 Affine mappings

Definition 1 (Affine mappings)

2.3 Finite element generation

Proposition 1

Proof

2.4 New parameters

Definition 2

Lemma 1

Proof

Remark 2

Theorem 1

Proof

Lemma 2

Proof

3 Scaling argument

Lemma 3

Proof

Lemma 4

Proof

Definition 3

Definition 4

Note 1

Note 2

Lemma 5

Proof

Remark 3

Note 3

Lemma 6

Proof

4 Remarks on anisotropic interpolation analysis

Lemma 7

5 Classical interpolation error estimates

Theorem

Proof

Remark 4

Theorem A

Proof

6 Anisotropic interpolation error estimates

6.1 Main theorem

Theorem B

Proof

Example 1

Remark 5

6.2 Examples satisfying conditions (54), (55), and (56) in Theorem B

Corollary 1

Proof

Corollary 2

Proof

7 Concluding remarks

7.1 Good elements or not for d=2,3d=2,3?

7.1.1 Isotropic mesh

7.1.2 Anisotropic mesh: two-dimensional case

Example 2 (Right-angled triangle)

Example 3 (Dagger)

Remark 6

Example 4 (Blade)

Example 5 (Dagger)

7.1.3 Anisotropic mesh: three-dimensional case

Example 6

Example 7

Remark 7

Example 8

7.2 Effect of the quantity |T0|1q−1p|T_{0}|^{\frac{1}{q}-\frac{1}{p}} on the interpolation error estimates for d=2,3d=2,3

7.2.1 Case in which q>pq\textgreater p

7.2.2 Case in which q<pq\textless p

7.3 Inverse inequalities

Lemma 8

Condition 1 (Case in which $d=2$ )

Condition 2 (Case in which $d=3$ )

7.1 Good elements or not for $d=2,3$ ?

7.2 Effect of the quantity $|T_{0}|^{\frac{1}{q}-\frac{1}{p}}$ on the interpolation error estimates for $d=2,3$

7.2.1 Case in which $q\textgreater p$

7.2.2 Case in which $q\textless p$