Error Analysis of Approximate Operators for a Particle Method based on Voronoi Diagram

TL;DR

This paper analyzes the errors of approximate gradient and Laplace operators in a Voronoi diagram-based particle method, providing error estimates and an example application.

Contribution

It introduces and studies new approximate operators for gradient and Laplace in a Voronoi-based particle method, with derived error estimates.

Findings

Error estimates for approximate operators are derived.

The integration region is divided into specific areas for analysis.

An example application demonstrates the main results.

Abstract

This paper considers several approximate operators used in a particle method based on a Voronoi diagram. We introduce and study our approximate operators on gradient and Laplace operators. We derive error estimates for these approximate operators by applying our weight functions. The key idea of deriving our error estimates is to divide the integration region into a ring-shaped area and some areas. In the Appendix, we give an example application of the main results of this paper.

Equations

$$B_{r,\ell }=\{y\in {\mathbb{R}}^2;\ell <|y|<r\},\text{ and }{B}_{r,\ell }(x)=\{y\in {\mathbb{R}}^2;\ell <|y-x|<r\}.$$

$$\Vert f{\Vert }_{C^m}=\Vert f{\Vert }_{C^m(\overline{U})}=\underset{j\in \{1,2,\cdots ,m\}}{\max }|f{|}_{C^j(\overline{U})},\Vert f{\Vert }_{L^1}=\Vert f{\Vert }_{L^1(U)}={\int }_U|f(x)|dx,$$

$$|f{|}_{C^0}=|f{|}_{C^0(\overline{U})}=\underset{x\in \overline{U}}{\max }|f(x)|,|f{|}_{C^j}=|f{|}_{C^j(\overline{U})}=\underset{|\alpha |=j}{\max }|D^{\alpha }f{|}_{C^0(\overline{U})},$$

$$\Vert |\cdot |f(\cdot ){\Vert }_{L^1(U)}={\int }_U|x||f(x)|dx,\Vert |\cdot {|}^{2}f(\cdot ){\Vert }_{L^1(U)}={\int }_U|x{|}^2|f(x){|}^{2}dx.$$

$$\Vert |a-\cdot {|}^{s}g(a-\cdot ){\Vert }_{L^1(B_{r,\ell }(a))}={\int }_{B_{r,\ell }(a)}|a-y{|}^s|g(a-y)|dy.$$

$${\Omega }_H=\{x\in {\mathbb{R}}^2;|x-y|<H,y\in \Omega \}.$$

$${\Omega }_{N,H}=\{a_i{\}}_{i=1}^N.$$

$${\sigma }_i:=\{x\in {\Omega }_H;|x-a_i|<|x-a_j|\text{ for each }{a}_j\in {\Omega }_{N,H}(j\ne i)\}.$$

$${\overline{\Omega }}_H=\bigcup _{i=1}^N{\overline{\sigma }}_i\text{ and }{\sigma }_i\cap {\sigma }_j=\varnothing (i\ne j),$$

$$r_{\sigma }=\underset{i\in \{1,\cdots ,N\}}{\max }\underset{y\in {\overline{\sigma }}_i}{\max }\{|y-a_i|\}.$$

$$r_{\sigma }<H.$$

$$a_k\in {\sigma }_k\cap \Omega .$$

$$\overline{\mathcal{R}}(x,h)$$

$$\mathcal{R}(x,h)$$

$$V_i(x):={\int }_{{\sigma }_i\cap B_{h,\delta }(x)}1dx.$$

$$w(x)\ge 0.$$

$$w(x)=0.$$

$$|w(x)-w(y)|\le L_w|x-y|.$$

$$w(x)=\hat{w}(|x|).$$

$$\sum _{i\in \mathcal{R}(x,h)}{\int }_{{\sigma }_i}w(x-y)dy$$

$$\sum _{j\in \mathcal{R}(x,h)}{V}_j(x)w(x-a_j)$$

$$w(x)=\{\begin{array}{ll}|x{|}^m,& x\in {\overline{B}}_{h,\delta },\\ 0,& x\in {\mathbb{R}}^2\setminus B_{h,\delta },\end{array}$$

$${\tilde{\Pi }}_{h}f(a_k)=\frac{\sum _{i\in \mathcal{R}(a_k,h)}{V}_i(a_k)f(a_i)w(a_k-a_i)}{\sum _{j\in \mathcal{R}(a_k,h)}{V}_j(a_k)w(a_k-a_j)},$$

$${\tilde{\nabla }}_{h}f(a_k)=2\frac{\sum _{i\in \mathcal{R}(a_k,h)}{V}_i(a_k)\frac{f(a_k)-f(a_i)}{|a_k-a_i|}\frac{a_k-a_i}{|a_k-a_i|}w(a_k-a_i)}{\sum _{j\in \mathcal{R}(a_k,h)}{V}_j(a_k)w(a_k-a_j)},$$

$${\tilde{\Delta }}_{h}f(a_k)=-4\frac{\sum _{i\in \mathcal{R}(a_k,h)}{V}_i(a_k)\{f(a_k)-f(a_i)\}w(a_k-a_i)}{\sum _{j\in \mathcal{R}(a_k,h)}{V}_j(a_k)|a_k-a_j{|}^{2}w(a_k-a_j)},$$

$${\tilde{\square }}_{h}f(a_k)=-4\frac{\sum _{i\in \mathcal{R}(a_k,h)}{V}_i(a_k)\frac{f(a_k)-f(a_i)}{|a_k-a_i{|}^2}w(a_k-a_i)}{\sum _{j\in \mathcal{R}(a_k,h)}{V}_j(a_k)w(a_k-a_j)}.$$

$$|f(a_k)-{\tilde{\Pi }}_{h}f(a_k)|\le (h+r_{\sigma })|f{|}_{C^1}+\{2c_1(a_k)+2c_2(a_k)\}|f{|}_{C^0}.$$

$$c_1(a_k)$$

$$c_2(a_k)$$

$$|\nabla f(a_k)-{\tilde{\nabla }}_{h}f(a_k)|\le 4h|f{|}_{C^2}+\left\{\frac{8r_{\sigma }}{\lambda h}+4c_1(a_k)+4c_2(a_k)+8c_3(a_k)\right\}|f{|}_{C^1}.$$

$$c_3(a_k):=\frac{\Vert w(a_k-\cdot ){\Vert }_{L^1(B_{\lambda h+r_{\sigma }}(a_k)\setminus {\sigma }_k)}}{\Vert w(a_k-\cdot ){\Vert }_{L^1(B_h(a_k)\setminus {\sigma }_k)}}.$$

$$|\Delta f(a_k)-{\tilde{\Delta }}_{h}f(a_k)|\le 24h|f{|}_{C^3}+\left\{4\sum _{\mathfrak{i}=4}^7{c}_{\mathfrak{i}}(a_k)\right\}|f{|}_{C^1}.$$

$$c_{4}(a_{k}):=\frac{\||a_{k}-\cdot|w(a_{k}-\cdot)\|_{L^{1}(\sigma_{k}\setminus B_{\delta}(a_{k}))}}{\||a_{k}-\cdot|^{2}w(a_{k}-\cdot)\|_{L^{1}(B_{h}(a_{k})\setminus B_{\delta}(a_{k}))}}\cdot\\ \bigg{(}1+r_{\sigma}\frac{\||a_{k}-\cdot|w(a_{k}-\cdot)\|_{L^{1}(B_{h}(a_{k})\setminus B_{\delta}(a_{k}))}}{\||a_{k}-\cdot|^{2}w(a_{k}-\cdot)\|_{L^{1}(B_{h}(a_{k})\setminus\sigma_{k})}}\bigg{)},$$

Full text

1. Introduction

We are interested in the error analysis of approximate operators for a moving particle semi-implicit method (MPS). A moving particle semi-implicit method is a numerical method developed by Koshizuka-Oka [4]. In [4], they introduced approximate operators of gradient and Laplace operators based on a Voronoi diagram. Ishijima-Kimura [3] considered the error on the approximate operators under assumptions on their weight function. In [3], they applied the spherical symmetry of their weight function to derive their error estimates. Imoto-Tagami [1, 2] modified the approximate operators introduced in [4], and studied their error estimates for their approximate operators. In [1, 2], they derived their convergence rates of their error estimates with respect to influence radius by using assumptions on their weight function.

This paper has three purposes. The first one is to generalize the approximate operators introduced in [4, 1, 2]. More precisely, this paper standardizes their approximate operators (see Definition 1.2 for details). The second is to derive error estimates for our approximate operators. We use some properties of our weight functions (Assumption 1.1) to study the error on our operators. The key idea of deriving our error estimates is to divide the integration region into a ring-shaped area and some areas. The third is to give an application of our main results (see the Appendix for details).

Let us first introduce notations. Let $x={}^{t}(x_{1},x_{2}),y={}^{t}(y_{1},y_{2}),z={}^{t}(z_{1},z_{2})\in\mathbb{R}^{2}$ be the spatial variables, and let $\alpha=(\alpha_{1},\alpha_{2})\in\mathbb{N}_{0}^{2}$ be a multi-index, where $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$. For each multi-index $\alpha=(\alpha_{1},\alpha_{2})$, $|\alpha|:=\alpha_{1}+\alpha_{2}$, $D^{\alpha}:=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$, and $\alpha!:=\alpha_{1}!\alpha_{2}!$, where $\partial_{j}:=\partial/{\partial x_{j}}$, $\partial_{j}^{0}:=1$, and $0!:=1$. For each $x\in\mathbb{R}^{2}$, $r,\ell>0$, the symbols $B_{r}$, $B_{r}(x)$, $B_{r,\ell}$, and $B_{r,\ell}(x)$ are defined by $B_{r}=\{y\in\mathbb{R}^{2};{\ }|y|<r\}$, $B_{r}(x)=\{y\in\mathbb{R}^{2};{\ }|x-y|<r\}$,

[TABLE]

Note that $B_{r,\ell}=B_{r}\setminus\overline{B}_{\ell}$. Let $U\subset\mathbb{R}^{n}$ be a domain, and $\overline{U}$ be the closure of $U$. For each smooth function $f=f(x)$ on $\overline{U}$, we define

[TABLE]

Morevoer, for each $g\in L^{1}(B_{r,\ell}(a))$ for some $a\in\mathbb{R}^{2}$ and $s\in\mathbb{R}$, we define

[TABLE]

The following notations are of particular importance in this study. Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain, and $H$ be a positive constant. Throughout this paper, we fix $\Omega$ and $H$. Define

[TABLE]

See Fig. 1. Let $N\in\mathbb{N}$ and $a_{1},a_{2},\cdots,a_{N}\in\Omega_{H}$ such that $a_{i}\neq a_{j}$ if $i\neq j$. Write

[TABLE]

For each $i\in\{1,2,\cdots,N\}$, we define

[TABLE]

See Fig. 1. In general, we call $\sigma_{i}$ a Voronoi region, and $\{\sigma_{i}\}_{i=1}^{N}$ a Voronoi diagram (see [5]). Since

[TABLE]

we call $\sigma_{i}$ a Voronoi cell and $\{\sigma_{i}\}_{i=1}^{N}$ a Voronoi decomposition. Write

[TABLE]

By definition, we see that $\sigma_{i}\subset\overline{B}_{r_{\sigma}}(a_{i})$. Throughout this paper, we assume that

[TABLE]

Let $h>0$ such that $r_{\sigma}<h<H$. Let $k\in\{1,\cdots,N\}$ be such that

[TABLE]

Let $\delta>0$ such that $B_{\delta}(a_{k})\subset\sigma_{k}\cap\Omega$ (see Fig. 1). Throughout this paper, we fix $r_{\sigma}$, $h$, $k$, and $\delta$. For each $x\in B_{\delta}(a_{k})$,

[TABLE]

and for each $i\in\{1,\cdots,N\}$,

[TABLE]

We assume that $B_{h+r_{\sigma}}(x)\subset\Omega_{H}$ for each $x\in B_{\delta}(a_{k})$. We call $h$ the radius of the interaction area (influence radius).

Next we introduce the assumptions of our weight functions and approximate operators.

Assumption 1.1** (Weight functions).**

Let $w\in L^{\infty}(\mathbb{R}^{2})\cap C(\overline{B}_{h,\delta})$. We call $w$ a weight function if the following five properties hold:

$(\mathrm{i})$ For almost all $x\in\mathbb{R}^{2}$,

[TABLE]

$(\mathrm{ii})$ For almost all $x\in\mathbb{R}^{2}\setminus\overline{B}_{h,\delta}$,

[TABLE]

$(\mathrm{iii})$ There is $L_{w}\geq 0$ such that for all $x,y\in B_{h,\delta}$

[TABLE]

$(\mathrm{iv})$ There is $\widehat{w}\in L^{1}(\delta,h)$ such that for almost all $x\in B_{h,\delta}$

[TABLE]

$(\mathrm{v})$ There is $C_{0}>0$ such that for each $x\in B_{\delta}(a_{k})$,

[TABLE]

Remark that several radial functions are our weight functions. In fact, we set

[TABLE]

for some $m\in\mathbb{Z}$. Since $|x|-|y|=(|x|^{2}-|y|^{2})/(|x|+|y|)\leq C(h,\delta)|x-y|$ for $x,y\in\overline{B}_{h,\delta}$, we easily check that $w$ satisfies the properties of Assumption 1.1. Applying the spherically symmetric property of our weight function, we derive our error estimates. See Sections 3-6 for details.

Definition 1.2** (Approximate operators).**

Let $w$ be a weight function satisfying the properties of Assumption 1.1. For each $f\in C(\overline{\Omega}_{H})$, define the approximate operators as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Note that $|\widetilde{\Pi}_{h}f(a_{k})|\leq|f|_{C^{0}}$, $|\widetilde{\nabla}_{h}f(a_{k})|\leq 4\delta^{-1}|f|_{C^{0}}$, $|\widetilde{\Delta}_{h}f(a_{k})|\leq 8\delta^{-2}|f|_{C^{0}}$, and $|\widetilde{\square}_{h}f(a_{k})|\leq 8\delta^{-2}|f|_{C^{0}}$.

The main results of this paper are as follows.

Theorem 1.3**.**

Let $w$ be a weight function satisfying the properties of Assumption 1.1. Then for each $f\in C^{1}(\overline{\Omega}_{H})$,

[TABLE]

Here

[TABLE]

Theorem 1.4**.**

Let $w$ be a weight function satisfying the properties of Assumption 1.1. Assume that $\mathcal{R}(a_{k},\lambda h)\neq\emptyset$ for some $0<\lambda<1$. Then for each $f\in C^{2}(\overline{\Omega}_{H})$,

[TABLE]

Here $c_{1}(a_{k})$, $c_{2}(a_{k})$ are the constants defined by Theorem 1.3, and

[TABLE]

Theorem 1.5**.**

Let $w$ be a weight function satisfying the properties of Assumption 1.1. Assume that $\mathcal{R}(a_{k},\lambda h)\neq\emptyset$ for some $0<\lambda<1$. Then for each $f\in C^{3}(\overline{\Omega}_{H})$,

[TABLE]

Here

[TABLE]

[TABLE]

and

[TABLE]

Theorem 1.6**.**

Let $w$ be a weight function satisfying the properties of Assumption 1.1. Assume that $\mathcal{R}(a_{k},\lambda h)\neq\emptyset$ for some $0<\lambda<1$. Then for each $f\in C^{3}(\overline{\Omega}_{H})$,

[TABLE]

Here

[TABLE]

and

[TABLE]

Remark 1.7**.**

We explain the application of our results to numerical analysis and simulation. In (MPS), $\Omega_{N,H}=\{a_{i}\}_{i=1}^{N}$ means the distribution of the particles in the domain $\Omega_{H}$, $a_{i}$ the position of a particle, and $N$ the number of the particles in $\Omega_{H}$. Using our approximate operators, we can study an approximation of a fluid system, however, we need advanced techniques. From Theorems 1.3-1.6, we see that our approximate operators become better approximations when the number of particles is sufficiently large.

In Section 2, we study some properties of our weight functions. We prove Theorem 1.3 in Section 3, Theorem 1.4 in Section 4, Theorem 1.5 in Section 5, and Theorem 1.6 in Section 6. In the Appendix, we give an application of the main results of this paper.

2. Preliminaries

In this section, we recall the Taylor theorem and study some fundamental properties of our weight functions.

Lemma 2.1** (Taylor’s theorem).**

Let $m\in\mathbb{N}$ and $f\in C^{m+1}(\overline{\Omega}_{H})$. Then for each $x,y\in\Omega_{H}$,

[TABLE]

Here

[TABLE]

Moreover, for each $x,y\in\Omega_{H}$,

[TABLE]

Proof.

(Lemma 2.1) We only derive (2.1). To this end, we show that for each $m\in\mathbb{N}$

[TABLE]

A direct calculation gives

[TABLE]

We now assume that $m\geq 3$. Let $\alpha=(\alpha_{1},\alpha_{2})\in\mathbb{N}_{0}^{2}$ such that $\alpha_{1}+\alpha_{2}=m$. It is easy to check that

[TABLE]

and that

[TABLE]

Thus, we have

[TABLE]

Since $m\geq 3$ and $\displaystyle{\sum_{|\alpha|=m}1=m+1}$, we find that

[TABLE]

Therefore, we obtain (2.2).

Let us now derive (2.1). Fix $m\in\mathbb{N}$ and $f\in C^{m+1}(\overline{\Omega}_{H})$. Since $ty+(1-t)x\in\overline{\Omega}_{H}$ for $0\leq t\leq 1$ and $x,y\in\Omega_{H}$, we apply (2.2) to check that

[TABLE]

Therefore, the lemma follows. ∎

To derive basic properties of our weight function, we prepare the following lemma.

Lemma 2.2**.**

$(\mathrm{i})$ For each $i\in\{1,\cdots,N\}$ and $f\in L^{1}(B_{r_{\sigma}}(a_{i}))$,

[TABLE]

$(\mathrm{ii})$ For each $x\in B_{\delta}(a_{k})$ and $g\in L^{1}(B_{\lambda h+r_{\sigma}}(x))$,

[TABLE]

$(\mathrm{iii})$ For each $x\in B_{\delta}(a_{k})$ and $0<\lambda<1$,

[TABLE]

Proof.

(Lemma 2.2) We first show $(\mathrm{i})$. By the definition of $r_{\sigma}$, we find that $y\in B_{r_{\sigma}}(a_{i})$ if $y\in\sigma_{i}$. Therefore, we see that for each $i=1,\cdots,N$ and $f\in L^{1}(B_{r_{\sigma}}(a_{i}))$,

[TABLE]

This is (2.3). Next, we derive (2.4). Let $x\in B_{\delta}(a_{k})$ and $g\in L^{1}(B_{\lambda h+r_{\sigma}}(x))$. By the definition of $\mathcal{R}(x,\lambda h)$, we find that $a_{i}\in B_{\lambda h}(x)$ if $i\in\mathcal{R}(x,\lambda h)$. Therefore, we observe that

[TABLE]

Finally, we prove $(\mathrm{iii})$. Let $i\in\mathcal{R}(x,h)$. Since $i\neq k$ and $x\in\sigma_{k}\cap\Omega$, we find that

[TABLE]

Let $j\in\mathcal{R}(x,h)\setminus\mathcal{R}(x,\lambda h)$. By definition, we see that

[TABLE]

This shows that

[TABLE]

Since

[TABLE]

we use (2.4) to check that

[TABLE]

Thus, we have (2.5). Therefore, the lemma follows. ∎

Lemma 2.3** (Properties of weight functions).**

Let $w$ be a weight function satisfying the properties of Assumption 1.1. Then the following six assertions hold:

$(\mathrm{i})$ For each $x\in B_{\delta}(a_{k})$ and $n\in\mathbb{Z}$,

[TABLE]

$(\mathrm{ii})$ For each $i=1,2$, $x\in B_{\delta}(a_{k})$, and $n\in\mathbb{Z}$,

[TABLE]

$(\mathrm{iii})$ For each $i,j=1,2$, $x\in B_{\delta}(a_{k})$, and $n\in\mathbb{Z}$,

[TABLE]

Here $\delta_{ij}$ denotes the Kronecker delta.

$(\mathrm{iv})$ For each $x\in B_{\delta}(a_{k})$,

[TABLE]

$(\mathrm{v})$ For each $x\in B_{\delta}(a_{k})$ and $0<\lambda<1$,

[TABLE]

$(\mathrm{vi})$ For each $x\in B_{\delta}(a_{k})$ and $0<\lambda<1$,

[TABLE]

Proof.

(Lemma 2.3) We first show $(\mathrm{i})$. Using a change of variables, we see that

[TABLE]

This gives

[TABLE]

Fix $x\in B_{\delta}(a_{k})$ and $n\in\mathbb{Z}$. Using a change of variables with (2.14), we observe that

[TABLE]

Therefore, we obtain $(\mathrm{i})$.

Next, we prove $(\mathrm{ii})$ and $(\mathrm{iii})$. Fix $i,j=1,2$, $x\in B_{\delta}(a_{k})$, and $n\in\mathbb{Z}$. From (2.15), we have

[TABLE]

Using a change of variables, we find that

[TABLE]

By (2.15), we check that

[TABLE]

Therefore, we have $(\mathrm{ii})$ and $(\mathrm{iii})$.

Now, we show $(\mathrm{iv})$. Since $\displaystyle{B_{h}(x)\subset\sum_{i\in\overline{\mathcal{R}}(x,h)}\overline{\sigma}_{i}}$ and $w(x)=0$ for $x\in\mathbb{R}^{2}\setminus\overline{B}_{h,\delta}$, we use the Lipschitz continuity of $w$ to observe that

[TABLE]

Thus, we have (2.11).

Finally, we prove $(\mathrm{v})$ and $(\mathrm{vi})$. Using (2.6), (2.7), and (2.4), we check that

[TABLE]

This gives (2.12). Similarly, we see (2.13). Therefore, the lemma follows. ∎

3. Error Estimate $(\mathrm{I})$

In this section, we study $|f(a_{k})-\widetilde{\Pi}_{h}f(a_{k})|$ to prove Theorem 1.3. To this end, we introduce some notations. Let $w$ be a weight function satisfying the properties of Assumption 1.1. For each $f\in C^{0}(\overline{\Omega}_{H})$, we define

[TABLE]

It is easy to check that $|\Pi_{h}f(a_{k})|\leq|f|_{C^{0}}$. Since

[TABLE]

we prove the following lemma.

Lemma 3.1**.**

For each $f\in C^{1}(\overline{\Omega}_{H})$,

[TABLE]

Proof.

(Lemma 3.1) We first show (3.1). Since

[TABLE]

we use the mean-value theorem to see that

[TABLE]

Therefore, we have (3.1).

Secondly, we derive (3.2). From

[TABLE]

we observe that

[TABLE]

Therefore, we see (3.2).

Next, we prove (3.3). Since

[TABLE]

we use the mean-value theorem and (2.11) to check that

[TABLE]

This is (3.3).

Finally, we show (3.4). By (2.11), we find that

[TABLE]

Therefore, we have (3.4), and the lemma follows. ∎

Finally, we prove Theorem 1.3.

Proof.

(Theorem 1.3) Using Lemma 3.1, we prove Theorem 1.3. ∎

4. Error Estimate $(\mathrm{II})$

In this section, we consider $|\nabla f(a_{k})-\widetilde{\nabla}_{h}f(a_{k})|$ to prove Theorem 1.4. Let $w$ be a weight function satisfying the properties of Assumption 1.1. For each $f\in C^{0}(\overline{\Omega}_{H})$, we define

[TABLE]

It is easy to check that

[TABLE]

It is clear that

[TABLE]

The aim of this section is to prove the following two lemmas.

Lemma 4.1**.**

For each $f\in C^{2}(\overline{\Omega}_{H})$,

[TABLE]

Lemma 4.2**.**

For each $f\in C^{1}(\overline{\Omega}_{H})$,

[TABLE]

We first show Lemma 4.1. Then we prove Lemma 4.2.

Proof.

(Lemma 4.1) Fix $f\in C^{2}(\overline{\Omega}_{H})$. From the Taylor expansion, we have

[TABLE]

where $y\in B_{h,\delta}(a_{k})$. Here

[TABLE]

Multiplying both sides of (4.5) by $2w(a_{k}-y)(a_{k}-y)/(|a_{k}-y|^{2}\|w\|_{L^{1}(B_{h,\delta})})$, and then integrating with respect to $y$, we have

[TABLE]

Here

[TABLE]

From (2.10), we see that

[TABLE]

By (2.1), we find that

[TABLE]

Combining (4.6)-(4.8) gives (4.1). Therefore, the lemma follows. ∎

Proof.

(Lemma 4.2) Let $f\in C^{1}(\overline{\Omega}_{H})$. We first show (4.2). By the mean-value theorem, we see that

[TABLE]

Therefore, we have (4.2). Note that $\|w(a_{k}-\cdot)\|_{L^{1}(\sigma_{k})}=\|w(a_{k}-\cdot)\|_{L^{1}(\sigma_{k}\setminus B_{\delta}(a_{k}))}$.

Next, we derive (4.3). A direct calculation gives

[TABLE]

Here

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Since

[TABLE]

we use the mean-value theorem and (2.12) to check that

[TABLE]

From (4.9) and (4.10), we have (4.3).

Finally, we show (4.4). From (2.11), we see that

[TABLE]

This is (4.4). Therefore, the lemma follows. ∎

Finally, we prove Theorem 1.4.

Proof.

(Theorem 1.4) Combining Lemmas 4.1 and 4.2 gives Theorem 1.4. ∎

5. Error Estimate $(\mathrm{III})$

In this section, we estimate $|\Delta f(a_{k})-\widetilde{\Delta}_{h}f(a_{k})|$ to prove Theorem 1.5. Let $w$ be a weight function satisfying the properties of Assumption 1.1. For each $f\in C^{0}(\overline{\Omega}_{H})$, we define

[TABLE]

We see at once that

[TABLE]

The proof of Theorem 1.5 makes use of the following two lemmas.

Lemma 5.1**.**

For each $f\in C^{3}(\overline{\Omega}_{H})$,

[TABLE]

Lemma 5.2**.**

For each $f\in C^{1}(\overline{\Omega}_{H})$,

[TABLE]

Here $c_{4}(a_{k}),\cdots,c_{8}(a_{k})$ are the constants defined by Theorem 1.5.

We first show Lemma 5.1. Then we prove Lemma 5.2.

Proof.

(Lemma 5.1) Fix $f\in C^{3}(\overline{\Omega}_{H})$. From the Taylor expansion, we have

[TABLE]

where $y\in B_{h,\delta}(a_{k})$. Here

[TABLE]

Multiplying both sides of (5.5) by $-4w(a_{k}-y)/\||\cdot|^{2}w(\cdot)\|_{L^{1}(B_{h,\delta})}$, and integrating with respect to $y$, we have

[TABLE]

Here

[TABLE]

Using (2.10), (2.9), and (2.1), we find that

[TABLE]

Combining (5.6)-(5.9), we have (5.1). Therefore, the lemma follows. ∎

Proof.

(Lemma 5.2) Let $f\in C^{1}(\overline{\Omega}_{H})$. We first show (5.2). Since

[TABLE]

for $y\in\sigma_{k}$, we use the mean-value theorem to see that

[TABLE]

Using $\sum_{i\in\mathcal{R}(a_{k},h)}=\sum_{i\in\mathcal{R}(a_{k},\lambda h)}+\sum_{i\in\mathcal{R}(a_{k},h)\setminus\mathcal{R}(a_{k},\lambda h)}$, we see that

[TABLE]

A direct calculation shows that

[TABLE]

Using (5.10), we check that

[TABLE]

Therefore, the lemma follows. ∎

Finally, we prove Theorem 1.5.

Proof.

(Theorem 1.5) Using Lemmas 5.1 and 5.2, we prove Theorem 1.5. ∎

6. Error Estimate $(\mathrm{IV})$

In this section, we consider $|\Delta f(a_{k})-\widetilde{\square}_{h}f(a_{k})|$ to prove Theorem 1.6. Let $w$ be a weight function satisfying the properties of Assumption 1.1. For each $f\in C^{0}(\overline{\Omega}_{H})$, we define

[TABLE]

It is easy to check that

[TABLE]

Let us attack the following two lemmas.

Lemma 6.1**.**

For each $f\in C^{3}(\overline{\Omega}_{H})$,

[TABLE]

Lemma 6.2**.**

For each $f\in C^{1}(\overline{\Omega}_{H})$,

[TABLE]

Here $c_{9}(a_{k}),\cdots,c_{12}(a_{k})$ are the constants defined by Theorem 1.6.

We first show Lemma 6.1. Then we prove Lemma 6.2.

Proof.

(Lemma 6.1) Fix $f\in C^{3}(\overline{\Omega}_{H})$. Multiplying both sides of (5.5) by $-4w(a_{k}-y)/(|a_{k}-y|^{2}\|w\|_{L^{1}(B_{h,\delta})})$, and then integrating with respect to $y$, we have

[TABLE]

Here

[TABLE]

By (2.10) and (2.9), we find that

[TABLE]

Applying (2.1), we see that

[TABLE]

Combining (6.5)-(6.8), we have (6.1). Therefore, the lemma follows. ∎

Proof.

(Lemma 6.2) Let $f\in C^{1}(\overline{\Omega}_{H})$. We first show (6.2). By the mean-value theorem, we see that

[TABLE]

Therefore, we have (6.2).

Next, we derive (6.3). A direct calculation gives

[TABLE]

Here

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Since

[TABLE]

we use the mean-value theorem and (2.13) to check that

[TABLE]

From (6.9) and (6.10), we have (6.3).

Finally, we derive (6.4). Using (2.5) and the mean-value theorem to see that

[TABLE]

This is (6.4). Therefore, the lemma follows. ∎

Finally, we prove Theorem 1.6.

Proof.

(Theorem 1.6) Using Lemmas 6.1 and 6.2, we prove Theorem 1.6. ∎

7. Appendix: Applications of Main Results

We state an application of the main results of this paper. We consider the following case:

[TABLE]

It is easy to check that $L_{w}=0$ and that for each $0<q<p$,

[TABLE]

In this section, we assume that $\delta=r_{\sigma}/2$,

[TABLE]

Suppose that there is $C_{*}>1$ such that

[TABLE]

Assume that $\mathcal{R}(a_{k},\lambda h)\neq\emptyset$ for some $0<\lambda<1$. Using Theorems 1.3-1.6 and $\|\cdot\|_{L^{1}(\sigma_{k}\setminus B_{\delta}(a_{k}))}\leq\|\cdot\|_{L^{1}(B_{r_{\sigma}}(a_{k})\setminus B_{\delta}(a_{k}))}$, we have the following corollary.

Corollary 7.1**.**

For each $f\in C^{3}(\overline{\Omega}_{H})$,

[TABLE]

[TABLE]

Moreover, the following two assertions hold:

$(\mathrm{i})$ If $r_{\sigma}=10^{-15}$, $C_{*}=10^{12}$, $\lambda=10^{-6}$, then

[TABLE]

$(\mathrm{ii})$ If $r_{\sigma}=10^{-2}$, $C_{*}=4$, $\lambda=1/2$, then

[TABLE]

Proof.

(Corollary 7.1) We only show (7.2) and (7.6) since other cases are similar. Fix $f\in^{3}(\overline{\Omega}_{H})$.

We first show (7.2). Since $L_{w}=0$, it follows from Theorem 1.4 to see that

[TABLE]

By the assumptions of Section 7, we check that

[TABLE]

Therefore, we obtain (7.2).

Next we show (7.6). A direct calculation shows that

[TABLE]

and that

[TABLE]

From (7.2), we have (7.6). Therefore, Corollary 7.1 is proved. ∎