Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier–Stokes equations

Abstract

To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements in accuracy are considered for a fully explicit particle method for incompressible Navier–Stokes equations. The explicit particle method is based on a penalty problem, which converges theoretically to the incompressible Navier–Stokes equations and is discretized in space by generalized approximate operators defined as a wider class of approximate operators than those of the smoothed particle hydrodynamics and moving particle semi-implicit methods. By considering an analytical derivation of the explicit particle method and truncation error estimates of the generalized approximate operators, sufficient conditions of convergence are conjectured. Under these conditions, the convergence of the explicit particle method is confirmed by numerically comparing errors between exact and approximate solutions. Moreover, by focusing on the truncation errors of the generalized approximate operators, an optimal weight function is derived by reducing the truncation errors over general particle distributions. The effectiveness of the generalized approximate operators with the optimal weight functions is confirmed using numerical results of truncation errors and driven cavity flow. As an application for flow problems with free surface effects, the explicit particle method is applied to a dam break flow.

Appendices

Appendix A: Notation

First, we summarize the computational rules of the multi-index. Let $\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _d)$ be the $d$th multi-index. For a vector $x\in \mathbb {R}^d$, we denote the kth element of x as ${x}^{(k)}$. Then, that operations for the multi-index are defined by

$$\begin{aligned} |\alpha |&=\sum _{j=1}^d\alpha _j, \end{aligned}$$

(102)

$$\begin{aligned} x^\alpha&=\prod _{j=1}^d({x}^{(j)})^{\alpha _j},\qquad x\in \mathbb {R}^d, \end{aligned}$$

(103)

$$\begin{aligned} \alpha !&= \prod _{i=1}^d\alpha _i!. \end{aligned}$$

(104)

Let $D^\alpha$ be the differential operator defined by

$$\begin{aligned} D^\alpha =\left( \frac{\partial }{\partial {x}^{(1)}}\right) ^{\alpha _1} \left( \frac{\partial }{\partial {x}^{(2)}}\right) ^{\alpha _2} \cdots \left( \frac{\partial }{\partial {x}^{(d)}}\right) ^{\alpha _d}, \end{aligned}$$

(105)

where $D^\alpha v=v$ if $|\alpha |=0$.

Next, we introduce some functional spaces. For a set $S\subset \mathbb {R}^d\,(d\in \mathbb {N})$, let $C(\overline{S})$ be the space of real continuous functions defined in $\overline{S}$, where $\overline{S}$ is the closure of S. The norm of $C(\overline{S})$ is defined by

$$\begin{aligned} \left\| \phi \right\| _{C(\overline{S})} := \max _{x\in \overline{S}}\left| \phi (x)\right| . \end{aligned}$$

(106)

For an open set S and positive integer k, let $C^k(\overline{S})$ be the space of functions in $C(\overline{S})$ with derivatives up to the kth order. The norm of $C^k(\overline{S})$ is defined as

$$\begin{aligned} \left\| \phi \right\| _{C^k(\overline{S})} := \max _{|\alpha |\le k}\left\| D^\alpha \phi \right\| _{C(\overline{S})}. \end{aligned}$$

(107)

Here, $\alpha$ is the multi-index. For a functional space $X(\overline{S})$, let $C([0,T];X(\overline{S}))$ be the space of functions on $\overline{S}\times [0,T]$ satisfying

$$\begin{aligned} \left\| \phi \right\| _{C([0,T];X(\overline{S}))} := \max _{t\in [0,T]}\left\| \phi (\cdot ,t)\right\| _{X(\overline{S})}<+\infty . \end{aligned}$$

(108)

Appendix B: Derivation of generalized approximate operators

We present the derivations of the generalized approximate operators in Sect. 2.2. The derivations we will show are characterized as deriving them uniformly. Let $x_{i}^{}\in \mathcal {X}_N\cap \Omega$. Let $B_r(x)$ be an open ball with the center at x and radius r:

$$\begin{aligned} B_r(x)\mathrel {\mathop :}=\{y\in \mathbb {R}^d;~|x-y|<r\}. \end{aligned}$$

(109)

Then, by Taylor expansion, for $y\in B_h(x_{i}^{})\setminus \{x_{i}^{}\}$ and $v\in C^n(\overline{\Omega }_{H})~(n\in \mathbb {N})$, we have

$$\begin{aligned} \phi (y) = \sum _{0\le |\alpha |\le n-1} \frac{D^{\alpha }\phi _i}{\alpha !}(y-x_{i}^{})^{\alpha } + R_{n,i}(y;\phi ). \end{aligned}$$

(110)

Here, $\alpha$ is a multi-index and $R_{n,i}(y;\phi )$ is the residual given by

$$\begin{aligned}&R_{n,i}(y;\phi ) \mathrel {\mathop :}=\sum _{|\alpha |=n}(y-x_{i}^{})^{\alpha } \frac{|\alpha |}{\alpha !} \nonumber \\&\quad \times \int _0^1 (1-s)^{|\alpha |-1} D^{\alpha }v(s y+(1-s) x_{i}^{})\,\mathrm{d}s. \end{aligned}$$

(111)

For $k=1,2,\ldots ,d$ and nonnegative integer l, let $\beta _{k,l}$ be a multi-index such that the kth element is l, while the others are 0. For $n=2, 3, 4$ and $k=1,2,\ldots ,d$, multiplying both the sides of (111) by

$$\begin{aligned} \frac{d(n-2)!}{h^{n-2}C_{n-2}(w)}\frac{(y-x_{i}^{})^{\,\beta _{k,n-2}}}{|y-x_{i}^{}|^{n-2}}w_h(|y-x_{i}^{}|) \end{aligned}$$

(112)

and integrating it over $\Omega _{H}$, we get

$$\begin{aligned}&\frac{d(n-2)!}{h^{n-2}C_{n-2}(w)}\int _{\Omega _{H}} \phi (y)\frac{(y-x_{i}^{})^{\,\beta _{k,n-2}}}{|y-x_{i}^{}|^{n-2}}w_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\quad =\frac{d(n-2)!}{h^{n-2}C_{n-2}(w)}\nonumber \\&\quad \times \sum _{0\le |\alpha |\le n-1} \frac{D^{\alpha }\phi _i}{\alpha !}\int _{\Omega _{H}}\frac{(y-x_{i}^{})^{\alpha +\beta _{k,n-2}}}{|y-x_{i}^{}|^{n-2}}w_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\quad +\, E_{i,k,n}. \end{aligned}$$

(113)

Here, $E_{i,k,n}$ is

$$\begin{aligned}&E_{i,k,n}\mathrel {\mathop :}=\nonumber \\&\quad \frac{d(n-2)!}{h^{n-2}C_{n-2}(w)}\int _{\Omega _{H}}R_{n,i}(y;\phi )\frac{(y-x_{i}^{})^{\,\beta _{k,n-2}}}{|y-x_{i}^{}|^{n-2}}w_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\qquad \quad \;=\mathcal {O}(h^2). \end{aligned}$$

(114)

By considering that

$$\begin{aligned}&\int _{\Omega _{H}}\frac{(y-x_{i}^{})^{\alpha +\beta _{k,n-2}}}{|y-x_{i}^{}|^{n-2}}w_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&= {\left\{ \begin{array}{ll} 0,&{} \begin{aligned} &{}\text{ one } \text{ or } \text{ more } \text{ elements } \text{ of } \\ &{}\alpha +\beta _{k, n-2} \text{ are } \text{ odd, } \end{aligned} \\ \dfrac{h^{n-2}C_{n-2}(w)}{d},&{}\alpha +\beta _{k, n-2}=\beta _{k, 2},\\ C_{0}(w),&{}n=2, \alpha =0, \end{array}\right. } \end{aligned}$$

(115)

for (113) with $n=2,3$, we obtain

$$\begin{aligned} \phi _i=\dfrac{1}{C_0(w)}\int _{\Omega _{H}}\phi (y)w_h(|y-x_{i}^{}|)\mathrm{\,d}y+\mathcal {O}(h^2) \end{aligned}$$

(116)

and

$$\begin{aligned}&{(\nabla \phi _i)}^{(k)}=\nonumber \\&\quad \frac{d}{hC_{1}(w)}\int _{\Omega _{H}}\{\phi (y)-\phi _i\}\frac{{(y-x_{i}^{})}^{(k)}}{|y-x_{i}^{}|}w_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\quad +\,\mathcal {O}(h^2). \end{aligned}$$

(117)

Moreover, when $n=4$, by

$$\begin{aligned} \sum _{k=1}^d\frac{(y-x_{i}^{})^{\,\beta _{k,2}}}{|y-x_{i}^{}|^{2}}=1 \end{aligned}$$

(118)

and

$$\begin{aligned}&\sum _{k=1}^d\int _{\Omega _{H}}(y-x_{i}^{})^{\alpha }w_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\quad = {\left\{ \begin{array}{ll} 0,&{}\text{ one } \text{ or } \text{ more } \text{ elements } \text{ of } \alpha \text{ are } \text{ odd },\\ \dfrac{h^{2}C_2(w)}{d},&{} \begin{aligned} &{}|\alpha |=2 \text{ and } \text{ all } \text{ elements }\\ &{}\text{ of } \alpha \text{ are } \text{ even }, \end{aligned} \end{array}\right. } \end{aligned}$$

(119)

we obtain

$$\begin{aligned} \Delta \phi _i&=\frac{2d}{h^2 C_{2}(w)}\int _{\Omega _{H}}\{\phi (y)-\phi _i\}w_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\quad +\mathcal {O}(h^2). \end{aligned}$$

(120)

By (4), the above integration can be approximated as

$$\begin{aligned} \int _{\Omega _{H}}\phi (y)\mathrm{\,d}y\approx \sum _{j=1}^NV_{j}\phi _j. \end{aligned}$$

(121)

Therefore, by (116) and (121), and replacing $w$ with $w^{\Pi }\in \mathcal {W}$, we derive the generalized interpolant (8) as follows:

$$\begin{aligned} \phi _i&=C_{\Pi }\int _{\Omega _{H}}\phi (y)w^{\Pi }_h(|y-x_{i}^{}|)\mathrm{\,d}y+\mathcal {O}(h^2) \nonumber \\&\approx C_{\Pi }\sum _{j=1}^NV_{j}\phi _jw^{\Pi }_h(|x_{j}^{}-x_{i}^{}|) = \Pi _{h}^{}\phi _i. \end{aligned}$$

(122)

By (117) and (121), and replacing $w$ with $w^{\nabla }\in \mathcal {W}$, we derive the generalized approximate gradient operator (9) as follows:

$$\begin{aligned} \nabla \phi _i&=\frac{C_{\nabla }}{h}\int _{\Omega _{H}}\{\phi (y)-\phi _i\}\frac{y-x_{i}^{}}{|y-x_{i}^{}|}w^{\nabla }_h(|y-x_{i}^{}|)\mathrm{\,d}y+\mathcal {O}(h^2) \nonumber \\&\approx \frac{C_{\nabla }}{h} \sum _{j\ne i}V_{j} (\phi _j-\phi _i)\frac{x_{j}^{}-x_{i}^{}}{|x_{j}^{}-x_{i}^{}|} w^{\nabla }_h(|x_{j}^{}-x_{i}^{}|) \nonumber \\&= \nabla _{h}^{} \phi _i. \end{aligned}$$

(123)

Moreover, by (120) and (121), and replacing $w$ with $w^{\Delta }\in \mathcal {W}$, we derive the generalized approximate Laplace operator (10) as follows:

$$\begin{aligned} \Delta \phi _i&=\frac{C_{\Delta }}{h^2}\int _{\Omega _{H}}\{\phi (y)-\phi _i\}w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y+\mathcal {O}(h^2) \nonumber \\&\approx \frac{C_{\Delta }}{h^2} \sum _{j\ne i}V_{j} (\phi _j-\phi _i)w^{\Delta }_h(|x_{j}^{}-x_{i}^{}|) \nonumber \\&= \Delta _{h}^{} \phi _i. \end{aligned}$$

(124)

The derivations are characterized as to be able to uniformly derive the generalized approximate operators by utilizing (113). The generalized approximate operators can be used as approximate operators of the conventional particle methods such as SPH and MPS by selecting the parameters of the generalized approximate operators appropriately; this is discussed further in “Appendix C”. Therefore, approximate operators of conventional particle methods can be derived using the abovementioned method.

Appendix C: Description of approximate operators in SPH and MPS using generalized approximate operators

We show that the generalized approximate operators (8)–(10) denote approximate operators in SPH and MPS if their parameters are selected appropriately. Let $w^\mathrm{SPH}\in \mathcal {W}$ be a reference weight function such that

$$\begin{aligned}&C_0(w^\mathrm{SPH})=\int _{\mathbb {R}^d} w^\mathrm{SPH}(|x|)\mathrm{\,d}x\nonumber \\&\qquad \qquad \quad \;= \int _{\mathbb {R}^d} w^\mathrm{SPH}_{h}(|x|)\mathrm{\,d}x= 1, \end{aligned}$$

(125)

$$\begin{aligned}&\dot{w}^\mathrm{SPH}(r)<0,\qquad 0<r<1, \end{aligned}$$

(126)

where $\dot{w}^\mathrm{SPH}$ is the first derivative of $w^\mathrm{SPH}$. Then, in SPH, the interpolant $\Pi _{h}^\mathrm{SPH}$, approximate gradient operator $\nabla _{h}^\mathrm{SPH}$, and approximate Laplace operator $\Delta _{h}^\mathrm{SPH}$ are defined as

$$\begin{aligned}&\Pi _{h}^\mathrm{SPH}\phi _i\mathrel {\mathop :}=\sum _{j=1}^N\dfrac{m_j}{\rho _j}\phi _{j}w^\mathrm{SPH}_{h}(|x_{j}^{}-x_{i}^{}|), \end{aligned}$$

(127)

$$\begin{aligned}&\nabla _{h}^\mathrm{SPH}\phi _i\mathrel {\mathop :}=\sum _{j\ne i}\dfrac{m_j}{\rho _j}(\phi _j-\phi _i)\nabla w^\mathrm{SPH}_{h}(|x_{j}^{}-x_{i}^{}|), \end{aligned}$$

(128)

$$\begin{aligned}&\Delta _{h}^\mathrm{SPH}\phi _i\mathrel {\mathop :}=\nonumber \\&\qquad 2 \sum _{j\ne i}\dfrac{m_j}{\rho _j}\dfrac{\phi _i-\phi _j}{|x_{j}^{}-x_{i}^{}|}\frac{x_{j}^{}-x_{i}^{}}{|x_{j}^{}-x_{i}^{}|}\cdot \nabla w^\mathrm{SPH}_{h}(|x_{j}^{}-x_{i}^{}|), \end{aligned}$$

(129)

respectively. Here, $m_j$ and $\rho _j$ are positive parameters referred to as the particle mass and particle density, respectively. The particle volume set $\mathcal {V}_N$ is given by $\mathcal {V}_N=\{V_{i}=m_{i}/\rho _{i}\mid i=1,\ldots ,N\}$. Then, from (125), the generalized interpolant (8) with $w^{\Pi }=w^\mathrm{SPH}$ is equivalent to the interpolant of SPH (127). From

$$\begin{aligned} -\int _{\mathbb {R}^d} |x| \dot{w}^\mathrm{SPH}(|x|)\mathrm{\,d}x&= \int _{\mathbb {R}^d} x\cdot \nabla w^\mathrm{SPH}(|x|)\mathrm{\,d}x\nonumber \\&= \int _{\mathbb {R}^d} (\nabla \cdot x) w^\mathrm{SPH}(|x|)\mathrm{\,d}x\nonumber \\&= d\int _{\mathbb {R}^d} w^\mathrm{SPH}(|x|)\mathrm{\,d}x\nonumber \\&= d, \end{aligned}$$

(130)

the generalized approximate gradient operator (9) with $w^{\nabla }=-\dot{w}^\mathrm{SPH}$ is equivalent to the approximate gradient operator of SPH (128). Moreover, from (130), the generalized approximate Laplace operator (10) with

$$\begin{aligned} w^{\Delta }(r) = -\dfrac{1}{r}\dot{w}^\mathrm{SPH}(r) \end{aligned}$$

(131)

is equivalent to the approximate Laplace operator of SPH (129).

Let $w^\mathrm{MPS}\in \mathcal {W}$ be a reference weight function defined by (74). A weight function $w^\mathrm{MPS}_{h}$ is set by (6). Then, in MPS, the approximate gradient operator $\nabla _{h}^\mathrm{MPS}$ and approximate Laplace operator $\Delta _{h}^\mathrm{MPS}$ are defined as

$$\begin{aligned}&\nabla _{h}^\mathrm{MPS}\phi _i\mathrel {\mathop :}=\nonumber \\&\qquad \qquad \frac{d}{n_0} \sum _{j\ne i}\dfrac{\phi _j-\phi _i}{|x_{j}^{}-x_{i}^{}|}\frac{x_{j}^{}-x_{i}^{}}{|x_{j}^{}-x_{i}^{}|} w_h^\mathrm{MPS} (|x_{j}^{}-x_{i}^{}|), \end{aligned}$$

(132)

$$\begin{aligned}&\Delta _{h}^\mathrm{MPS}\phi _i\mathrel {\mathop :}=\frac{2d}{n_0\lambda _0} \sum _{j\ne i}(\phi _j-\phi _i) w_h^\mathrm{MPS} (|x_{j}^{}-x_{i}^{}|), \end{aligned}$$

(133)

respectively. Here, $n_0$ and $\lambda _0$ are parameters that depend on both $w^\mathrm{MPS}$ and $h$. In general, $\lambda _0$ is given by $\lambda _0=C_2(w^\mathrm{MPS}_{h})$. Then, the particle volume set $\mathcal {V}_N$ is given by $\mathcal {V}_N=\{V_{i}=C_0(w^\mathrm{MPS}_{h})/n_0\mid i=1,\ldots ,N\}$. Further, the generalized approximate gradient operator (9) with

$$\begin{aligned} w^{\nabla }(r)=\dfrac{1}{r}w^\mathrm{MPS}(r) \end{aligned}$$

(134)

is equivalent to the approximate gradient operator of MPS (132). Furthermore, the generalized approximate Laplace operator (10) with $w^{\Delta }=w^\mathrm{MPS}$ is equivalent to the approximate Laplace operator of MPS (133) with $\lambda _0=C_2(w^\mathrm{MPS}_{h})$.

Appendix D: Truncation error estimates of generalized approximate operators

We show truncation error (26) of the generalized Laplace operators using their derivations. We have established a more precise theorem to estimate truncation errors, which have been reported in the literature [8, 11, 12]. However, because the formulations of approximate operators are a little different for that in previous studies, we simplistically show the proof. We assume $x_{i}^{}\in \mathcal {X}_N\cap \Omega$, $\phi \in C^4(\overline{\Omega }_{H})$, and $w^{\Delta }\in \mathcal {W}\cap C^1([0,\infty ))$. From the derivation of the generalized approximate Laplace operator (124), we estimate its truncation error as

$$\begin{aligned} |\Delta \phi _i-\Delta _{h}^{}\phi _i| \le |\widetilde{E}_{i}| + |\widehat{E}_{i}|. \end{aligned}$$

(135)

Here,

$$\begin{aligned} \widetilde{E}_{i} \mathrel {\mathop :}=\,&2\sum _{k=1}^dE_{i,k,4} \nonumber \\ =\,&2\frac{C_{\Delta }}{h^{2}}\int _{\Omega _{H}}R_{4,i}(y;\phi )w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\ =\,&\mathcal {O}(h^2), \end{aligned}$$

(136)

$$\begin{aligned} \widehat{E}_{i} \mathrel {\mathop :}=&\frac{C_{\Delta }}{h^2}\int _{\Omega _{H}}\{\phi (y)-\phi _i\}w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&- \frac{C_{\Delta }}{h^2} \sum _{j=1}^NV_{j} (\phi _j-\phi _i)w^{\Delta }_h(|x_{j}^{}-x_{i}^{}|). \end{aligned}$$

(137)

Note that the estimate $\widetilde{E}_{i}=\mathcal {O}(h^2)$ is derived from (114). Now, we estimate the error $\widehat{E}_{i}$, which consists of the integration and the numerical integration, which are the first and second terms on the right-hand side of (137), respectively. For a $C^1$ class function $g:\Omega _{H}\rightarrow \mathbb {R}$ and generators $y_i\in \Omega _{H}\,(i=1,2,\ldots ,N)$, we assume a numerical integration for the integration of g over $\Omega _{H}$ given by

$$\begin{aligned} \sum _{i=1}^N|\sigma _i|g(y_i). \end{aligned}$$

(138)

Here, $\sigma =\{\sigma _i\}_{i=1}^{N}$ is a decomposition of $\Omega _{H}$ satisfying

$$\begin{aligned} \bigcup _{i=1}^{N} \overline{\sigma }_i= \overline{\Omega }_{H}, \qquad \sigma _i \cap \sigma _j =\emptyset \quad (i\ne j), \end{aligned}$$

(139)

where $\overline{\sigma }_i$ is the closure of $\sigma _i$. Then, as an estimate of the Riemann sum, we can estimate the numerical integration as

$$\begin{aligned}&\left| \int _{\Omega _{H}} g(y)dy -\sum _{i=1,2,\ldots ,N}|\sigma _i|g(y_i)\right| \nonumber \\&\quad =\mathcal {O}\left( \max _{i=1,2,\ldots ,N}\mathrm{rad}(\sigma _i)\right) . \end{aligned}$$

(140)

Here, $\mathrm{rad}(\sigma _i):=\sup \left\{ |y_i-z|\;\big |\;z\in \sigma _i\right\}$. Furthermore, because $\sigma$ is arbitrary, we can estimate the numerical integration as

$$\begin{aligned}&\left| \int _{\Omega _{H}} g(y)dy -\sum _{i=1,2,\ldots ,N}|\sigma _i|g(y_i)\right| \nonumber \\&\quad =\mathcal {O}\left( \inf _{\sigma }\max _{i=1,2,\ldots ,N}\mathrm{rad}(\sigma _i)\right) . \end{aligned}$$

(141)

Let any $\sigma =\{\sigma _i\}_{i=1}^{N}$ such that (21)–(23). Furthermore, we assume $\delta _\infty \le h$. Then, by Taylor’s theorem, we can estimate the following:

$$\begin{aligned} |\widehat{E}_{i}|&=\frac{C_{\Delta }}{h^2}\bigg |\int _{\Omega _{H}}\{\phi (y)-\phi _i\}w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\qquad -\sum _{j=1}^NV_{j} (\phi _j-\phi _i)w^{\Delta }_h(|x_{j}^{}-x_{i}^{}|)\bigg | \nonumber \\&\le \frac{C_{\Delta }}{h^2}\bigg |\int _{\Omega _{H}}\{\phi (y)-\phi _i\}w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\qquad -\sum _{j=1}^N(\phi _j-\phi _i)\int _{\sigma _j}w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y\bigg | \nonumber \\&\qquad +\frac{C_{\Delta }}{h^2}\bigg |\sum _{j=1}^N(\phi _j-\phi _i)\int _{\sigma _j}w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y\nonumber \\&\qquad -\sum _{j=1}^NV_{j} (\phi _j-\phi _i)w^{\Delta }_h(|x_{j}^{}-x_{i}^{}|)\bigg | \nonumber \\&\le \frac{C_{\Delta }}{h^2}\bigg |\sum _{j=1}^N\int _{\sigma _j}\{\phi (y)-\phi _j\}w^{\Delta }_h(|y-x_{i}^{}|)\mathrm{\,d}y\bigg | \nonumber \\&\qquad +\frac{C_{\Delta }}{h^2}\bigg |\sum _{j=1}^N(\phi _j-\phi _i)\int _{\sigma _j}\{w^{\Delta }_h(|y-x_{i}^{}|) \nonumber \\&\qquad -\,w^{\Delta }_h(|x_{j}^{}-x_{i}^{}|)\}\mathrm{\,d}y\bigg | \nonumber \\&\le \frac{\delta _{\sigma }}{h^2}|\phi |_{C^1(\overline{\Omega }_{H})}C_{\Delta }C_{1}(w^{\Delta })\nonumber \\&\qquad +\frac{h+\delta _{\sigma }}{h^2}|\phi |_{C^1(\overline{\Omega }_{H})}C_{\Delta }\sum _{j\in \{k;\,|x_{k}^{}-x_{i}^{}|<h+\delta _{\sigma }\}} \nonumber \\&\qquad \times \int _{\sigma _j}|w^{\Delta }_h(|y-x_{i}^{}|)-w^{\Delta }_h(|x_{j}^{}-x_{i}^{}|)|\mathrm{\,d}y\nonumber \\&=\frac{\delta _{\sigma }}{h^2}|\phi |_{C^1(\overline{\Omega }_{H})}C_{\Delta }C_{1}(w^{\Delta })\nonumber \\&\qquad +\left( 1+\frac{\delta _{\sigma }}{h}\right) \frac{\delta _{\sigma }}{h}|\phi |_{C^1(\overline{\Omega }_{H})}C_{\Delta }\nonumber \\&\qquad \times \int _{\mathbb {R}^d}\bigg | \frac{\mathrm{d}}{\mathrm{d}{r}} w^{\Delta }_h(|y-x_{i}^{}|)\bigg |\mathrm{\,d}y\nonumber \\&\qquad +\,\mathcal {O}(\delta _{\sigma }^2h^{-3}) \nonumber \\&=C_{\Delta }\left( C_{0}(w^{\Delta })+2\int _{\mathbb {R}^d}\bigg | \frac{\mathrm{d}}{\mathrm{d}{r}} w^{\Delta }(|y|)\bigg |\mathrm{\,d}y\right) \nonumber \\&\qquad \times \frac{\delta _{\sigma }}{h^2}|\phi |_{C^1(\overline{\Omega }_{H})}+\mathcal {O}(\delta _{\sigma }^2h^{-3}) \end{aligned}$$

(142)

Because $\sigma$ is arbitrary, we obtain

$$\begin{aligned}&|\widehat{E}_{i}|=\nonumber \\&C_{\Delta }\left( C_{0}(w^{\Delta })+2\int _{\mathbb {R}^d}\left| \frac{\mathrm{d}}{\mathrm{d}{r}} w^{\Delta }(|y|)\right| \mathrm{\,d}y\right) \frac{\delta _{\infty }}{h^2}|\phi |_{C^1(\overline{\Omega }_{H})}\nonumber \\&\quad +\,\mathcal {O}(\delta _{\infty }^2h^{-3}). \end{aligned}$$

(143)

Hence, $\delta _\infty \le h$ yields

$$\begin{aligned} |\widehat{E}_{i}| = \mathcal {O}(\delta _{\infty }h^{-2}). \end{aligned}$$

(144)

Consequently, by (135) and (136), and (144), we establish

$$\begin{aligned} |\Delta \phi _i-\Delta _{h}^{}\phi _i| = \mathcal {O}(h^2+\delta _{\infty }h^{-2}). \end{aligned}$$

(145)