Improving Numerical Accuracy of the Localized Oscillatory Radial Basis Functions Collocation Method for Solving Elliptic Partial Differential Equations in 2D

Abstract

Recently, the localized oscillatory radial basis functions collocation method (L-ORBFs) has been introduced to solve elliptic partial differential equations in 2D with a large number of computational nodes. The research clearly shows that the L-ORBFs is very convenient and useful for solving large-scale problems, but this method is numerically less accurate. In this paper, we propose a numerical scheme to improve the accuracy of the L-ORBFs by adding low-degree polynomials in the localized collocation process. The numerical results validate that the proposed numerical scheme is highly accurate and clearly outperforms the results of the L-ORBFs.

1. Introduction

The state-of-the-art meshless methods using radial basis functions (RBFs) continue to evolve as researchers have been exploring new applications and techniques for decades. RBFs are a type of mathematical function commonly used in interpolation, approximation, and machine learning applications. The most commonly used RBFs in the literature that are non-oscillatory in nature are the multiquadric, inverse multiquadric, Gaussian, and polyharmonic splines. Researchers have also employed similar collocation methods for solving various types of problems, such as fractional chemical clock reactions, logarithmic integro-differential equations, and the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation, with nonlocal conditions [1,2,3]. In 2006, Fornberg et al. [4] introduced and discussed various properties of a new class of radial basis functions known as oscillatory radial basis functions (ORBFs). ORBFs are a particular class of RBFs that are characterized by their oscillatory behavior, which allows them to capture the oscillatory behavior of functions. They are often used in problems that involve functions with oscillatory behavior, such as signal processing, image processing, and computational electromagnetics.

Chen et al. [5,6,7] introduced an RBFs-based method known as the method of approximate particular solutions (MAPS), which has been shown to be very effective in solving various problems in fluid mechanics, heat transfer, and solid mechanics. Among many other particular solution-based methods, the MAPS requires particular solutions of the corresponding RBFs. To that end, Lamichhane et al. [8] derived the particular solutions of Poisson’s equation in 2D using ORBFs and successfully implemented the derived particular solutions in the method of approximate particular solutions for solving various elliptic partial differential equations (PDEs). This method is known as the ORBFs collocation method, which outperformed the multiquadric RBF in terms of accuracy. However, global methods have become prohibitively expensive or inefficient when the size of the domain is very large. Therefore, researchers in this field have employed localized methods for decades to solve large-scale problems. Localized methods in computational mathematics are numerical techniques that focus on approximating solutions to PDEs in a specified (small) region or domain of interest rather than the entire domain. By focusing on the behavior of the solution in a specific region of the domain, localized methods can handle a quite large number of computational nodes as compared to the global methods that attempt to solve the entire domain at once [9,10,11,12,13,14].

Lamichhane et al. [9] recently introduced and successfully implemented the localized oscillatory radial basis functions collocation method (L-ORBFs) for solving elliptic PDEs in 2D in large-scale problems. While implementing such localized collocation methods, improving numerical accuracy is of great concern. Researchers have employed various techniques to improve the accuracy of such numerical methods. One of the techniques we have seen in the literature is to augment polynomial terms in the localization process [15,16], which significantly improves the numerical accuracy while solving large-scale problems. In this article, we propose a numerical scheme in which we add low-degree polynomial terms in the localized collocation process to improve the accuracy of the L-ORBFs. We demonstrate the applicability of the proposed numerical method for the solutions of linear elliptic PDEs with constant and variable coefficients in 2D. The numerical results clearly show that our method is more highly accurate than the L-ORBFs.

The outline of the paper is as follows. In Section 2, we present the formulation of our proposed numerical scheme on the elliptic boundary value problem. In Section 3, numerical experiments are given and discussed in detail. A comparison study of the proposed scheme with the L-ORBFs is performed in our numerical experiments. A summary of our work and a future direction for research are furnished in Section 4.

2. L-ORBFs with Augmented Polynomial Terms

In this section, we present the proposed numerical approach, L-ORBFs with augmented polynomials, in the localized collocation process. For simplicity, we consider following elliptic PDEs with a Dirichlet boundary condition:

$$\begin{array}{cc}\hfill {\alpha }_1\left(\mathsf{\xi }\right)\Delta v\left(\mathsf{\xi }\right)+{\alpha }_2\left(\mathsf{\xi }\right)v_{x_1}\left(\mathsf{\xi }\right)+{\alpha }_3\left(\mathsf{\xi }\right)v_{x_2}\left(\mathsf{\xi }\right)+{\alpha }_4\left(\mathsf{\xi }\right)v\left(\mathsf{\xi }\right)& =f\left(\mathsf{\xi }\right),\mathsf{\xi }\in {\Gamma }_I,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill v\left(\mathsf{\xi }\right)& \hfill =g\left(\mathsf{\xi }\right),\mathsf{\xi }\in {\Gamma }_B,\end{array}$$

(2)

where ${\alpha }_i\left(\mathsf{\xi }\right):={\alpha }_i(x_1,x_2),i=1,2,3,4$ are constant or variable coefficients; $\mathsf{\xi }=(x_1,x_2)$, $v\left(\mathsf{\xi }\right):=v(x_1,x_2)$ is the analytical solution; $v_{x_1},v_{x_2}$ are partial derivatives of v with respect to $x_1,x_2$, respectively; ${\Gamma }_I$ and ${\Gamma }_B$ are, respectively, the interior and boundary of the computational domain; and $f\left(\mathsf{\xi }\right)$ and $g\left(\mathsf{\xi }\right)$ are the known functions.

Consider ${\mathsf{\xi }}_k,k=1,\dots ,\mathcal{N}$ computational nodes with $n_i$ interior nodes in ${\Gamma }_I$ and $n_b$ boundary nodes on ${\Gamma }_B$ such that $\mathcal{N}=n_i+n_b$. We choose one of the ORBFs [4] defined by

$${\varphi }_{d_k}\left(\mathsf{\xi }\right):={\varphi }_d\left(\right||\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|)={\displaystyle \frac{J_{d-1/2}\left(\epsilon \right||\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|)}{\left(\epsilon \right||\mathsf{\xi }-{\mathsf{\xi }}_k{\left|\right|)}^{d-1/2}}},\epsilon >0,d=0,1,2,\dots ,$$

(3)

where $\left|\right|.\left|\right|$ is the Euclidean norm, $\epsilon$ is the shape parameter, $J_{d-1/2}$ signifies the J Bessel function of the first kind of order $d-1/2$, and ${\varphi }_d$ is called the ORBF of order d. To solve the elliptic PDEs (1) and (2) using the proposed scheme, we require the particular solutions for $\Delta {\Phi }_d={\varphi }_d$, derived in [8], as follows:

For $\left|\right|\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|\ne 0,$

$$\begin{array}{c}{\Phi }_{d_k}\left(\mathsf{\xi }\right):={\Phi }_d\left(\right||\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|)={\displaystyle \frac{1}{{\epsilon }^2}}\left[\sum _{i=1}^{d-1}{\displaystyle \frac{(2d-2i-3)!!}{(2d-3)!!}}{\varphi }_{d-i}\left(\right||\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|)\right.\hfill \\ \hfill \left.-\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{1}{(2d-3)!!}}\left(\mathrm{Ci}\left(\epsilon \right||\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|)-\ln \left(\right||\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|)\right)\right],\end{array}$$

where $!!$ is the double factorial defined as

$$\left(a\right)!!=\left\{\begin{array}{cc}a·(a-2)\dots 5·3·1,\hfill & a:\mathrm{positive} \mathrm{odd} \mathrm{integer},\hfill \\ a·(a-2)\dots 6·4·2,\hfill & a:\mathrm{positive} \mathrm{even} \mathrm{integer},\hfill \\ 1,\hfill & a=-1,0.\hfill \end{array}\right.$$

and

$$Ci\left(\epsilon \right||\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|)=-{\int }_{\epsilon \left|\right|\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|}^{\infty }\frac{\cos t}{t}dt$$

is the cosine integral function.

For $\left|\right|\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|=0,$ we use ${\displaystyle \underset{\left|\right|\mathsf{\xi }-{\mathsf{\xi }}_k\left|\right|\to 0}{\lim }{\Phi }_{d_k}\left(\mathsf{\xi }\right).}$ Details on deriving the particular solutions using ORBFs can be found in [8].

Next, we choose $n_q$ polynomials $p_s\left(\mathsf{\xi }\right)=x_1^{m-s}{x}_2^s$ up to a degree of m as follows:

$$\begin{array}{c}\mathrm{degree} 0: 1\\ \mathrm{degree} 1: x_1{x}_2\\ \mathrm{degree} 2: x_1^2{x}_1{x}_2{x}_2^2\\ \mathrm{degree} 3: x_1^3{x}_1^2{x}_2{x}_1{x}_2^2{x}_2^3\\ \mathrm{degree} \mathrm{m}: x_1^m{x}_1^{m-1}{x}_2\dots x_1^{m-s}{x}_2^s\dots x_1{x}_2^{m-1}{x}_2^m\end{array}$$

Note that $n_q={\displaystyle \frac{(m+1)(m+2)}{2}}.$ For example, if we choose polynomials up to a degree of three, then we augment 10 polynomial terms $\{1,x_1,x_2,x_1^2,x_1{x}_2,x_2^2,x_1^3,x_1^2{x}_2,x_1{x}_2^2,x_2^3\}$.

Now, we choose $n_l$ nearest nodes ${\mathsf{\xi }}_j^{\left[k\right]},j=1,2,\dots ,n_l$ at each node ${\mathsf{\xi }}_k$ to form a corresponding local domain ${\Gamma }_k$. Then, at each node $\mathsf{\xi }\in {\Gamma }_k,$ we approximate $v\left(\mathsf{\xi }\right)$ by

$$\widehat{v}\left(\mathsf{\xi }\right)=\sum _{j=1}^{n_l}{\beta }_j^{\left[k\right]}{\Phi }_{d_j}\left(\mathsf{\xi }\right)+\sum _{s=1}^{n_q}{\beta }_{n_l+s}^{\left[k\right]}{p}_s\left(\mathsf{\xi }\right),$$

(4)

where ${\Phi }_{d_j}\left(\mathsf{\xi }\right)={\Phi }_d\left(\right||\mathsf{\xi }-{\mathsf{\xi }}_j^{\left[k\right]}\left|\right|)$ and $p_s\left(\mathsf{\xi }\right)$ are the additional polynomial terms. Because of the addition of $n_q$ terms in (4), we need to add the following $n_q$ constraints in the system of equations:

$$\sum _{j=1}^{n_l}{\beta }_j^{\left[k\right]}{p}_s\left({\mathsf{\xi }}_j\right)=0,s=1,2,3,\dots ,n_q.$$

(5)

For ${\mathsf{\xi }}_j^{\left[k\right]},j=1,2,\dots ,n_l,$ (4) and (5) imply a matrix equation

$$\left[\begin{array}{c}{\widehat{\mathbf{v}}}^{\left[k\right]}\\ {\mathbf{0}}_{n_q}\end{array}\right]=\left[\begin{array}{cc}{\mathsf{\Psi }}_{n_{ll}}& {\mathbf{P}}_{n_{lq}}\\ {\mathbf{P}}_{n_{lq}}^T& {\mathbf{0}}_{n_{qq}}\end{array}\right]{\mathsf{\beta }}^{\left[k\right]},$$

(6)

where

$${\mathsf{\Psi }}_{n_{ll}}=\left[\begin{array}{ccc}{\Phi }_1\left({\mathsf{\xi }}_1^{\left[k\right]}\right)& \cdots & {\Phi }_{n_l}\left({\mathsf{\xi }}_1^{\left[k\right]}\right)\\ {\Phi }_1\left({\mathsf{\xi }}_2^{\left[k\right]}\right)& \cdots & {\Phi }_{n_l}\left({\mathsf{\xi }}_2^{\left[k\right]}\right)\\ ⋮& \ddots & ⋮\\ {\Phi }_1\left({\mathsf{\xi }}_{n_l}^{\left[k\right]}\right)& \cdots & {\Phi }_{n_l}\left({\mathsf{\xi }}_{n_l}^{\left[k\right]}\right)\end{array}\right],{\mathbf{P}}_{n_{lq}}=\left[\begin{array}{ccc}{p}_1\left({\mathsf{\xi }}_1^{\left[k\right]}\right)& \cdots & p_{n_q}\left({\mathsf{\xi }}_1^{\left[k\right]}\right)\\ p_1\left({\mathsf{\xi }}_2^{\left[k\right]}\right)& \cdots & p_{n_q}\left({\mathsf{\xi }}_2^{\left[k\right]}\right)\\ ⋮& \ddots & ⋮\\ p_1\left({\mathsf{\xi }}_{n_l}^{\left[k\right]}\right)& \cdots & p_{n_q}\left({\mathsf{\xi }}_{n_l}^{\left[k\right]}\right)\end{array}\right],$$

$${\widehat{\mathbf{v}}}^{\left[k\right]}=\left[\begin{array}{c}\widehat{v}\left({\mathsf{\xi }}_1^{\left[k\right]}\right)\\ \widehat{v}\left({\mathsf{\xi }}_2^{\left[k\right]}\right)\\ ⋮\\ \widehat{v}\left({\mathsf{\xi }}_{n_l}^{\left[k\right]}\right)\end{array}\right],{\beta }^{\left[k\right]}=\left[\begin{array}{c}{\beta }_1^{\left[k\right]}\\ {\beta }_2^{\left[k\right]}\\ ⋮\\ {\beta }_{n_l}^{\left[k\right]}\\ {\beta }_{n_l+1}^{\left[k\right]}\\ ⋮\\ {\beta }_{n_l+n_q}^{\left[k\right]}\end{array}\right],$$

${\mathbf{0}}_{n_q}$ and ${\mathbf{0}}_{n_{qq}}$ are zero matrices of sizes $n_q\times 1$ and $n_q\times n_q$, respectively.

Solving (6), we obtain

$${\mathsf{\beta }}^{\left[k\right]}={\left[\begin{array}{cc}{\mathsf{\Psi }}_{n_{ll}}& {\mathbf{P}}_{n_{lq}}\\ {\mathbf{P}}_{n_{lq}}^T& {\mathbf{0}}_{n_{qq}}\end{array}\right]}^{-1}\left[\begin{array}{c}{\widehat{\mathbf{v}}}^{\left[k\right]}\\ {\mathbf{0}}_{n_q}\end{array}\right].$$

(7)

To solve (1) and (2), we assume that for each ${\mathsf{\xi }}_k,k=1,\dots ,\mathcal{N}$, (4) satisfies (1) and (2). This implies that for $k=1,\dots ,n_i,$

$$\begin{array}{ccc}\hfill f\left({\mathsf{\xi }}_k\right)& =& {\alpha }_1\left({\mathsf{\xi }}_k\right)\Delta \widehat{v}\left({\mathsf{\xi }}_k\right)+{\alpha }_2\left({\mathsf{\xi }}_k\right){\widehat{v}}_{x_1}\left({\mathsf{\xi }}_k\right)+{\alpha }_3\left({\mathsf{\xi }}_k\right){\widehat{v}}_{x_2}\left({\mathsf{\xi }}_k\right)+{\alpha }_4\left({\mathsf{\xi }}_k\right)\widehat{v}\left({\mathsf{\xi }}_k\right)\hfill \\ & =& \left[{\alpha }_1\left({\mathsf{\xi }}_k\right)\Delta +{\alpha }_2\left({\mathsf{\xi }}_k\right)\partial /\partial x_1+{\alpha }_3\left({\mathsf{\xi }}_k\right)\partial /\partial x_2+{\alpha }_4\left({\mathsf{\xi }}_k\right)\right]\widehat{v}\left({\mathsf{\xi }}_k\right)\hfill \\ & =& \left[{\alpha }_1\left({\mathsf{\xi }}_k\right)\Delta +{\alpha }_2\left({\mathsf{\xi }}_k\right)\partial /\partial x_1+{\alpha }_3\left({\mathsf{\xi }}_k\right)\partial /\partial x_2+{\alpha }_4\left({\mathsf{\xi }}_k\right)\right]\left[\sum _{j=1}^{n_l}{\beta }_j^{\left[k\right]}{\Phi }_j\left({\mathsf{\xi }}_k\right)+\sum _{s=1}^{n_q}{\beta }_{n_l+s}^{\left[k\right]}{p}_s\left({\mathsf{\xi }}_k\right)\right].\hfill \end{array}$$

Using the linearity of a differential operator, we obtain

$$\begin{array}{ccc}\hfill f\left({\mathsf{\xi }}_k\right)& =& \sum _{j=1}^{n_l}{\beta }_j^{\left[k\right]}\mathbb{L}{\Phi }_j\left({\mathsf{\xi }}_k\right)+\sum _{s=1}^{n_q}{\beta }_{n_l+s}^{\left[k\right]}\mathbb{L}{p}_s\left({\mathsf{\xi }}_k\right),\hfill \end{array}$$

where $\mathbb{L}:={\alpha }_1\left({\mathsf{\xi }}_k\right)\Delta +{\alpha }_2\left({\mathsf{\xi }}_k\right)\partial /\partial x_1+{\alpha }_3\left({\mathsf{\xi }}_k\right)\partial /\partial x_2+{\alpha }_4\left({\mathsf{\xi }}_k\right)$. This implies that

$$\begin{array}{ccc}\hfill f\left({\mathsf{\xi }}_k\right)& =& \mathbb{L}{\mathsf{\Phi }}^{\left[k\right]}{\mathsf{\beta }}^{\left[k\right]},\hfill \\ & & \mathrm{where}\mathbb{L}{\mathsf{\Phi }}^{\left[k\right]}={[\mathbb{L}{\Phi }_1\left({\mathsf{\xi }}_k\right)\dots \mathbb{L}{\Phi }_{n_l}\left({\mathsf{\xi }}_k\right)\mathbb{L}{p}_1\left({\mathsf{\xi }}_k\right)\dots \mathbb{L}{p}_{n_q}\left({\mathsf{\xi }}_k\right)]}_{1\times (n_l+n_q)}\hfill \\ & =& \mathbb{L}{\mathsf{\Phi }}^{\left[k\right]}{\left[\begin{array}{cc}{\mathsf{\Psi }}_{n_{ll}}& {\mathbf{P}}_{n_{lq}}\\ {\mathbf{P}}_{n_{lq}}^T& {\mathbf{0}}_{n_{qq}}\end{array}\right]}^{-1}\left[\begin{array}{c}{\widehat{\mathbf{v}}}^{\left[k\right]}\\ {\mathbf{0}}_{n_q}\end{array}\right]\hfill \\ & =& {\mathsf{\Lambda }}^{\left[k\right]}\left[\begin{array}{c}{\widehat{\mathbf{v}}}^{\left[k\right]}\\ {\mathbf{0}}_{n_q}\end{array}\right]\hfill \\ & =& \mathsf{\Lambda }\left({\mathsf{\xi }}_k\right)\widehat{\mathbf{v}},\hfill \end{array}$$

where

$${\mathsf{\Lambda }}^{\left[k\right]}=\mathbb{L}{\mathsf{\Phi }}^{\left[k\right]}{\left[\begin{array}{cc}{\mathsf{\Psi }}_{n_{ll}}& {\mathbf{P}}_{n_{lq}}\\ {\mathbf{P}}_{n_{lq}}^T& {\mathbf{0}}_{n_{qq}}\end{array}\right]}^{-1}$$

and $\mathsf{\Lambda }\left({\mathsf{\xi }}_k\right)$ is obtained by inserting $\mathcal{N}-n_l$ zeros in ${\mathsf{\Lambda }}^{\left[k\right]}$ based on mapping between ${\widehat{\mathbf{v}}}^{\left[k\right]}$ and $\widehat{\mathbf{v}}.$ Thus,

$$f\left({\mathsf{\xi }}_k\right)=\mathsf{\Lambda }\left({\mathsf{\xi }}_k\right)\widehat{\mathbf{v}},1\le k\le n_i.$$

(8)

For $k=n_i+1,\dots ,\mathcal{N},$ in general, if we have any boundary operator $\mathbb{B}$, we obtain

$$\begin{array}{ccc}\hfill g\left({\mathsf{\xi }}_k\right)=\mathbb{B}\widehat{v}\left({\mathsf{\xi }}_k\right)& =& \sum _{j=1}^{n_l}{\beta }_j^{\left[k\right]}\mathbb{B}{\Phi }_j\left({\mathsf{\xi }}_k\right)+\sum _{s=1}^{n_q}{\beta }_{n_l+s}^{\left[k\right]}\mathbb{B}{p}_s\left({\mathsf{\xi }}_k\right)\hfill \\ \\ & =& \mathbb{B}{\Phi }^{\left[k\right]}{\mathsf{\beta }}^{\left[k\right]},\hfill \\ & & \mathrm{where}\mathbb{B}{\mathsf{\Phi }}^{\left[k\right]}={[\mathbb{B}{\Phi }_1\left({\mathbf{x}}_k\right)\dots \mathbb{B}{\Phi }_{n_l}\left({\mathbf{x}}_k\right)\mathbb{B}{p}_1\left({\mathsf{\xi }}_k\right)\dots \mathbb{B}{p}_{n_q}\left({\mathsf{\xi }}_k\right)]}_{1\times (n_l+n_q)}\hfill \\ & =& \mathbb{B}{\mathsf{\Phi }}^{\left[k\right]}{\left[\begin{array}{cc}{\mathsf{\Psi }}_{n_{ll}}& {\mathbf{P}}_{n_{lq}}\\ {\mathbf{P}}_{n_{lq}}^T& {\mathbf{0}}_{n_{qq}}\end{array}\right]}^{-1}\left[\begin{array}{c}{\widehat{\mathbf{v}}}^{\left[k\right]}\\ {\mathbf{0}}_{n_q}\end{array}\right]\hfill \\ & =& {\mathsf{{\rm Y}}}^{\left[k\right]}\left[\begin{array}{c}{\widehat{\mathbf{v}}}^{\left[k\right]}\\ {\mathbf{0}}_{n_q}\end{array}\right]\hfill \\ & =& \mathsf{{\rm Y}}\left({\mathsf{\xi }}_k\right)\widehat{\mathbf{v}},\hfill \end{array}$$

where

$${\mathsf{{\rm Y}}}^{\left[k\right]}=\mathbb{B}{\Phi }^{\left[k\right]}{\left[\begin{array}{cc}{\mathsf{\Psi }}_{n_{ll}}& {\mathbf{P}}_{n_{lq}}\\ {\mathbf{P}}_{n_{lq}}^T& {\mathbf{0}}_{n_{qq}}\end{array}\right]}^{-1}$$

and $\mathsf{{\rm Y}}\left({\mathsf{\xi }}_k\right)$ is obtained by inserting $\mathcal{N}-n_l$ zeros in ${\mathsf{{\rm Y}}}^{\left[k\right]}$ based on mapping between ${\widehat{\mathbf{v}}}^{\left[k\right]}$ and $\widehat{\mathbf{v}}.$

For the identity boundary operator in (2), we can create the vector $\mathsf{{\rm Y}}\left({\mathsf{\xi }}_k\right)$ whose $k^{th}$ entry is 1 and other entries are 0.

Thus,

$$g\left({\mathsf{\xi }}_k\right)=\mathsf{{\rm Y}}\left({\mathsf{\xi }}_k\right)\widehat{\mathbf{v}},n_i+1\le k\le \mathcal{N}.$$

(9)

The approximate solution $\widehat{v}$ of (1) and (2) at each of the given computational nodes is obtained by solving Equations (8) and (9).

3. Numerical Results

We perform three numerical experiments to verify our proposed numerical approach. The ${\mathrm{MATLAB}}_{\copyright }$ R2021a built-in functions KDTreeSearcher and knnsearch are used to find the nearest neighborhood nodes on the local computational domain ${\Gamma }_k$. These built-in functions are available in the ${\mathrm{MATLAB}}_{\copyright }$ statistics and machine learning toolbox.

The numerical accuracy is measured by the root mean square error (RMSE) and $l_{\infty }$ error, which are defined as

$$\begin{array}{c}\hfill \mathrm{RMSE}=\sqrt{\frac{1}{\mathcal{N}}\sum _{k=1}^{\mathcal{N}}{\left[v\left({\mathsf{\xi }}_k\right)-\widehat{v}\left({\mathsf{\xi }}_k\right)\right]}^2}\end{array}$$

and

$$\begin{array}{c}\hfill l_{\infty }=\underset{k}{\max }|v\left({\mathsf{\xi }}_k\right)-\widehat{v}\left({\mathsf{\xi }}_k\right)|.\end{array}$$

Unless otherwise specified, on most of the numerical experiments, we choose the first order of ORBFs ${\varphi }_1$, and the shape parameter $\epsilon$ is chosen by a heuristic approach. Each local domain ${\Gamma }_k$ contains $n_l=25$ nearest neighborhood nodes, and the augmented polynomial terms are of a degree up to $m=4$. Along with the unit square domain $[0,1]\times [0,1]$, we have used some of the irregular domains bounded by the following parametric curves:

$$\left\{(x_1,x_2)\right|x_1=r\cos t,x_2=r\sin t,0\le t<2\pi \},$$

where

• $r=e^{\sin t}{\sin }^2\left(2t\right)+e^{\cos t}{\cos }^2\left(2t\right),$ amoeba-shaped curve (Figure 1a);
• $r={\left(\cos \left(4t\right)+\sqrt{18/5-{\sin }^2\left(4t\right)}\right)}^{1/3},$ Cassini-shaped curve (Figure 1b);
• $r=(1+{\cos }^2\left(4t\right)),$ star-shaped curve (Figure 1c);
• $r=1+\frac{1}{15}\tanh \left[15\sin \left(15t\right)\right],$ gear curve (Figure 1d) with 15 cogs.

Another type of gear curve (Figure 1e) with six cogs used in the experiment is defined as

$$\left\{(x_1,x_2)\right|x_1=r\cos (t+1/2\sin \left(6t\right)),x_2=r\sin (t+1/2\sin \left(6t\right)),0\le t<2\pi \},$$

where $r=(2+1/2\sin (6t\left)\right).$

Figure 1. Irregularly shaped curves used in the experiments to bound the computational domain.

Figure 1. Irregularly shaped curves used in the experiments to bound the computational domain.

Example 1.

We consider the elliptic PDEs (1) and (2) with ${\alpha }_1\left(\mathsf{\xi }\right)=1 and{\alpha }_i\left(\mathsf{\xi }\right)=0(\mathrm{for}i=2,3,4)$; $f\left(\mathsf{\xi }\right)$ and $g\left(\mathsf{\xi }\right)$ are defined based on the analytical solution given below

$$\begin{array}{c}\hfill v\left(\mathsf{\xi }\right):=v(x_1,x_2)=e^{{\omega }_1{x}_1+{\omega }_2{x}_2},\end{array}$$

(10)

where ${\omega }_1,{\omega }_2$ are constants.

First, we investigate the numerical accuracy of the L-ORBFs and the L-ORBFs with augmented polynomial terms in the localized collocation process for solving the Poisson’s equation given in this example. For this numerical experiment, a family of exponential-type analytical solutions is chosen. In

Table 1. Example 1: Comparison of RMSE obtained by L-ORBFs and proposed approach for the family of exponential functions $v(x_1,x_2)=e^{{\omega }_1{x}_1+{\omega }_2{x}_2}$.

		L-ORBFs		Proposed Approach
${\mathsf{\omega }}_{\mathbf{1}}$	${\mathsf{\omega }}_{\mathbf{2}}$	$\mathsf{\epsilon }$	RMSE	$\mathsf{\epsilon }$	RMSE
1	1	10.3	1.99 × 10−5	$18.5$	8.12 × 10−10
2	1	10.3	7.18 × 10−5	19	1.60 × 10−8
2	2	8.8	1.78 × 10−4	$15.9$	1.11 × 10−7
2	3	7.8	7.86 × 10−4	$16.9$	1.76 × 10−6
3	3	12.9	1.27 × 10−3	$15.9$	6.20× 10−6
4	4	12.9	9.71 × 10−3	$16.9$	1.71 × 10−4
1	5	7.8	6.49 × 10−3	$18.0$	2.69 × 10−5
5	5	12.9	8.25 × 10−2	$16.9$	2.35 × 10−3

, we present the RMSE of the L-ORBFs and the proposed approach with different values of ${\omega }_1$ and ${\omega }_2$ in the analytical solution $v(x_1,x_2)=e^{{\omega }_1{x}_1+{\omega }_2{x}_2}$. A unit square domain is considered with $n_i=20,000$ interior nodes and $n_b=880$ boundary nodes. Generally, the L-ORBFs is suitable for approximating the functions that are oscillatory in nature, but the result shows that the proposed approach not only performs better than the L-ORBFs but also solves PDEs with high accuracy when the analytical solutions are exponential in nature.

Without loss of generality, we choose ${\omega }_1=2,{\omega }_2=1$. In Figure 2, we observe the highly accurate approximate solution $\widehat{v}$ of the analytical solution $v(x_1,x_2)=e^{2x_1+x_2}$ over a unit square domain. The error distribution plot in Figure 2c shows that the proposed approach is highly accurate everywhere in the domain except for a small region with $l_{\infty }=1$ × 10${}^{-8}$.

As meshless methods are renowned for dealing with irregular domains, we have compared our proposed approach with the L-ORBFs for solving the given Poisson’s equation defined over the domain bounded by irregular curves. Figure 3a depicts the profile of the analytical solution $v(x_1,x_2)=e^{2x_1+x_2}$ over the computational domain bounded by the gear-shaped curve with six cogs. We choose $n_i=\mathrm{20,393}$ interior nodes and $n_b=800$ boundary nodes over the domain. The RMSE vs. the error profile of the shape parameter $\epsilon$ depicted in Figure 3b clearly shows that the proposed approach produces better numerical accuracy (RMSE = $4.72$ × 10${}^{-7}$ at $\epsilon =14$) than the L-ORBFs (RMSE = $3.14$ × 10${}^{-4}$ at $\epsilon =10.8$).

Figure 4 depicts the numerical results obtained over a computational domain ($n_i=\mathrm{20,431}$, $n_b=880$) bounded by the Cassini-shaped curve. In Figure 4a, we observe that the proposed approach with the fourth degree augmented polynomial terms has performed better than the L-ORBFs in terms of the numerical accuracy. A similar phenomenon can be observed in

Table 2. Example 1: Comparison of RMSE with L-ORBFs and proposed approach for domains bounded by different irregular curves.

	L-ORBFs		Proposed Approach
Domain	$\mathsf{\epsilon }$	RMSE	$\mathsf{\epsilon }$	RMSE
Cassini-shaped	9.7	5.79 × 10−5	14.4	1.57 × 10−8
star-shaped	13.2	1.62 × 10−4	10.5	4.07 × 10−8
gear-shaped (6 cogs)	10.8	3.14 × 10−4	14	4.72 × 10−7
amoeba-shaped	10.8	1.32 × 10−3	14	1.64 × 10−7

, which provides the comparison of the RMSEs between the L-ORBFs and the proposed approach for the domains bounded by different irregular curves. Figure 5 portrays the profile of the RMSE vs. the shape parameter for the star-shaped and amoeba-shaped domains. In both cases, our proposed approach undoubtedly surpassed the accuracy of the L-ORBFs. Figure 4b shows the profile of the RMSE vs. the degree m of the augmented polynomials. We clearly notice from Figure 4b that when we increase the degree m of the augmented polynomials, we obtain an increasingly accurate solution. During our experiment, we also observed that when the total number of polynomial terms is closer to or surpasses the $n_l$ nearest neighborhood nodes, then the method is unstable and ceases to produce results. Therefore, we have chosen low-degree polynomials in our proposed approach. If more polynomial terms are to be used, then we need to increase the size of the local computational domain, which provides highly accurate solutions; however, this will be computationally more expensive.

Next, we solve Example 1 with a large number of computational nodes. A unit square domain is considered using $\mathrm{100,000}$ as the interior and 2080 as the boundary nodes.

Table 3. Example 1: Comparison of the RMSE obtained by the L-ORBFs and the proposed approach for varied number $n_l$ of the nearest neighborhood nodes on the local domains.

	L-ORBFs		Proposed Approach
${\mathit{n}}_{\mathit{l}}$	$\mathsf{\epsilon }$	RMSE	$\mathsf{\epsilon }$	RMSE
25	$19.8$	1.10 × 10−4	$48.5$	1.89 × 10−10
35	$39.6$	3.09 × 10−5	$42.1$	3.09 × 10−10
45	$13.4$	6.89 × 10−5	$43.1$	1.59 × 10−10
55	$12.9$	4.17 × 10−5	$37.6$	5.23 × 10−10
65	$26.7$	4.89 × 10−5	$28.7$	7.19 × 10−10
75	$49.0$	3.50 × 10−5	$27.2$	2.06 × 10−10

presents the numerical results for the varied number $n_l$ of nearest neighborhood nodes on the local domains. We clearly observed that the numerical accuracy is consistently around 1 × 10${}^{-10}$ for a varied number of local computational nodes.

In

Table 4. Example 1: The RMSE and $l_{\infty }$ error for the proposed method with varied highest degree m of the augmented polynomials.

m	$\mathit{\epsilon }$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
0	$15.4$	6.50 × 10−5	2.40 × 10−4	80
1	$37.6$	4.54 × 10−6	1.50 × 10−5	115
2	$47.5$	6.53 × 10−7	2.35 × 10−6	114
3	$25.3$	3.32 × 10−8	1.12 × 10−7	111
4	$48.5$	1.53 × 10−10	2.34 × 10−9	133

, we investigate the numerical accuracy of the proposed approach by varying the highest degree m of the augmented polynomial terms. We observed that the increased m produces a more accurate solution while increasing the computational cost. We note that the RMSE using the L-ORBFs is equal to $1.10$ × 10${}^{-4}$ in less than 80 s, but, using the proposed approach, the accuracy increases to $1.53$ × 10${}^{-10}$ in 133 s.

Example 2.

Next, we choose a modified Helmholtz problem with ${\alpha }_1\left(\mathsf{\xi }\right)=1$, ${\alpha }_2\left(\mathsf{\xi }\right)={\alpha }_3\left(\mathsf{\xi }\right)=0$, and ${\alpha }_4\left(\mathsf{\xi }\right)=-{\lambda }^2,$ in (1) and (2). The functions $f\left(\mathsf{\xi }\right)$ and $g\left(\mathsf{\xi }\right)$ are considered based on the following analytical solution

$$\begin{array}{c}\hfill v\left(\mathsf{\xi }\right):=v(x_1,x_2)=\sin \left(\pi x_1\right)\cos \left(\frac{\pi x_2}{2}\right).\end{array}$$

(11)

A star-shaped domain is considered throughout this example to observe the accuracy of the proposed method. The profile of the analytical solution over the star-shaped domain is presented in Figure 6.

Table 5. Example 2: RMSE and $l_{\infty }$ error of the proposed approach for various sets of interior and boundary nodes with different degrees of the augmented polynomial terms.

$({\mathit{n}}_{\mathit{i}},{\mathit{n}}_{\mathit{b}})$	m	$\mathit{ϵ}$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
$(\mathrm{51,278},1000)$	2	41.2	7.24 × 10−8	1.65 × 10−6	131
	3	42.1	8.77 × 10−9	2.23 × 10−7	150
	4	47.4	1.46 × 10−9	7.04× 10−8	151
	5	41.2	3.97 × 10−10	2.02 × 10−8	142
	L-ORBFs	39.3	2.42 × 10−6	9.62 × 10−5	133
$(\mathrm{107,235},3000)$	2	51.9	2.46 × 10−7	9.69 × 10−6	280
	3	51.0	8.56 × 10−9	3.47 × 10−7	286
	4	56.3	4.37 × 10−10	1.42 × 10−8	280
	5	54.2	1.47 × 10−10	1.11 × 10−8	293
	L-ORBFs	56.5	2.09 × 10−6	1.81 × 10−4	297
$(\mathrm{186,540},4000)$	2	60.0	1.08 × 10−7	4.15 × 10−6	296
	3	58.8	2.00 × 10−9	7.80 × 10−8	289
	4	58.8	7.19 × 10−10	3.43 × 10−8	304
	5	59.4	1.83 × 10−11	7.39 × 10−10	310
	L-ORBFs	19.8	$5.49$ × 10${}^{-6}$	$6.84$ × 10${}^{-5}$	308

displays the numerical results for various sets of large-scale computational nodes. The size of the local computational domain is chosen to be $n_l=35$ and the wavelength is $\lambda =10$ in this experiment. The RMSE and $l_{\infty }$ error for different degrees ($m=2-5$) of the augmented polynomials are presented in this table. It is clearly observable that the accuracy of the proposed method significantly improves when we increase the degree of the augmented polynomials. However, the computational cost becomes higher with the increase in the degree of the augmented polynomials. A descending numerical accuracy is obtained by adding low-degree polynomials to the ORBFs, and we are able to validate our claim that the polynomial augmentations process on the L-ORBFs clearly outperforms the accuracy of the L-ORBFs.

Furthermore,

Table 6. Example 2: RMSE and $l_{\infty }$ error for different values of wavelength $\lambda$.

$\mathit{\lambda }$	$\mathit{ϵ}$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
1	47.4	$8.74$ × 10${}^{-9}$	$1.00$ × 10${}^{-6}$	135
10	47.4	$1.46$ × 10${}^{-9}$	$7.04$ × 10${}^{-8}$	151
50	33.1	$1.92$ × 10${}^{-10}$	$1.10$ × 10${}^{-8}$	122
100	42.9	$6.79$ × 10${}^{-11}$	$4.32$ × 10${}^{-9}$	133
500	39.3	$3.71$ × 10${}^{-12}$	$6.34$ × 10${}^{-10}$	134
1000	42.9	$4.99$ × 10${}^{-13}$	$5.06$ × 10${}^{-11}$	149

presents the numerical accuracy of our proposed scheme for different values of the wavelength $\lambda$ in the modified Helmholz problem. The interior and boundary nodes on the star-shaped domain are taken as 51,278 and 1000, respectively. The number of local computational nodes is $n_l=35$, and the highest degree of the augmented polynomials is $m=4$ for this numerical experiment. It is clear from the results that the numerical accuracy improves as we increase the wavelength in the problem. Therefore, we conclude that our method is very promising and highly accurate.

Example 3.

Finally, we choose the elliptic PDEs with ${\alpha }_1\left(\mathsf{\xi }\right)=1,{\alpha }_2\left(\mathsf{\xi }\right)=x_2^2+cos\left(x_1\right),{\alpha }_3\left(\mathsf{\xi }\right)=x_{2}sin\left(x_1\right), and{\alpha }_4\left(\mathsf{\xi }\right)=x_1^2{x}_2$ in (1) and (2); $f\left(\mathsf{\xi }\right)$ and $g\left(\mathsf{\xi }\right)$ are considered based on the following analytical solution

$$\begin{array}{c}\hfill v\left(\mathsf{\xi }\right):=v(x_1,x_2)=sin(x_2^2+x_1)-cos(x_2-x_1^2).\end{array}$$

(12)

First, we choose the computational domain bounded by the amoeba-shaped curve (Figure 1a). The profile of the analytical solution $v(x_1,x_2)=sin(x_2^2+x_1)-cos(x_2-x_1^2)$ over an amoeba-shaped domain is depicted in Figure 7a.

In this experiment, we have chosen $n_i=\mathrm{19,635}$ interior nodes and $n_b=1000$ boundary nodes in the amoeba-shaped domain. In

Table 7. Example 3: RMSE and $l_{\infty }$ error obtained by L-ORBFs for different $n_l$ in amoeba-shaped domain.

${\mathit{n}}_{\mathit{l}}$	$\mathit{\epsilon }$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
25	$15.4$	$2.53$ × 10${}^{-5}$	$1.33$ × 10${}^{-4}$	21
35	$23.6$	$7.07$ × 10${}^{-5}$	$7.64$ × 10${}^{-4}$	32
45	$7.2$	$1.53$ × 10${}^{-5}$	$4.26$ × 10${}^{-4}$	29
55	$9.2$	$4.14$ × 10${}^{-5}$	$1.20$ × 10${}^{-4}$	41
65	$21.5$	$1.67$ × 10${}^{-5}$	$6.60$ × 10${}^{-5}$	69
75	$21.5$	$7.27$ × 10${}^{-5}$	$2.29$ × 10${}^{-4}$	83

,

Table 8. Example 3: RMSE and $l_{\infty }$ error obtained by proposed approach.

${\mathit{n}}_{\mathit{l}}$	m	$\mathit{\epsilon }$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
25	0	$15.3$	$1.86$ × 10${}^{-5}$	$1.87$ × 10${}^{-4}$	21
	1	$14.5$	$5.63$ × 10${}^{-6}$	$5.81$ × 10${}^{-5}$	22
	2	$18.8$	$4.20$ × 10${}^{-6}$	$5.75$ × 10${}^{-5}$	30
	3	$14.9$	$2.38$ × 10${}^{-6}$	$4.63$ × 10${}^{-5}$	31
	4	$12.9$	$1.08$ × 10${}^{-7}$	$6.21$ × 10${}^{-5}$	28
35	0	$24.0$	$6.59$ × 10${}^{-6}$	$5.52$ × 10${}^{-5}$	33
	1	$22.0$	$1.62$ × 10${}^{-6}$	$3.56$ × 10${}^{-5}$	32
	2	$29.0$	$1.56$ × 10${}^{-6}$	$9.06$ × 10${}^{-6}$	35
	3	$24.0$	$1.34$ × 10${}^{-7}$	$1.09$ × 10${}^{-6}$	33
	4	$36.0$	$2.74$ × 10${}^{-8}$	$4.84$ × 10${}^{-7}$	40

and

Table 9. Example 3: RMSE and $l_{\infty }$ error obtained by proposed approach for varying $n_l$ in an amoeba-shaped domain.

${\mathit{n}}_{\mathit{l}}$	$\mathit{\epsilon }$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
45	37	$9.94$ × 10${}^{-9}$	$1.25$ × 10${}^{-7}$	123
55	43	$4.71$ × 10${}^{-9}$	$7.94$ × 10${}^{-8}$	138
65	51	$2.80$ × 10${}^{-9}$	$2.14$ × 10${}^{-8}$	175
75	62	$5.35$ × 10${}^{-9}$	$6.07$ × 10${}^{-8}$	243

, we present the RMSE and $l_{\infty }$ error for solving elliptic PDEs with variable coefficients. Table 7 presents the numerical results of the L-ORBFs for various sizes of the local computational domain $n_l$, whereas Table 8 and Table 9 present the numerical results of the L-ORBFs with augmented polynomials.

Comparing numerical results from Table 7, Table 8 and Table 9, we observe significant improvement in the numerical accuracy while employing the proposed scheme. When the number of local computational nodes increases in the L-ORBFs, it provides consistent accuracy around 1 × 10${}^{-5}$ (Table 7). On the other hand, when $n_l=45,55,65,75$ are chosen, the proposed scheme reaches accuracy around 1 × 10${}^{-9}$ (Table 9). Although the results in these tables are obtained for a specific $\epsilon$ chosen by a heuristic approach, we can observe from Figure 7b that the proposed scheme is more highly accurate than the L-ORBFs.

Table 8 depicts the numerical results using the proposed scheme corresponding to the local computational nodes $n_l=25$ and 35. Various degrees of augmented polynomial terms ($m=0,1,2,3,4$) are chosen. As expected, adding more polynomial terms not only increases accuracy but also increases computational cost so it is not a wise idea to add a lot of polynomial terms in the localized collocation process. This indicates that if we would like to improve the accuracy, it is better to add only a few more polynomial terms. In the table, with $n_l=35$, we only need to add polynomial terms up to $m=4$ to achieve accuracy up to 1 × 10${}^{-8}$, which means by only adding $n_q=15$ more polynomial terms on the localized collocation process, we can achieve a highly accurate solution, and it only takes a time of 40 s.

Next, we choose the computational domain bounded by the gear curve with 15 cogs (Figure 1d). The surface plot of the analytical solutions $v(x_1,x_2)=\sin (x_2^2+x_1)-\cos (x_2-x_1^2)$ over a gear-curve domain is depicted in Figure 8.

Let us implement our numerical scheme using $\mathrm{110,765}$ interior nodes, and 6000 boundary nodes in the gear-curve domain. For a large number of computational nodes, we believe that it makes sense to take more local computational nodes to capture the local calculus of the domain.

Table 10. Example 3: RMSE and $l_{\infty }$ error obtained by L-ORBFs on a gear-curve domain for varying $n_l$.

${\mathit{n}}_{\mathit{l}}$	$\mathit{\epsilon }$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
25	21	$3.10$ × 10${}^{-5}$	$3.99$ × 10${}^{-4}$	90
35	68	$4.21$ × 10${}^{-6}$	$8.01$ × 10${}^{-5}$	171
45	21	$7.79$ × 10${}^{-6}$	$3.02$ × 10${}^{-5}$	154
55	15	$7.02$ × 10${}^{-6}$	$2.67$ × 10${}^{-5}$	196
65	18	$6.76$ × 10${}^{-6}$	$2.08$ × 10${}^{-5}$	265
75	6	$4.58$ × 10${}^{-6}$	$1.63$ × 10${}^{-5}$	261

presents the numerical results obtained by the L-ORBFs, and

Table 11. Example 3: RMSE and $l_{\infty }$ error obtained by proposed approach on a gear-curve domain for varying $n_l$.

${\mathit{n}}_{\mathit{l}}$	$\mathit{\epsilon }$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
25	19	$1.88$ × 10${}^{-8}$	$8.58$ × 10${}^{-8}$	110
35	84	$3.01$ × 10${}^{-11}$	$1.31$ × 10${}^{-9}$	195
45	75	$2.98$ × 10${}^{-10}$	$1.72$ × 10${}^{-9}$	246
55	68	$1.44$ × 10${}^{-10}$	$5.51$ × 10${}^{-10}$	301
65	89	$2.32$ × 10${}^{-10}$	$9.89$ × 10${}^{-10}$	452
75	98	$9.19$ × 10${}^{-11}$	$2.99$ × 10${}^{-10}$	569

shows the results of the proposed scheme. In this experiment, we have only included the polynomial terms up to the degree $m=4$. We notice that even with the large number of computational nodes, similar conclusions are perceived. The proposed scheme is more highly accurate than the L-ORBFs. Comparing the results presented in Table 10 and Table 11 for the different number of local computational nodes, the L-ORBFs only produce accuracy around 1 × 10${}^{-6}$ while our proposed approach reaches accuracy closer to 1 × 10${}^{-11}$. This clearly validates our claim.

Finally, we would like to comment on the varying order d of ORBFs. Earlier research shows that if we choose an optimal shape parameter for each order d, it produces a reasonably accurate solution in the L-ORBFs. Without loss of generality, throughout the paper, we have fixed the order of ORBFs to $d=1$. Ultimately, we want to verify that this is not only true in the L-ORBFs but also in this proposed scheme.

Table 12. Example 3: RMSE and $l_{\infty }$ error obtained by the proposed approach for various order d of ORBFs.

d	$\mathit{\epsilon }$	RMSE	${\mathit{l}}_{\mathit{\infty }}$	Elapsed Time (s)
1	84	$3.01$ × 10${}^{-11}$	$1.31$ × 10${}^{-9}$	195
2	87	$9.30$ × 10${}^{-11}$	$7.33$ × 10${}^{-9}$	210
3	116	$8.25$ × 10${}^{-11}$	$5.56$ × 10${}^{-9}$	241
4	110	$2.84$ × 10${}^{-11}$	$1.37$ × 10${}^{-9}$	242
5	128	$3.26$ × 10${}^{-11}$	$1.99$ × 10${}^{-9}$	265

presents the RMSE and $l_{\infty }$ error for a various order d of ORBFs on a gear-curve domain with $n_i=\mathrm{110,765}$ and $n_b=6000$. For this experiment, we have chosen $n_l=35$ and $m=4$. We observed that varying the order d of ORBFs does not change the final outcome of the solution significantly, but it definitely changes the computational time. For $d=5$, to achieve the RMSE = $3.26$ × 10${}^{-11}$ and $l_{\infty }=1.99$ × 10${}^{-9}$, it takes 265 s. On the other hand, similar accuracy can be achieved by choosing $d=1$ within 195 s. Note that these computational costs vary by order due to the evaluation cost of ORBFs and their corresponding particular solutions.

4. Conclusions and Future Work

The localized oscillatory radial basis function collocation method (L-ORBFs) is one of the most effective and powerful numerical methods for solving large-scale problems. In this article, we introduce and implement a numerical scheme in which we add low-degree polynomial terms on the localized collocation process of the L-ORBFs to achieve high accuracy. We compare the numerical results of the proposed method with the L-ORBFs, and the results clearly indicate that the proposed method is very promising and outperformed the results of the L-ORBFs. At the meantime, we also observe that the L-ORBFs with augmented polynomial terms is computationally more expensive than the L-ORBFs. One of our future works in this context will be to introduce a numerical method to minimize the computational cost. We also want to have a comparison study of this approach with other RBFs in the literature. Furthermore, this method heavily depends on the choice of the parameter shape while solving elliptic PDEs similar to other RBF collocation methods. Therefore, we want to explore a new strategy to find the optimal shape parameter for RBF-based collocation methods in the near future.