Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations

Abstract

In this paper, the symmetric radial basis function method is utilized for the numerical solution of two- and three-dimensional elliptic PDEs. Numerical results are obtained by using a set of uniform or random points. Numerical tests are accomplished to demonstrate the efficacy and accuracy of the method on both regular and irregular domains. Furthermore, the proposed method is tested for the solution of elliptic PDE in the case of various frequencies.

1. Introduction

A considerable number of science and engineering problems with suitable boundary conditions are managed by elliptic partial differential equations (EPDEs). EPDEs arise in many time-independent physical problems such as steady-state problems involving incompressible fluids, the steady-state distribution of heat in a plane region, while microwaves, as well as acoustic waves could be simulated by the Helmholtz equation. Generally, it is hard to attain the closed-form solutions of such equations.

Owing to this fact, several proficient and precise methods have been developed for the numerical solution of EPDEs. Contemporary contributions in this regard comprise spline collocation methods [1,2], the finite element method [3], finite difference methods [4,5], the wavelet collocation method [6], meshless methods [7,8,9], etc.

A broad variety of meshless methods underway is based on radial basis functions (RBF) [9,10,11,12]. The simplest among them is unsymmetric RBF collocation or the Kansa method [13]. When the goal is to approximate dispersed data in several dimensions, the RBF methods in the case of multivariate approximation turn out to be the most frequently-applied methodology. The global, non-polynomial, RBF method is a valuable substitute for obtaining exponential accuracy where classical methods are hard to apply or fail. In multifaceted problems, non-rectangular domains comprise an instance. To compose a univariate function with the Euclidean norm, the RBF methods turns out to perform well in extremely common settings. Consequently, a multidimensional problem can be effectively altered into one-dimensional.

A vast range of problems has been successfully solved by the Kansa method. However, there is no prescribed guarantee for the standard formulation that the collocation unsymmetric matrix will be non-singular [14]. In particular, to get a non-singular collocation matrix, a small perturbation of functional center locations is also useful [15]; whereas, numerical ill-conditioning issues may be possible in the perturbed collocation matrix.

Fasshauer [16] proposed an alternate method, the PDE operator, which has been applied to the RBF as the basis function, known as the Hermitian or symmetric RBF (SRBF) method, which yields a symmetric collocation matrix (thus attractive for the benefits of storage) and was shown by Wu [17] to be nonsingular only if no two collocation points sharing the same operator were placed in the same location.

The SRBF method has several advantages over the Kansa method, but Robert Shaback [18] has shown that both methods experience the same basic problem, the uncertainty relation; good conditioning is related to inferior accuracy, and inferior conditioning is related to better accuracy. In the case of increasing the system size, this problem turns out to be more distinct. To avoid this problem, much of the research has been dedicated in recent years to the formulation of RBF-specific pre conditioners [15], the adaptive selection of functional centers, and collocation points [14].

The SRBF method has a benefit over the Kansa method that the collocation matrices are symmetric. Therefore, the computation cost can be significantly decreased, which is vital for larger problems. Different problems have been solved using the SRBF method [19,20,21,22].

2. Governing Equations

We consider the following governing equations.

The Poisson equation:

$$\mathfrak{L}u\left(\mathit{x}\right)=\Delta u\left(\mathit{x}\right)=f\left(\mathit{x}\right),\mathit{x}\in \Omega \subset {\Re }^n,$$

(1)

with boundary conditions:

$${\mathbb{B}}_1\left(\mathit{x}\right)=f_1\left(\mathit{x}\right),\mathit{x}\in \partial \Omega ,$$

(2)

where $\mathfrak{L}=\Delta$.

The Helmholtz equation:

$$\mathfrak{L}u\left(\mathit{x}\right)=\Delta u\left(\mathit{x}\right)+ku\left(\mathit{x}\right)=g\left(\mathit{x}\right),\mathit{x}\in \Omega \subset {\Re }^n,$$

(3)

with boundary conditions:

$${\mathbb{B}}_2\left(\mathit{x}\right)=g_1\left(\mathit{x}\right),\mathit{x}\in \partial \Omega ,$$

(4)

where $\mathfrak{L}=\Delta +k$.

$\Delta$ is the Laplace operator, and n is the dimension of the spatial variable in both cases.

3. Numerical Scheme

In the SRBF method, an approximate solution is sought of the form [16]:

$$u\left(x\right)={\lambda }_1\varphi \left(\right|x-x_1\left|\right)+{\displaystyle \sum _{i=2}^{N-1}}{\lambda }_i{\mathfrak{L}}^2\varphi \left(\right|x-x_i\left|\right)+{\lambda }_N\varphi \left(\right|x-x_N\left|\right),$$

(5)

where ${\mathfrak{L}}^2$ is a linear differential operator similar to $\mathfrak{L}$, but acting on $\varphi$ as a function of the second variable $x_i$. Using Equation (5) on the one-dimensional elliptic PDE with boundary conditions given by:

$$\begin{array}{cc}\hfill \mathfrak{L}u\left(x\right)& =f\left(x\right),a<x<b,\hfill \\ \hfill u\left(a\right)& =u_a,\hfill \\ \hfill u\left(b\right)& =u_b,\hfill \end{array}$$

(6)

enforcing the collocation conditions at the interior points, $i=2,3,\dots ,N-1$, we get the system of linear equations:

$$\begin{array}{cc}\hfill {\lambda }_1\varphi \left(\right|a-x_1\left|\right)+{\displaystyle \sum _{i=2}^{N-1}}{\lambda }_i{\mathfrak{L}}^2\varphi \left(\right|a-x_i\left|\right)+{\lambda }_N\varphi \left(\right|a-x_N\left|\right)& =u_a,\hfill \\ \hfill {\lambda }_1\mathfrak{L}\varphi \left(\right|x-x_1\left|\right)+{\displaystyle \sum _{i=2}^{N-1}}{\lambda }_i\mathfrak{L}{\mathfrak{L}}^2\varphi \left(\right|x-x_i\left|\right)+{\lambda }_N\mathfrak{L}\varphi \left(\right|x-x_N\left|\right)& =f\left(x\right),x\in [a,b],\hfill \\ \hfill {\lambda }_1\varphi \left(\right|b-x_1\left|\right)+{\displaystyle \sum _{i=2}^{N-1}}{\lambda }_i{\mathfrak{L}}^2\varphi \left(\right|b-x_i\left|\right)+{\lambda }_N\varphi \left(\right|b-x_N\left|\right)& =u_b.\hfill \end{array}$$

(7)

In matrix notation, we can write:

$$\mathbf{A}\mathit{\lambda }=\mathbf{B},$$

where:

$$\begin{array}{cc}\hfill \mathbf{A}& =\left[\begin{array}{cc}{A}_{\mathfrak{L}{\mathfrak{L}}^2}& A_{\mathfrak{L}}\\ A_{{\mathfrak{L}}^2}& A\end{array}\right],\hfill \\ \hfill \mathit{\lambda }& ={[{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N]}^T,\hfill \\ \hfill \mathbf{B}& ={[u_a,f\left(x_2\right),f\left(x_3\right),\dots ,f\left(x_{N-1}\right),u_b]}^T.\hfill \end{array}$$

(8)

Here, the four blocks in matrix $\mathbf{A}$ are generated as follows:

$$\begin{array}{cc}\hfill {\left(A_{\mathfrak{L}{\mathfrak{L}}^2}\right)}_{ij}& =\mathfrak{L}{\mathfrak{L}}^2\varphi (\parallel x_i-x_j\parallel ),x_i,x_j\in {\Omega }_1,\hfill \\ \hfill {\left(A_{\mathfrak{L}}\right)}_{ij}& =\mathfrak{L}\varphi (\parallel x_i-x_j\parallel ),x_i\in {\Omega }_1,x_j\in {\Omega }_2,\hfill \\ \hfill {\left(A_{{\mathfrak{L}}^2}\right)}_{ij}& ={\mathfrak{L}}^2\varphi (\parallel x_i-x_j\parallel ),x_i\in {\Omega }_2,x_j\in {\Omega }_1,\hfill \\ \hfill A_{ij}& =\varphi (\parallel x_i-x_j\parallel ),x_i,x_j\in {\Omega }_2,\hfill \end{array}$$

where ${\Omega }_1$ and ${\Omega }_2$ are interior and boundary nodes, respectively.

4. Numerical Results

In this section, we apply the SRBF method for the numerical solution of two-dimensional Poisson and Helmholtz and three-dimensional Poisson elliptic PDEs with Dirichlet boundary conditions. We used MQ radial basis functions. The accuracy of the scheme is measured in terms of the error norms $L_{abs}$, $L_{\infty }$, and Root Mean Square error (RMS).

$$\begin{array}{cc}\hfill L_{abs}& {=|u\left(i\right)}_{ex}-u_i|,i=1,2,\dots ,N^n.\hfill \\ \hfill L_{\infty }& =max\left(L_{abs}\right),\hfill \\ \hfill \mathrm{RMS}& =\sqrt{\frac{{\sum }_{i=1}^N{\left(u_i\right)}^2}{N}},\hfill \end{array}$$

where $u_{ex}$ represents the exact and u represents the approximate solution.

In this paper, the SRBF method is applied on both regular and irregular domains, as well as on uniform and non-uniform (Chebyshev, random, and Halton) nodal points. Numerical results of the paper are summarized as follows. Numerical results of two-dimensional Poisson equations with uniform nodes are shown in

Table 1. Error norms for Problem 1.

N	c	${\mathit{L}}_{\mathbf{\infty }}$	RMS
81	0.124	$1.6296\times {10}^{-11}$	$4.4250\times {10}^{-11}$
256	0.125	$6.1207\times {10}^{-11}$	$2.3986\times {10}^{-11}$
400	0.125	$4.6252\times {10}^{-11}$	$2.8510\times {10}^{-10}$
900	0.129	$1.6221\times {10}^{-10}$	$1.3108\times {10}^{-9}$
1600	0.126	$8.8824\times {10}^{-11}$	$1.4803\times {10}^{-9}$
2500	0.126	$2.5943\times {10}^{-10}$	$2.9648\times {10}^{-9}$

and

Table 2. Error norms for Problems 2.

N	c	${\mathit{L}}_{\mathbf{\infty }}$	RMS
81	0.406	$5.3081\times {10}^{-7}$	$1.5419\times {10}^{-6}$
256	0.52	$3.1206\times {10}^{-7}$	$1.1868\times {10}^{-6}$
400	0.96	$5.3085\times {10}^{-8}$	$3.0250\times {10}^{-7}$
900	1.67	$2.3015\times {10}^{-8}$	$2.6587\times {10}^{-7}$
1600	2.15	$2.1188\times {10}^{-8}$	$3.7525\times {10}^{-7}$
2500	2.52	$1.4109\times {10}^{-8}$	$4.6924\times {10}^{-7}$

and in Figure 1 and Figure 2, and those with non-uniform nodes are shown in Figure 3, whereas numerical results over different types of irregular domains are shown in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. Numerical results of two-dimensional Helmholtz equations are shown in

Table 3. Error norms for Problem 3.

t	N	c	${\mathit{L}}_{\mathbf{\infty }}$	RMS
1	81	0.3	$5.0926\times {10}^{-8}$	$1.3014\times {10}^{-7}$
	400	0.8	$1.5166\times {10}^{-8}$	$8.7744\times {10}^{-8}$
	900	1.59	$4.9779\times {10}^{-9}$	$3.4086\times {10}^{-8}$
5	81	0.43	$7.7247\times {10}^{-5}$	$1.8988\times {10}^{-4}$
	400	1.1	$7.5979\times {10}^{-7}$	$3.6823\times {10}^{-6}$
	900	1.69	$1.7120\times {10}^{-7}$	$9.8686\times {10}^{-7}$
10	81	1.15	$1.8200\times {10}^{-2}$	$3.7200\times {10}^{-2}$
	400	1.22	$1.5067\times {10}^{-5}$	$6.9115\times {10}^{-5}$
	900	1.93	$4.1883\times {10}^{-8}$	$2.5267\times {10}^{-5}$
20	81	4.13	$2.4950\times {10}^{-1}$	$4.4920\times {10}^{-1}$
	400	1.97	$2.9000\times {10}^{-3}$	$8.2000\times {10}^{-3}$
	900	2.22	$1.2416\times {10}^{-4}$	$5.4952\times {10}^{-4}$

and

Table 4. Error norms for Problem 4.

N	c	${\mathit{L}}_{\mathbf{\infty }}$	RMS
81	0.73	$8.5033\times {10}^{-6}$	$2.2291\times {10}^{-5}$
256	0.9	$3.5236\times {10}^{-7}$	$1.8206\times {10}^{-6}$
400	1.31	$7.2561\times {10}^{-8}$	$4.5238\times {10}^{-7}$
900	2.02	$1.8554\times {10}^{-8}$	$1.4499\times {10}^{-7}$
1600	2.42	$4.5357\times {10}^{-9}$	$4.9253\times {10}^{-8}$
2500	2.92	$1.9209\times {10}^{-9}$	$1.5399\times {10}^{-8}$

and in Figure 9 and Figure 10. Numerical results of three-dimensional Poisson equations are shown in

Table 5. Error norms for Problem 5.

N	c	${\mathit{L}}_{\mathbf{\infty }}$	RMS
343	0.06	$3.6458\times {10}^{-6}$	$1.7514\times {10}^{-5}$
1331	0.06	$3.7429\times {10}^{-6}$	$2.9703\times {10}^{-5}$
3375	0.06	$5.1290\times {10}^{-6}$	$6.9616\times {10}^{-5}$

and in Figure 11, Figure 12 and Figure 13.

Problem 1.

Consider the two-dimensional Poisson equation:

$$u_{xx}+u_{yy}=f(x,y),(x,y)\in {[-1,1]}^2,$$

(9)

where function$f(x,y)$is specified so that the exact solution is [23]:

$$u(x,y)=\frac{65}{65+{(x-0.2)}^2+{(y+0.1)}^2}.$$

(10)

Numerical results in terms of $L_{\infty }$ and RMS for various values of nodes N and corresponding shape parameters value c for Problem 1 are reported in Table 1. One can see that better accuracy is obtained for each N.

Numerical results of the proposed meshless method for $N=256$ and $c=0.125$ are shown in Figure 1 (left), whereas the $L_{abs}$ error norm is shown in Figure 1 (right).

Problem 2.

Consider the two-dimensional Poisson equation:

$$u_{xx}+u_{yy}=-5/4{\pi }^{2}sin\left(\pi x\right)cos(\pi y/2),(x,y)\in {[a,b]}^2.$$

(11)

The exact solution [7] is given by:

$$u(x,y)=sin\left(\pi x\right)cos(\pi y/2).$$

(12)

Numerical results of Problem 2 are obtained using the proposed meshless method, and in this case, the method is analyzed up to the maximum nodes $N=2500$. The $L_{\infty }$ and RMS error norms are calculated for different values of N and the corresponding value of c and are shown in Table 2. It can be observed from the table that the accuracy increases with the increase in the number of nodes.

The proposed method is truly meshless and eliminates the need for meshing. Numerical solutions for $N=400$ and $c=0.96$ are shown in Figure 2 (left), whereas the $L_{abs}$ error norm is shown in Figure 2 (right). Figure 3 shows the numerical results in terms of $L_{\infty }$ error norms using random nodes, Chebyshev nodes, and Halton nodes. Figure 3 is the evidence that better accuracy is obtained in these cases, as well.

Numerical results using the SRBF method over different types of irregular domains are shown in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. One can see that a better approximate solution can be obtained by the proposed method.

Problem 3.

Consider the two-dimensional Helmholtz equation:

$$u_{xx}+u_{yy}+ku(x,y)=cos\left(tx\right)sin\left(ty\right)(-2t^2+k),(x,y)\in {[0,1]}^2,$$

(13)

where$k=5$and t is the frequency variable; the exact solution is:

$$u(x,y)=cos\left(tx\right)sin\left(ty\right).$$

(14)

Numerical results for Problem 3 for various frequencies t and nodes N are shown in Table 3. Figure 9 shows the numerical result and absolute error of the proposed method with t = 10, c = 1.22 and N = 400. It can be seen from Figure 9 that reasonably good accuracy can be achieved by the method. Table 3 shows that the proposed method can easily capture the solution up to high frequency at $t=20$. One can also see from the table that accuracy increases with the increase in the number of nodes N in this case as well.

Problem 4.

Consider the two-dimensional Helmholtz equation:

$$u_{xx}+u_{yy}+ku(x,y)=f(x,y),(x,y)\in {[0,1]}^2,$$

(15)

with$k=-5$[24], and the function$f(x,y)$is specified, so that the exact solution is:

$$u(x,y)=e^{xy}{(x^2-x)}^2{(y^2-y)}^2.$$

(16)

Numerical results for Problem 4 are reported in Table 4 for different N and its corresponding shape parameter value c in terms of $L_{\infty }$ and $RMS$ error norms. Better results are obtained in this case, as well, and the accuracy increases with increase in N. The accuracy of the proposed method is also shown in Figure 10.

Problem 5.

Consider the three-dimensional Poisson equation:

$$u_{xx}+u_{yy}+u_{zz}=6,(x,y)\in {[0,1]}^2.$$

(17)

The exact solution is [25]:

$$u(x,y,z)=x^2+y^2+z^2.$$

(18)

To check the efficiency of the SRBF method in the three-dimensional case, we have considered the Poisson equation. The results of Problem 5 are shown in Table 5. The proposed meshless method in the three-dimensional case is also tested over different irregular domains, as shown in Figure 11, Figure 12 and Figure 13. The performance of the method is satisfactory in these cases as well.

5. Conclusions

In this study, we have used the symmetric radial basis function method for the simulation of two- and three-dimensional elliptic PDEs. For this purpose, we have employed uniform and non-uniform nodes. Furthermore, the accuracy of the method has been testified in regular, as well as irregular domains.