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

Phatiphat Thounthong 1 , Muhammad Nawaz Khan 2, Iltaf Hussain 2, Imtiaz Ahmad 2 and Poom Kumam 3,* 1 Renewable Energy Research Centre & Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut’s University of Technology North Bangkok (KMUTNB), Wongsawang, Bangsue, Bangkok 10800, Thailand; phatiphat.t@fte.kmutnb.ac.th 2 Department of Basic Sciences, University of Engineering and Technology, Peshawar 25000, Pakistan; mnawaz77@gmail.com (M.N.K.); iltafhussain@uetpeshawar.edu.pk (I.H.); imtiazkakakhil@gmail.com (I.A.) 3 KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand * Correspondence: poom.kum@kmutt.ac.th; Tel.: +66-2-4708994


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.
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].

Governing Equations
We consider the following governing equations.
The Poisson equation: with boundary conditions: where L = ∆.
The Helmholtz equation: with boundary conditions: where L = ∆ + k. ∆ is the Laplace operator, and n is the dimension of the spatial variable in both cases.

Numerical Scheme
In the SRBF method, an approximate solution is sought of the form [16]: where L 2 is a linear differential operator similar to L, but acting on φ as a function of the second variable x i .Using Equation ( 5) on the one-dimensional elliptic PDE with boundary conditions given by: enforcing the collocation conditions at the interior points, i = 2, 3, ..., N − 1, we get the system of linear equations: In matrix notation, we can write: where: Here, the four blocks in matrix A are generated as follows: where Ω 1 and Ω 2 are interior and boundary nodes, respectively.

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 ∞ , and Root Mean Square error (RMS).
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 Tables 1 and 2 and in Figures 1 and 2, and those with non-uniform nodes are shown in Figure 3, whereas numerical results over different types of irregular domains are shown in Figures 4-8.Numerical results of two-dimensional Helmholtz equations are shown in Tables 3  and 4 and in Figures 9 and 10.Numerical results of three-dimensional Poisson equations are shown in Table 5 and in Figures 11-13.Problem 1.Consider the two-dimensional Poisson equation: where function f (x, y) is specified so that the exact solution is [23]: Numerical results in terms of L ∞ 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).
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 ∞ 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 ∞ 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 Figures 4-8.One can see that a better approximate solution can be obtained by the proposed method.Problem 3. Consider the two-dimensional Helmholtz equation: where k = 5 and t is the frequency variable; the exact solution is: u(x, y) = cos(tx) sin(ty).
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.
with k = −5 [24], and the function f (x, y) is specified, so that the exact solution is: Numerical results for Problem 4 are reported in Table 4 for different N and its corresponding shape parameter value c in terms of L ∞ 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.  ( 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 Figures 11-13.The performance of the method is satisfactory in these cases as well.

Figure 11 .
Figure 11.A three-dimensional computational domain for Problem 5.

Figure 12 .
Figure 12.Computational domain (left) and L abs error norm on the line x = y = z (right) for Problem 5.

Figure 13 .
Figure 13.Computational domain (left) and L abs error norm on the line x = y = z (right) for Problem 5.