Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

Abstract

This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

1. Introduction

In the past years, finding numerical solutions to elliptic partial differential equations (PDEs) has been an active area of interest in numerical mathematics. This is due to their valuable applications in engineering and science disciplines, where elliptic PDEs emerge in areas such as fracture and fluid mechanics, elastodynamics, electromagnetics and acoustics [1]. Most physical processes, including steady-state thermal conduction and convection equations that model phenomena of transport, are described using elliptic PDEs [2,3,4]. Numerous numerical approaches have been developed and used to solve elliptic PDEs. These approaches constitute traditional methods such as finite-difference [5,6,7] and finite element [8] methods that have been reported to demonstrate poor accuracy and computational inefficiency due to the usage of much more grid points for numerical discretization. Other numerical methods that have been later used to solve elliptic PDEs include meshless methods [9,10,11,12,13], spline collocation methods [14,15,16], the domain decomposition method [17], Sinc–Galerkin method [18], wavelets collocation methods [19,20,21,22] and spectral collocation methods [23].

The use of spectral methods has been favored in giving superior accuracy and minimizing calculation time. Such superiority in accuracy and computational performance has been evident in problems possessing smooth solutions. However, spectral methods have been found to be ineffective in solving problems defined over irregular domains and those exhibiting discontinuity. Pfeiffer et al. [24] handled nonlinear elliptic PDEs via the domain decomposition spectral method that uses Chebyshev basis functions. The method was reported to be superior in execution time and accuracy, with many collocation points of up to 1000 and for problems with smooth solutions. Wang et al. [25] discussed the two-level Legendre/Chebyshev spectral collocation approach for solving semilinear and nonlinear elliptic PDEs possessing multiple solutions. The two-level spectral method was found to be computationally cheaper and proved high-order accuracy for the semilinear elliptic PDEs. Gu and Shen [26] solved elliptic PDEs defined on complex domains via an accurate and effective fictitious domain spectral method and disclosed that spectral convergence rests upon the smoothness of the solution. Tao et al. [27] presented a Garlekin spectral method for handling the optimal control problem modeled by elliptic PDEs and noted a quick exponential decay in errors. Wang et al. [28] developed the Fourier spectral collocation method for solving inhomogeneous elliptic PDEs. The numerical approach is based on the use of Fourier basis functions and Gauss–Lobatto nodes as collocation points. Liu et al. [29] introduced the Chebyshev spectral method that uses the quasi-inverse approach to directly solve linear elliptic PDEs with homogeneous Robin boundary conditions. Oh [30] presented a numerical approach that uses quasi-inverse matrix diagonalization method along with Chebyshev–Galerkin method to solve linear PDEs spectrally. Ghamire et al. [31] used the Chebyshev basis function in the implementation of a pseudo-spectral collocation method to find numerical solutions to nonlinear elliptic PDEs. With the use of many grid points, these methods were found to be effective and produced highly accurate results.

The use of a high number of grid points and full differentiation matrices means more computer memory will be required to store the matrices, and more computational time will be required to run the algorithm. However, using a higher number of grid points does not ensure good accuracy some times [32]. This is because when the number of collocation points used is many, the size of the resulting coefficient matrix in the linear system is also increased, making the matrix ill-conditioned in nature. Computation using an ill-conditioned matrix is very likely to give numerical solutions with big errors and poor accuracy. In the current work, we attempt to develop a numerical method that can overcome such challenges to some extent. Thus, the purpose of this work is to introduce a spectral collocation method that is based on the use of the quasi-linearization method (QLM) [33] to linearize nonlinear elliptic boundary value problems, triple Lagrange interpolation polynomials established via the Chebyshev–Gauss–Lobatto (CGL) nodes [34] to approximate solutions of elliptic PDEs and Kronecker tensor product [22] to discretize the differentiation matrices. The choice of Gauss–Lobatto nodes is to ensure convergence of the algorithm. To minimize the errors associated with the use of large matrices, the collocation process in the current work is carried out at the interior points, thus leading to a reduction in the size of the differentiation matrices. Consequently, the size of the resulting coefficient matrix in the linear system will also be reduced. Thus the use of fewer grid points, less computer memory and small runtime will be required in the numerical computation.

The remainder of the paper is comprised of the following sections. In Section 2, a description of the trivariate spectral quasi-linearization method (TSQLM) is provided for the general single and system of three-dimensional nonlinear elliptic PDEs. Section 3 entails typical examples that are worked out to demonstrate the applicability and accuracy of the numerical approach. Such typical examples emerge in diverse applications in structural engineering, image processing, thin beams and plates. Section 3 comprises error norms, computing time and stability results, which are provided and discussed for the TSQLM approach. In the same section, a comparison between approximate solutions and exact solutions is presented and discussed. The paper ends with Section 5, which contains the conclusion and possible forthcoming works emanating from the current study.

2. Numerical Procedure

In this section, we provide the solution procedure for second-order three-dimensional nonlinear elliptic PDEs. To achieve that, we first consider a nonlinear elliptic PDE of the form

$$\begin{array}{ccc}& & \mathsf{\Gamma }\left(\frac{{\partial }^{2}u}{\partial x^2},\frac{{\partial }^{2}u}{\partial y^2},\frac{{\partial }^{2}u}{\partial z^2},\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z},u\right)=g(x,y,z),(x,y,z)\in [a,b]\times [c,d]\times [e,f],\hfill \end{array}$$

(1)

where $\mathsf{\Gamma }$ represents a nonlinear operator functioning on the unknown function $u(x,y,z)$, its first and second-order space derivatives and $g(x,y,z)$ is a known function. This single elliptic PDE (1) is to be solved subject to the following Dirichlet boundary conditions

$$\begin{array}{ccc}& & u(a,y,z)=h_a(y,z),u(b,y,z)=h_b(y,z),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & u(x,c,z)=h_c(x,z),u(x,d,z)=h_d(x,z),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & u(x,y,e)=h_e(x,y),u(x,y,f)=h_f(x,y),\hfill \end{array}$$

(4)

where $h_a(y,z),h_b(y,z),h_c(x,z),h_d(x,z),h_e(x,y)$ and $h_f(x,y)$ are known functions.

Secondly, we consider a system of two nonlinear elliptic PDEs of the form

$$\begin{array}{ccc}& & {\mathsf{\Gamma }}_1\left(\frac{{\partial }^{2}u}{\partial x^2},\frac{{\partial }^{2}u}{\partial y^2},\frac{{\partial }^{2}u}{\partial z^2},\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z},\frac{{\partial }^{2}v}{\partial x^2},\frac{{\partial }^{2}v}{\partial y^2},\frac{{\partial }^{2}v}{\partial z^2},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y},\frac{\partial v}{\partial z},u,v\right)=g_1(x,y,z),\hfill \\ & & {\mathsf{\Gamma }}_2\left(\frac{{\partial }^{2}u}{\partial x^2},\frac{{\partial }^{2}u}{\partial y^2},\frac{{\partial }^{2}u}{\partial z^2},\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z},\frac{{\partial }^{2}v}{\partial x^2},\frac{{\partial }^{2}v}{\partial y^2},\frac{{\partial }^{2}v}{\partial z^2},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y},\frac{\partial v}{\partial z},u,v\right)=g_2(x,y,z),\hfill \end{array}$$

(5)

where ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ denote nonlinear operators. The system of Equation (5) is solved subject to the following Dirichlet boundary conditions

$$\begin{array}{ccc}& & u(a,y,z)=h_a(y,z),u(b,y,z)=h_b(y,z),u(x,c,z)=h_c(x,z),u(x,d,z)=h_d(x,z),\hfill \\ & & u(x,y,e)=h_e(x,y),u(x,y,f)=h_f(x,y),v(a,y,z)=h_a^{*}(y,z),v(b,y,z)=h_b^{*}(y,z),\hfill \\ & & v(x,c,z)=h_c^{*}(x,z),v(x,d,z)=h_d^{*}(x,z),v(x,y,e)=h_e^{*}(x,y),v(x,y,f)=h_f^{*}(x,y),\hfill \end{array}$$

(6)

where $h_a(y,z),$ $h_b(y,z),$ $h_c(x,z),$ $h_d(x,z),$ $h_e(x,y),$ $h_f(x,y),$ $h_a^{*}(y,z),$ $h_b^{*}(y,z),$ $h_c^{*}(x,z),$ $h_d^{*}(x,z),$ $h_e^{*}(x,z)$ and $h_f^{*}(x,z)$ are known functions.

2.1. Solving the Single Nonlinear Elliptic PDE

To solve Equation (1), we first simplify the boundary value problem by linearizing the equation using quasilinearization method (QLM) [33]. The QLM algorithm is based on the assumption that if the difference $u^{\iota +1}-u^{\iota }$ is hugely small, then the nonlinear components can be approximated by utilizing the linear terms of the Taylor series. Applying QLM to Equation (1) gives the subsequent linear iterative scheme

$${\sigma }_6^{\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial x^2}+{\sigma }_5^{\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial y^2}+{\sigma }_4^{\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial z^2}+{\sigma }_3^{\iota }\frac{\partial u^{\iota +1}}{\partial x}+{\sigma }_2^{\iota }\frac{\partial u^{\iota +1}}{\partial y}+{\sigma }_1^{\iota }\frac{\partial u^{\iota +1}}{\partial z}+{\sigma }_0^{\iota }{u}^{\iota +1}=R^{\iota },$$

(7)

where the variable coefficients are given by

$$\begin{array}{ccc}& & {\sigma }_6^{\iota }=\frac{\partial \mathsf{\Gamma }}{\partial u_{xx}^{\iota }},{\sigma }_5^{\iota }=\frac{\partial \mathsf{\Gamma }}{\partial u_{yy}^{\iota }},{\sigma }_4^{\iota }=\frac{\partial \mathsf{\Gamma }}{\partial u_{zz}^{\iota }},{\sigma }_3^{\iota }=\frac{\partial \mathsf{\Gamma }}{\partial u_x^{\iota }},{\sigma }_2^{\iota }=\frac{\partial \mathsf{\Gamma }}{\partial u_y^{\iota }},{\sigma }_1^{\iota }=\frac{\partial \mathsf{\Gamma }}{\partial u_z^{\iota }},\hfill \\ & & {\sigma }_0^{\iota }=\frac{\partial \mathsf{\Gamma }}{\partial u^{\iota }},R^{\iota }={\sigma }_6^{\iota }\frac{{\partial }^2{u}^{\iota }}{\partial x^2}+{\sigma }_5^{\iota }\frac{{\partial }^2{u}^{\iota }}{\partial y^2}+{\sigma }_4^{\iota }\frac{{\partial }^2{u}^{\iota }}{\partial z^2}+{\sigma }_3^{\iota }\frac{\partial u^{\iota }}{\partial x}+{\sigma }_2^{\iota }\frac{\partial u^{\iota }}{\partial y}+{\sigma }_1^{\iota }\frac{\partial u^{\iota }}{\partial z}+{\sigma }_0^{\iota }{u}^{\iota }-{\mathsf{\Gamma }}^{\iota },\hfill \end{array}$$

(8)

with $u_{xx}^{\iota }=\frac{{\partial }^2{u}^{\iota }}{\partial x^2},u_{yy}^{\iota }=\frac{{\partial }^2{u}^{\iota }}{\partial y^2},u_{zz}^{\iota }=\frac{{\partial }^2{u}^{\iota }}{\partial z^2},u_x^{\iota }=\frac{\partial u^{\iota }}{\partial x},u_y^{\iota }=\frac{\partial u^{\iota }}{\partial y},u_z^{\iota }=\frac{\partial u^{\iota }}{\partial z},$ $\iota$ and $\iota +1$ symbolizes respective previous and current iterations. To carry out the solution process, the approximate solution of the nonlinear PDE (1) is obtained by expanding the unknown function $u(x,y,z)$ using the trivariate Lagrange interpolating polynomial in all space variables. Thus, the function $u(x,y,z)$ can be approximated as

$$\begin{array}{ccc}& & u(x,y,z)=\sum _{k=0}^{N_z}\sum _{j=0}^{N_y}\sum _{i=0}^{N_t}{u}_{kji}{L}_k\left(z\right)L_j\left(y\right)L_i\left(x\right),\hfill \end{array}$$

(9)

where the unknown coefficient $u_{ijk}=u(x_i,y_j,z_k)$ is referred to as a spectral coefficient and the functions $L_i\left(x\right),L_j\left(y\right)$ and $L_k\left(z\right)$ signify the characteristic Lagrange cardinal polynomials [35] that are assembled as a set of Lagrange interpolation functions associated with the respective grid points ${\left\{x_i\right\}}_{i=0}^{N_x},{\left\{y_j\right\}}_{j=0}^{N_y}$ and ${\left\{z_j\right\}}_{k=0}^{N_z}$. The function $u(x,y,z)$ is interpolated via Chebyshev–Gauss–Lobatto (CGL) points [34] given by

$$\begin{array}{ccc}& & {\widehat{x}}_i={\left\{cos\left(\frac{i\pi }{N_x}\right)\right\}}_{i=0}^{N_x},{\widehat{y}}_j={\left\{cos\left(\frac{j\pi }{N_y}\right)\right\}}_{j=0}^{N_y},{\widehat{z}}_k={\left\{cos\left(\frac{k\pi }{N_z}\right)\right\}}_{k=0}^{N_z}.\hfill \end{array}$$

(10)

Prior to the application of the spectral collocation method, the spatial domains $x\in [a,b],y\in [c,d]$ and $z\in [e,f]$ are transformed into $\widehat{x}\in [-1,1],\widehat{y}\in [-1,1]$ and $\widehat{z}\in [-1,1]$, upon utilizing the linear mappings

$$\begin{array}{ccc}& & x=\frac{1}{2}(b-a)\widehat{x}+\frac{1}{2}(b+a),y=\frac{1}{2}(d-c)\widehat{y}+\frac{1}{2}(d+c),z=\frac{1}{2}(f-e)\widehat{z}+\frac{1}{2}(f+e).\hfill \end{array}$$

(11)

The space derivative matrices in $x,y$ and z are approximated at the collocation points $(\widehat{x},\widehat{y},\widehat{z})$ as follows

$$\begin{array}{ccc}& & \frac{\partial u}{\partial x}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)\approx \sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)L_m\left({\widehat{y}}_j\right)L_l^{\prime }\left({\widehat{x}}_i\right)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)L_m\left({\widehat{y}}_j\right)D_{li},\hfill \\ & & \frac{{\partial }^{2}u}{\partial x^2}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)L_m\left({\widehat{y}}_j\right)D_{li}^{\left(2\right)},\frac{{\partial }^{s}u}{\partial x^s}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)L_m\left({\widehat{y}}_j\right)D_{li}^{\left(s\right)}\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & \frac{\partial u}{\partial y}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)\approx \sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)L_m^{\prime }\left({\widehat{y}}_j\right)L_l\left({\widehat{x}}_i\right)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)D_{mj}{L}_l\left({\widehat{x}}_i\right),\hfill \\ & & \frac{{\partial }^{2}u}{\partial y^2}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)D_{mj}^{\left(2\right)}{L}_l\left({\widehat{x}}_i\right),\frac{{\partial }^{s}u}{\partial y^s}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n\left({\widehat{z}}_k\right)D_{mj}^{\left(s\right)}{L}_l\left({\widehat{x}}_i\right),\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & \frac{\partial u}{\partial z}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)\approx \sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{L}_n^{\prime }\left({\widehat{z}}_k\right)L_m^{\prime }\left({\widehat{y}}_j\right)L_l\left({\widehat{x}}_i\right)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{D}_{nk}{L}_m\left({\widehat{y}}_j\right)L_l\left({\widehat{x}}_i\right),\hfill \\ & & \frac{{\partial }^{2}u}{\partial z^2}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{D}_{nk}^{\left(2\right)}{L}_m\left({\widehat{y}}_j\right)L_l\left({\widehat{x}}_i\right),\frac{{\partial }^{s}u}{\partial z^s}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\sum _{n=0}^{N_z}\sum _{m=0}^{N_y}\sum _{l=0}^{N_x}{u}_{kji}{D}_{nk}^{\left(s\right)}{L}_m\left({\widehat{y}}_j\right)L_l\left({\widehat{x}}_i\right),\hfill \end{array}$$

(14)

where D represent the differentiation matrix, and $D_{li}=\frac{d{L}_l}{d\widehat{x}}\left({\widehat{x}}_i\right)$, $D_{mj}=\frac{d{L}_m}{d\widehat{y}}\left({\widehat{y}}_j\right)$ and $D_{nk}=\frac{d{L}_n}{d\widehat{z}}\left({\widehat{z}}_k\right)$ are entries of the standard first-order Chebyshev differentiation matrices [34,36] with sizes $(N_x+1)(N_x+1),(N_y+1)(N_y+1)$ and $(N_z+1)(N_z+1),$ respectively. Note that the letters $i,j,k,$ indicate the grid points in $x,y$ and z variables, while $l,m$ and n are employed in indexing the Lagrange cardinal functions in $x,y$ and z variables, respectively. To save computational processing time and ensure easy application of the solution algorithm, the Kronecker tensor product [22,34] is used to represent the matrix form of the numerical solution. The definitions of the Kronecker product between two and three matrices is outlined below. Let $\mathbf{A}$ be a matrix of size p and $\mathbf{B}$ be a matrix of size $q,$ then the product of $\mathbf{A}$ and $\mathbf{B}$ can be denoted by $\mathbf{A}\otimes \mathbf{B}$, which can be expressed in the form

$${\displaystyle \mathbf{A}\otimes \mathbf{B}=\left[\begin{array}{cccc}{\alpha }_{11}\mathbf{B}& {\alpha }_{12}\mathbf{B}& \dots & {\alpha }_{1q}\mathbf{B}\\ {\alpha }_{21}\mathbf{B}& {\alpha }_{22}\mathbf{B}& \dots & {\alpha }_{2q}\mathbf{B}\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_{p1}\mathbf{B}& {\alpha }_{p2}\mathbf{B}& \dots & {\alpha }_{pq}\mathbf{B}\end{array}\right]}.$$

(15)

Since the numerical scheme is implemented on MATLAB, the Kronecker tensor product of the matrices $\mathbf{A}$ and $\mathbf{B}$ are directly invoked using the command $kron(\mathbf{A},\mathbf{B}).$ The Kronecker tensor product of three matrices $\mathbf{A},\mathbf{B}$ and $\mathbf{C}$ each of sizes $p\times q$ is calculated as

$${\displaystyle \mathbf{A}\otimes \mathbf{B}\otimes \mathbf{C}=\left[\begin{array}{cccc}{\alpha }_{11}\mathbf{E}& {\alpha }_{12}\mathbf{E}& \dots & {\alpha }_{1q}\mathbf{E}\\ {\alpha }_{21}\mathbf{E}& {\alpha }_{22}\mathbf{E}& \dots & {\alpha }_{2q}\mathbf{E}\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_{p1}\mathbf{E}& {\alpha }_{p2}\mathbf{E}& \dots & {\alpha }_{pq}\mathbf{E}\end{array}\right]},$$

(16)

where $\mathbf{E}$ is expressed in the form

$${\displaystyle \mathbf{E}=\mathbf{B}\otimes \mathbf{C}=\left[\begin{array}{cccc}{\beta }_{11}\mathbf{C}& {\beta }_{12}\mathbf{C}& \dots & {\beta }_{1q}\mathbf{C}\\ {\beta }_{21}\mathbf{C}& {\beta }_{22}\mathbf{C}& \dots & {\beta }_{2q}\mathbf{C}\\ ⋮& ⋮& \ddots & ⋮\\ {\beta }_{p1}\mathbf{C}& {\beta }_{p2}\mathbf{C}& \dots & {\beta }_{pq}\mathbf{C}\end{array}\right]}.$$

(17)

In MATLAB, the $kron$ function is generalized to come up with the function $superkron$ for computing the product of more than two matrices. Thus, the Kronecker product of the three matrices is evaluated in one-shot using the function $superkron(\mathbf{A},\mathbf{B},\mathbf{C}).$ Consequently, (9) can be written in the form

$$\begin{array}{ccc}& & u({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\left(L\left({\widehat{z}}_k\right)\otimes L\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right)\right)U,\hfill \end{array}$$

(18)

where

$$\begin{array}{c}L\left({\widehat{z}}_k\right)=\left[L_0\left({\widehat{z}}_k\right),L_1\left({\widehat{z}}_k\right),\dots ,L_{N_z}\left({\widehat{z}}_k\right)\right],L\left({\widehat{y}}_j\right)=\left[L_0\left({\widehat{y}}_j\right),L_1\left({\widehat{y}}_j\right),\dots ,L_{N_y}\left({\widehat{y}}_j\right)\right],\hfill \\ L\left({\widehat{x}}_i\right)=\left[L_0\left({\widehat{x}}_i\right),L_1\left({\widehat{x}}_i\right),\dots ,L_{N_x}\left({\widehat{x}}_i\right)\right],\hfill \end{array}$$

(19)

$$U={[u_{000},\dots ,u_{00N_x},u_{010},\dots ,u_{01N_x},\dots ,u_{0N_y{N}_x},u_{100},\dots ,u_{1,N_y{N}_x},\dots ,u_{N_z{N}_y{N}_x}]}^T$$

(20)

$$\begin{array}{cc}& L\left({\widehat{z}}_k\right)\otimes L\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right)=[L_0\left({\widehat{z}}_k\right)\otimes L_0\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right),\dots ,L_0\left({\widehat{z}}_k\right)\otimes L_{N_y}\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right),\hfill \\ & L_1\left({\widehat{z}}_k\right)\otimes L_0\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right),\dots ,L_1\left({\widehat{z}}_k\right)\otimes L_{N_y}\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right),\dots ,L_{N_z}\left({\widehat{z}}_k\right)\otimes L_{N_y}\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right)].\hfill \end{array}$$

(21)

The first and sth derivatives of the approximate solution are approximated at the collocation points $(\widehat{x},\widehat{y},\widehat{z})$ as follows

$$\begin{array}{ccc}& & \frac{\partial u}{\partial x}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\left(L\left({\widehat{z}}_k\right)\otimes L\left({\widehat{y}}_j\right)\otimes \widehat{\mathbf{D}}\right)U,\frac{{\partial }^{s}u}{\partial x^s}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\left(L\left({\widehat{z}}_k\right)\otimes L\left({\widehat{y}}_j\right)\otimes {\widehat{\mathbf{D}}}^{\left(s\right)}\right)U,\hfill \\ & & \frac{\partial u}{\partial y}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\left(L\left({\widehat{z}}_k\right)\otimes \widehat{\widehat{\mathbf{D}}}\otimes L\left({\widehat{x}}_i\right)\right)U,\frac{{\partial }^{s}u}{\partial y^s}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\left(L\left({\widehat{z}}_k\right)\otimes {\widehat{\widehat{\mathbf{D}}}}^{\left(s\right)}\otimes L\left({\widehat{x}}_i\right)\right)U,\hfill \\ & & \frac{\partial u}{\partial z}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\left(\widehat{\widehat{\widehat{\mathbf{D}}}}\otimes L\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right)\right)U,\frac{{\partial }^{s}u}{\partial z^s}({\widehat{x}}_i,{\widehat{y}}_j,{\widehat{z}}_k)=\left({\widehat{\widehat{\widehat{\mathbf{D}}}}}^{\left(s\right)}\otimes L\left({\widehat{y}}_j\right)\otimes L\left({\widehat{x}}_i\right)\right)U.\hfill \end{array}$$

(22)

We note that the hat in $\widehat{\mathbf{D}}$ of size $(N_x+1)\times (N_x+1)$ has been used to distinguish the differentiation matrix in x from those in y and $z.$ The same holds for the double and triple hats in the differentiation matrices $\widehat{\widehat{\mathbf{D}}}$ and $\widehat{\widehat{\widehat{\mathbf{D}}}}$ of sizes $(N_y+1)\times (Ny+1)$ and $(N_z+1)\times (Nz+1),$ respectively.

Applying the spectral collocation method by making use of the definition of discrete derivatives on the QLM iterative scheme, we get a system of linear algebraic equations with size $(N_x+1)(N_y+1)(N_z+1),$ which is given by

$$\begin{array}{ccc}& & [\mathrm{diag}\left[{\sigma }_6^{\iota }\right]\left({\mathbf{I}}_{N_z+1}\otimes {\mathbf{I}}_{N_y+1}\otimes {\mathsf{\Omega }}_x^2{\widehat{\mathbf{D}}}^{\left(2\right)}\right)+\mathrm{diag}\left[{\sigma }_5^{\iota }\right]\left({\mathbf{I}}_{N_z+1}\otimes {\mathsf{\Omega }}_y^2{\widehat{\widehat{\mathbf{D}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_x+1}\right)\hfill \\ & & +\mathrm{diag}\left[{\sigma }_4^{\iota }\right]\left({\mathsf{\Omega }}_z^2{\widehat{\widehat{\widehat{\mathbf{D}}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_y+1}\otimes {\mathbf{I}}_{N_x+1}\right)+\mathrm{diag}\left[{\sigma }_3^{\iota }\right]\left({\mathbf{I}}_{N_z+1}\otimes {\mathbf{I}}_{N_y+1}\otimes {\mathsf{\Omega }}_x\widehat{\mathbf{D}}\right)\hfill \\ & & +\mathrm{diag}\left[{\sigma }_2^{\iota }\right]\left({\mathbf{I}}_{N_z+1}\otimes {\mathsf{\Omega }}_y\widehat{\widehat{\mathbf{D}}}\otimes {\mathbf{I}}_{N_x+1}\right)+\mathrm{diag}\left[{\sigma }_1^{\iota }\right]\left({\mathsf{\Omega }}_z\widehat{\widehat{\widehat{\mathbf{D}}}}\otimes {\mathbf{I}}_{N_y+1}\otimes {\mathbf{I}}_{N_x+1}\otimes \right)+\mathrm{diag}\left[{\sigma }_0^{\iota }\right]]U^{\iota +1}=R^{\iota },\hfill \end{array}$$

(23)

where $U^{\iota +1}={[u_{000}^{\iota +1},\dots ,u_{00N_x}^{\iota +1},u_{010},\dots ,u_{01N_x}^{\iota +1},\dots ,u_{0N_y{N}_x}^{\iota +1},u_{100}^{\iota +1},\dots ,u_{1,N_y{N}_x}^{\iota +1},\dots ,u_{N_z{N}_y{N}_x}^{\iota +1}]}^T,$${\mathsf{\Omega }}_x=\frac{2}{b-a},{\mathsf{\Omega }}_y=\frac{2}{d-c}$ and ${\mathsf{\Omega }}_z=\frac{2}{f-e}$ are used for scaling the respective derivative matrices, and $I_{N_x+1},I_{N_y+1}$ and $I_{N_z+1}$ are identity matrices of sizes $(N_x+1)\times (N_x+1),(N_y+1)\times (N_y+1)$ and $(N_z+1)\times (N_z+1),$ respectively. From the boundary conditions (2)–(4), we note that numerical values of the unknown U are given at the boundaries. Thus, we have to solve Equation (23) for the interior points. Collocating at the interior points, with $\left\{x_1,x_2,\dots ,x_{N_x-1}\right\}$, $\left\{y_1,y_2,\dots ,y_{N_y-1}\right\}$ and $\left\{z_1,z_2,\dots ,z_{N_z-1}\right\}$ as our computational grid, and applying the corresponding Dirichlet boundary conditions at the respective boundary points $x_0=b,x_{N_x}=a,y_0=d,y_{N_y}=c,z_0=f,z_{N_z}=e,$ we obtain the following equation

$$\begin{array}{ccc}& & [\mathrm{diag}\left[{\widehat{\sigma }}_6^{\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x^2{\left[{\widehat{\mathbf{D}}}^{\left(2\right)}\right]}_{2:N_x,:}\right)+\mathrm{diag}\left[{\widehat{\sigma }}_5^{\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y^2{\left[{\widehat{\widehat{\mathbf{D}}}}^{\left(2\right)}\right]}_{2:N_y,:}\otimes {\mathbf{I}}_{N_x-1}\right)\hfill \\ & & +\mathrm{diag}\left[{\widehat{\sigma }}_4^{\iota }\right]\left({\mathsf{\Omega }}_z^2{\left[{\widehat{\widehat{\widehat{\mathbf{D}}}}}^{\left(2\right)}\right]}_{2:N_z,:}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\widehat{\sigma }}_3^{\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x{\left[\widehat{\mathbf{D}}\right]}_{2:N_x,:}\right)\hfill \\ & & +\mathrm{diag}\left[{\widehat{\sigma }}_2^{\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y{\left[\widehat{\widehat{\mathbf{D}}}\right]}_{2:N_y,:}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\widehat{\sigma }}_1^{\iota }\right]\left({\mathsf{\Omega }}_z{\left[\widehat{\widehat{\widehat{\mathbf{D}}}}\right]}_{2:N_z,:}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)\hfill \\ & & +\mathrm{diag}\left[{\widehat{\sigma }}_0^{\iota }\right]]{\widehat{U}}^{\iota +1}={\widehat{R}}^{\iota },\hfill \end{array}$$

(24)

where $I_{N_x-1},I_{N_y-1}$ and $I_{N_z-1}$ are identity matrices of sizes $(N_x-1)\times (N_x-1),(N_y-1)\times (N_y-1)$ and $(N_z-1)\times (N_z-1),$ respectively, $\widehat{U}=U_{2:N_x,2:N_y,2:N_z},{\widehat{\sigma }}_{r,\iota }(r=1,2,3,4,5,6)={\left[{\sigma }_{r,\iota }\right]}_{2:N_x,2:N_y,2:N_z},$${\widehat{R}}_{\iota }={\left[R_{\iota }\right]}_{2:N_x,2:N_y,2:N_z}-\left({\varphi }_{ijk}+{\widehat{\varphi }}_{ijk}\right)-\left({\phi }_{ijk}+{\widehat{\phi }}_{ijk}\right)-\left({\vartheta }_{ijk}+{\widehat{\vartheta }}_{ijk}\right),$ and

$$\begin{array}{ccc}& & {\varphi }_{ijk}={\sigma }_{6,\iota }{\left[{\widehat{D}}^{\left(2\right)}\right]}_{2:N_x,1}\otimes U_{1,2:N_y,2:N_z}+{\sigma }_{3,\iota }{\left[\widehat{D}\right]}_{2:N_x,1}\otimes U_{1,2:N_y,2:N_z},\hfill \\ & & {\widehat{\varphi }}_{ijk}={\sigma }_{6,\iota }{\left[{\widehat{D}}^{\left(2\right)}\right]}_{2:N_x,N_x+1}\otimes U_{N_x+1,2:N_y,2:N_z}+{\sigma }_{3,\iota }{\left[\widehat{D}\right]}_{2:N_x,N_x+1}\otimes U_{N_x+1,2:N_y,2:N_z},\hfill \\ & & {\phi }_{ijk}={\sigma }_{5,\iota }{\left[{\widehat{\widehat{D}}}^{\left(2\right)}\right]}_{2:N_y,1}\otimes U_{2:N_x,1,2:N_z}+{\sigma }_{2,\iota }{\left[\widehat{\widehat{D}}\right]}_{2:N_y,1}\otimes U_{2:N_x,1,2:N_z},\hfill \\ & & {\widehat{\phi }}_{ijk}={\sigma }_{5,\iota }{\left[{\widehat{\widehat{D}}}^{\left(2\right)}\right]}_{2:N_y,N_y+1}\otimes U_{2:N_x,N_y+1,2:N_z}+{\sigma }_{2,\iota }{\left[\widehat{\widehat{D}}\right]}_{2:N_y,N_y+1}\otimes U_{2:N_x,N_y+1,2:N_z},\hfill \\ & & {\vartheta }_{ijk}={\sigma }_{4,\iota }{\left[{\widehat{\widehat{\widehat{D}}}}^{\left(2\right)}\right]}_{2:N_z,1}\otimes U_{2:N_x,2:N_y,1}+{\sigma }_{1,\iota }{\left[\widehat{\widehat{\widehat{D}}}\right]}_{2:N_z,1}\otimes U_{2:N_x,2:N_y,1},\hfill \\ & & {\widehat{\vartheta }}_{ijk}={\sigma }_{4,\iota }{\left[{\widehat{\widehat{\widehat{D}}}}^{\left(2\right)}\right]}_{2:N_z,N_z+1}\otimes U_{2:N_x,2:N_y,N_z+1}+{\sigma }_{1,\iota }{\left[\widehat{\widehat{\widehat{D}}}\right]}_{2:N_z,N_z+1}\otimes U_{2:N_x,2:N_y,N_z+1}.\hfill \end{array}$$

It is worth noting that the derivative coefficient matrices $\widehat{\mathbf{D}},\widehat{\widehat{\mathbf{D}}},\widehat{\widehat{\widehat{\mathbf{D}}}}$ in Equation (24) are of respective orders $(N_x-1)\times (N_x-1),(N_y-1)\times (N_y-1)$ and $(N_z-1)\times (Nz-1),$ whereas $\widehat{D},\widehat{\widehat{D}},\widehat{\widehat{\widehat{D}}}$ are of orders $(N_x-1)\times 1,(N_y-1)\times 1$ and $(N_z-1)\times 1,$ respectively. Equation (24) is a system of $(N_x-1)(N_y-1)(N_z-1)$ linear equations in $(N_x-1)(N_y-1)(N_z-1)$ unknowns that can be written compactly as an $(N_x-1)(N_y-1)(N_z-1)\times (N_x-1)(N_y-1)(N_z-1))$ matrix system of the form

$${\displaystyle \left[\begin{array}{c}\mathbf{A}\end{array}\right]}{\displaystyle \left[\begin{array}{c}\mathbf{U}\end{array}\right]}={\displaystyle \left[\begin{array}{c}\mathbf{R}\end{array}\right]},$$

(25)

where $\mathbf{A}$ is a $(N_x-1)(N_y-1)(N_z-1)\times (N_x-1)(N_y-1)(N_z-1)$ square matrix, and $\mathbf{R}$ is the $(N_x-1)(N_y-1)(N_z-1)\times 1$ array corresponding to the unknown $\mathbf{U}.$ After modifying the matrix system by eliminating solutions corresponding to given boundary conditions, the approximate solutions of $u(x,y,z)$ are then obtained iteratively from solving

$$\begin{array}{ccc}& & \mathbf{U}={\mathbf{A}}^{-1}\mathbf{R}\hfill \end{array}$$

(26)

at the interior nodes.

2.2. Solving the System of Nonlinear Elliptic PDEs

Applying QLM to the system of Equations (5) gives the following linearized system of PDEs

$$\begin{array}{ccc}& & {\sigma }_6^{1,\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial x^2}+{\sigma }_5^{1,\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial y^2}+{\sigma }_4^{1,\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial z^2}+{\sigma }_3^{1,\iota }\frac{\partial u^{\iota +1}}{\partial x}+{\sigma }_2^{1,\iota }\frac{\partial u^{\iota +1}}{\partial y}+{\sigma }_1^{1,\iota }\frac{\partial u^{\iota +1}}{\partial z}+{\delta }_6^{1,\iota }\frac{{\partial }^2{v}^{\iota +1}}{\partial x^2}\hfill \\ & & +{\delta }_5^{1,\iota }\frac{{\partial }^2{v}^{\iota +1}}{\partial y^2}+{\delta }_4^{1,\iota }\frac{{\partial }^2{v}^{\iota +1}}{\partial z^2}+{\delta }_3^{1,\iota }\frac{\partial u^{\iota +1}}{\partial x}+{\delta }_2^{1,\iota }\frac{\partial v^{\iota +1}}{\partial y}+{\delta }_1^{1,\iota }\frac{\partial v^{\iota +1}}{\partial z}+{\delta }_0^{1,\iota }{v}^{\iota +1}+{\sigma }_0^{1,\iota }{u}^{\iota +1}=R_1^{\iota },\hfill \\ & & {\sigma }_6^{2,\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial x^2}+{\sigma }_5^{2,\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial y^2}+{\sigma }_4^{2,\iota }\frac{{\partial }^2{u}^{\iota +1}}{\partial z^2}+{\sigma }_3^{2,\iota }\frac{\partial u^{\iota +1}}{\partial x}+{\sigma }_2^{2,\iota }\frac{\partial u^{\iota +1}}{\partial y}+{\sigma }_1^{2,\iota }\frac{\partial u^{\iota +1}}{\partial z}+{\delta }_6^{2,\iota }\frac{{\partial }^2{v}^{\iota +1}}{\partial x^2}\hfill \\ & & +{\delta }_5^{2,\iota }\frac{{\partial }^2{v}^{\iota +1}}{\partial y^2}+{\delta }_4^{2,\iota }\frac{{\partial }^2{v}^{\iota +1}}{\partial z^2}+{\delta }_3^{2,\iota }\frac{\partial v^{\iota +1}}{\partial x}+{\delta }_2^{2,\iota }\frac{\partial v^{\iota +1}}{\partial y}+{\delta }_1^{2,\iota }\frac{\partial v^{\iota +1}}{\partial z}+{\sigma }_0^{2,\iota }{u}^{\iota +1}+{\delta }_0^{2,\iota }{v}^{\iota +1}=R_2^{\iota },\hfill \end{array}$$

(27)

where the variable coefficients are given by

$$\begin{array}{ccc}& & {\sigma }_6^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial u_{xx}^{\iota }},{\sigma }_5^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial u_{yy}^{\iota }},{\sigma }_4^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial u_{zz}^{\iota }},{\sigma }_3^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial u_x^{\iota }},{\sigma }_2^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial u_y^{\iota }},{\sigma }_1^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial u_z^{\iota }},{\sigma }_0^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial u^{\iota }},\hfill \\ & & {\delta }_6^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial v_{xx}^{\iota }},{\delta }_5^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial v_{yy}^{\iota }},{\delta }_4^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial v_{zz}^{\iota }},{\delta }_3^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial v_x^{\iota }},{\delta }_2^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial v_y^{\iota }},{\delta }_1^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial v_z^{\iota }},{\delta }_0^{\lambda ,\iota }=\frac{\partial {\mathsf{\Gamma }}_{\lambda }}{\partial v^{\iota }},\hfill \\ & & R_{\lambda }^{\iota }={\sigma }_6^{\lambda ,\iota }\frac{{\partial }^2{u}^{\iota }}{\partial x^2}+{\sigma }_5^{\lambda ,\iota }\frac{{\partial }^2{u}^{\iota }}{\partial y^2}+{\sigma }_4^{\lambda ,\iota }\frac{{\partial }^2{u}^{\iota }}{\partial z^2}+{\sigma }_3^{\lambda ,\iota }\frac{\partial u^{\iota }}{\partial x}+{\sigma }_2^{\lambda ,\iota }\frac{\partial u^{\iota }}{\partial y}+{\sigma }_1^{\lambda ,\iota }\frac{\partial u^{\iota }}{\partial z}+{\delta }_6^{\lambda ,\iota }\frac{{\partial }^2{v}^{\iota }}{\partial x^2}\hfill \\ & & +{\delta }_5^{\lambda ,\iota }\frac{{\partial }^2{v}^{\iota }}{\partial y^2}+{\delta }_4^{\lambda ,\iota }\frac{{\partial }^2{v}^{\iota }}{\partial z^2}+{\delta }_3^{\lambda ,\iota }\frac{\partial v^{\iota }}{\partial x}+{\delta }_2^{\lambda ,\iota }\frac{\partial v^{\iota }}{\partial y}+{\delta }_1^{\lambda ,\iota }\frac{\partial v^{\iota }}{\partial z}+{\sigma }_0^{\lambda ,\iota }{u}^{\iota }+{\delta }_0^{\lambda ,\iota }{v}^{\iota }-{\mathsf{\Gamma }}_{\lambda }^{\iota },\hfill \end{array}$$

(28)

with $u_{xx}^{\iota }=\frac{{\partial }^2{u}^{\iota }}{\partial x^2},u_{yy}^{\iota }=\frac{{\partial }^2{u}^{\iota }}{\partial y^2},u_{zz}^{\iota }=\frac{{\partial }^2{u}^{\iota }}{\partial z^2},u_x^{\iota }=\frac{\partial u^{\iota }}{\partial x},u_y^{\iota }=\frac{\partial u^{\iota }}{\partial y},u_z^{\iota }=\frac{\partial u^{\iota }}{\partial z},$$v_{xx}^{\iota }=\frac{{\partial }^2{v}^{\iota }}{\partial x^2},v_{yy}^{\iota }=\frac{{\partial }^2{v}^{\iota }}{\partial y^2},v_{zz}^{\iota }=\frac{{\partial }^2{v}^{\iota }}{\partial z^2},v_x^{\iota }=\frac{\partial v^{\iota }}{\partial x},u_y^{\iota }=\frac{\partial v^{\iota }}{\partial y},v_z^{\iota }=\frac{\partial v^{\iota }}{\partial z}.$ The process of discretization, domain transformation and trivariate Lagrange interpolation are performed as in the linearized single elliptic PDEs. Applying the Chebyshev spectral collocation method by using the derivative matrices at the interior collocation points and applying the boundary conditions at the boundary points, Equation (27) can be expressed using the Kronecker product in a matrix form given by

$$\begin{array}{ccc}& & [\mathrm{diag}\left[{\sigma }_6^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x^2{\widehat{\mathbf{D}}}^{\left(2\right)}\right)+\mathrm{diag}\left[{\sigma }_5^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y^2{\widehat{\widehat{\mathbf{D}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_x-1}\right)\hfill \\ & & +\mathrm{diag}\left[{\sigma }_4^{1,\iota }\right]\left({\mathsf{\Omega }}_z^2{\widehat{\widehat{\widehat{\mathbf{D}}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\sigma }_3^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}{\mathsf{\Omega }}_x\widehat{\mathbf{D}}\right)\hfill \\ & & +\mathrm{diag}\left[{\sigma }_2^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y\widehat{\widehat{\mathbf{D}}}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\sigma }_1^{1,\iota }\right]\left({\mathsf{\Omega }}_z\widehat{\widehat{\widehat{\mathbf{D}}}}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\sigma }_0^{1,\iota }\right]]{\widehat{U}}^{\iota +1}\hfill \\ & & +[\mathrm{diag}\left[{\delta }_6^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x^2{\widehat{\mathbf{D}}}^{\left(2\right)}\right)+\mathrm{diag}\left[{\delta }_5^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y^2{\widehat{\widehat{\mathbf{D}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_x-1}\right)\hfill \\ & & +\mathrm{diag}\left[{\delta }_4^{1,\iota }\right]\left({\mathsf{\Omega }}_z^2{\widehat{\widehat{\widehat{\mathbf{D}}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\delta }_3^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x\widehat{\mathbf{D}}\right)\hfill \\ & & +\mathrm{diag}\left[{\delta }_2^{1,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y\widehat{\widehat{\mathbf{D}}}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\delta }_1^{1,\iota }\right]\left({\mathsf{\Omega }}_z\widehat{\widehat{\widehat{\mathbf{D}}}}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\delta }_0^{1,\iota }\right]]V^{\iota +1}={\widehat{R}}_1^{\iota },\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & [\mathrm{diag}\left[{\sigma }_6^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x^2{\widehat{\mathbf{D}}}^{\left(2\right)}\right)+\mathrm{diag}\left[{\sigma }_5^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y^2{\widehat{\widehat{\mathbf{D}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_x-1}\right)\hfill \\ & & +\mathrm{diag}\left[{\sigma }_4^{2,\iota }\right]\left({\mathsf{\Omega }}_z^2{\widehat{\widehat{\widehat{\mathbf{D}}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\sigma }_3^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}{\mathsf{\Omega }}_x\widehat{\mathbf{D}}\right)\hfill \\ & & +\mathrm{diag}\left[{\sigma }_2^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y\widehat{\widehat{\mathbf{D}}}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\sigma }_1^{2,\iota }\right]\left({\mathsf{\Omega }}_z\widehat{\widehat{\widehat{\mathbf{D}}}}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\sigma }_0^{1,\iota }\right]]{\widehat{U}}^{\iota +1}\hfill \\ & & +[\mathrm{diag}\left[{\delta }_6^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x^2{\widehat{\mathbf{D}}}^{\left(2\right)}\right)+\mathrm{diag}\left[{\delta }_5^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y^2{\widehat{\widehat{\mathbf{D}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_x-1}\right)\hfill \\ & & +\mathrm{diag}\left[{\delta }_4^{2,\iota }\right]\left({\mathsf{\Omega }}_z^2{\widehat{\widehat{\widehat{\mathbf{D}}}}}^{\left(2\right)}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\delta }_3^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathsf{\Omega }}_x\widehat{\mathbf{D}}\right)\hfill \\ & & +\mathrm{diag}\left[{\delta }_2^{2,\iota }\right]\left({\mathbf{I}}_{N_z-1}\otimes {\mathsf{\Omega }}_y\widehat{\widehat{\mathbf{D}}}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\delta }_1^{2,\iota }\right]\left({\mathsf{\Omega }}_z\widehat{\widehat{\widehat{\mathbf{D}}}}\otimes {\mathbf{I}}_{N_y-1}\otimes {\mathbf{I}}_{N_x-1}\right)+\mathrm{diag}\left[{\delta }_0^{2,\iota }\right]]V^{\iota +1}={\widehat{R}}_2^{\iota },\hfill \end{array}$$

(30)

which can be expressed compactly as $2(N_x-1)(N_y-1)(N_z-1)\times 2(N_x-1)(N_y-1)(N_z-1))$ matrix system given as

$${\displaystyle \left[\begin{array}{cc}{\mathbf{A}}_{1,1}& {\mathbf{A}}_{1,2}\\ {\mathbf{A}}_{2,1}& {\mathbf{A}}_{2,2}\end{array}\right]}\left[\begin{array}{c}\widehat{\mathbf{U}}\\ \widehat{\mathbf{V}}\end{array}\right]=\left[\begin{array}{c}{\widehat{\mathbf{R}}}_1\\ {\widehat{\mathbf{R}}}_2\end{array}\right],$$

(31)

or

$$\begin{array}{ccc}& & {\mathbf{C}}^{\iota }{\mathsf{\Gamma }}^{\iota +1}={\mathbf{K}}^{\iota }.\hfill \end{array}$$

(32)

In Equation (32), $\mathbf{K}$ is the right-hand side corresponding to the unknown $\mathsf{\Gamma }$ with size $2(N_x-1)(N_y-1)(N_z-1)\times 1.$ Equation (32) is to be solved iteratively to obtain the numerical solution of the system of elliptic PDEs.

3. Error Bound Theorem in Trivariate Polynomial Interpolation

This segment of the current work provides error bound theorems along with their proofs for the interpolation error in the triple Lagrange interpolating polynomial, which is established via the CGL nodes. These nodes are relative extremes of the Chebyshev polynomial of the first kind given as $T_{N_x}\left(\overline{x}\right)=cos[N_{x}arccos\left(\widehat{x}\right)],$ where $N_x$ is the order of the polynomial and $\overline{x}\in [-1,1].$ The full set of the CGL points are the roots of the polynomial with order $N_x+1$ expressed in the form

$$\begin{array}{ccc}& & L_{N_x+1}\left(\overline{x}\right)=\left(1-{\overline{x}}^2\right)T_{N_x}^{\prime }\left(\overline{x}\right).\hfill \end{array}$$

(33)

Below, we provide a theorem that validates the derivation of the error bound theorems of trivariate polynomial interpolation.

Theorem 1

([37]). Suppose that $u(x,y,z)\in C^{N_x+N_y+N_z+3}$ is smooth enough in such a way that at least the partial derivative of order $N_x+1$ with reference to $x,$ the partial derivative of order $N_y+1$ with reference to $y,$ the partial derivative of order $N_z+1$ with reference to $z,$ the mixed partial derivative of order $N_x+N_y+N_z+3$ with reference to $x,y,z$ exist and are all continuous, then ∃ values ${\omega }_x,{\omega }_x^{\prime }\in (a,b),{\omega }_y,{\omega }_y^{\prime }\in (c,d)$ and ${\omega }_z,{\omega }_z^{\prime }\in (e,f)$ such that

$$\begin{array}{ccc}& & u(x,y,z)-U(x,y,z)\hfill \\ & & =\frac{{\partial }^{N_x+1}u({\omega }_x,y,z)}{{\partial }^{N_x+1}(N_x+1)!}\prod _{i=0}^{N_x}(x-x_i)+\frac{{\partial }^{N_y+1}u(x,{\omega }_y,z)}{{\partial }^{N_y+1}(N_y+1)!}\prod _{j=0}^{N_y}(y-y_j)+\frac{{\partial }^{N_z+1}u(x,y,{\omega }_z)}{{\partial }^{N_z+1}(N_z+1)!}\prod _{k=0}^{N_z}(z-z_k)\hfill \\ & & +\frac{{\partial }^{N_x+N_y+N_z+3}u({\omega }_x^{\prime },{\omega }_y^{\prime },{\omega }_z^{\prime })}{{\partial }^{N_x+1}{\partial }^{N_y+1}{\partial }^{N_z+1}(N_x+1)!(N_y+1)!(N_z+1)!}\prod _{i=0}^{N_x}(x-x_i)\prod _{j=0}^{N_y}(y-y_j)\prod _{k=0}^{N_z}(z-z_k),\hfill \end{array}$$

(34)

where $U(x,y,t)$ is a trivariate interpolating polynomial of $u(x,y,z)$ at grid points ${\left\{x_i\right\}}_{i=0}^{N_x}$ in the x-variable, grid points ${\left\{y_j\right\}}_{j=0}^{N_y}$ in the y-variable and grid points ${\left\{z_k\right\}}_{k=0}^{N_z}$ in the z-variable.

Equation (34) emanates from the use of the mean value theorem and is derived recursively using the univariate formula [38] for the sufficiently smooth function $u(x,y,z).$ When taking into consideration the absolute values, we get

$$\begin{array}{ccc}& & u(x,y,z)-U(x,y,z)\le \underset{(x,y,z)\in \mathsf{{\rm Y}}}{\max }\left|\frac{{\partial }^{N_x+1}u({\omega }_x,y,z)}{{\partial }^{N_x+1}}\right|\frac{\left|{\prod }_{i=0}^{N_x}(x-x_i)\right|}{(N_x+1)!}\hfill \\ & & +\underset{(x,y,z)\in \mathsf{{\rm Y}}}{\max }\left|\frac{{\partial }^{N_y+1}u(x,{\omega }_y,z)}{{\partial }^{N_y+1}}\right|\frac{\left|{\prod }_{j=0}^{N_y}(y-y_j)\right|}{(N_y+1)!}+\underset{(x,y,z)\in \mathsf{{\rm Y}}}{\max }\left|\frac{{\partial }^{N_z+1}u(x,y,{\omega }_z)}{{\partial }^{N_z+1}}\right|\frac{\left|{\prod }_{k=0}^{N_z}(z-z_k)\right|}{(N_z+1)!}\hfill \\ & & +\underset{(x,y,z)\in \mathsf{{\rm Y}}}{\max }\left|\frac{{\partial }^{N_x+N_y+N_z+3}u({\omega }_x^{{}^{\prime }},{\omega }_x^{{}^{\prime }},{\omega }_z^{{}^{\prime }})}{{\partial }^{N_x+1}{\partial }^{N_y+1}{\partial }^{N_z+1}}\right|\frac{\left|{\prod }_{i=0}^{N_x}(x-x_i)\right|\left|{\prod }_{j=0}^{N_y}(y-y_j)\right|\left|{\prod }_{k=0}^{N_z}(z-z_k)\right|}{(N_x+1)!(N_y+1)!(N_z+1)!},\hfill \end{array}$$

(35)

where $\mathsf{{\rm Y}}=[a,b]\times [c,d]\times [e,f].$ Since the function $u(x,y,z)$ is taken to be smooth on the domain of interest, then its derivatives are bounded and ∃ constants $C_1,C_2,C_3,$$C_4$ such that

$$\begin{array}{ccc}& & \underset{(x,y,z)\in {\rm Y}}{\max }\left|\frac{{\partial }^{N_x+1}u({\omega }_x,y,z)}{{\partial }^{N_x+1}}\right|\le C_1,\underset{(x,y,z)\in {\rm Y}}{\max }\left|\frac{{\partial }^{N_y+1}u(x,{\omega }_y,z)}{{\partial }^{N_y+1}}\right|\le C_2\hfill \\ & & \underset{(x,y,z)\in {\rm Y}}{\max }\left|\frac{{\partial }^{N_z+1}u(x,y,{\omega }_z)}{{\partial }^{N_z+1}}\right|\le C_3,\underset{(x,y,z)\in {\rm Y}}{\max }\left|\frac{{\partial }^{N_x+N_y+N_z+3}u({\omega }_x^{\prime },{\omega }_y^{\prime },{\omega }_z^{\prime })}{{\partial }^{N_x+1}{\partial }^{N_y+1}{\partial }^{N_z+1}}\right|\le C_4.\hfill \end{array}$$

(36)

The following theorem validates the error bound theorem for trivariate polynomial interpolation via CGL nodes when a single domain of approximation is regarded.

Theorem 2.

The resultant error bound theorem when CGL nodes ${\left\{x_i\right\}}_{i=0}^{N_x}$, ${\left\{y_j\right\}}_{j=0}^{N_y}$ and ${\left\{z_k\right\}}_{k=0}^{N_z}$ in the $x,y$ and z variables, respectively, are employed in trivariate polynomial interpolation is provided as

$$\begin{array}{ccc}& & E(x,y,z)\le C_1\frac{8{\left(\frac{b-a}{4}\right)}^{N_x+1}}{(N_x+1)!}+C_2\frac{8{\left(\frac{d-c}{4}\right)}^{N_y+1}}{(N_y+1)!}+C_3\frac{8{\left(\frac{f-e}{4}\right)}^{N_z+1}}{(N_z+1)!}\hfill \\ & & +C_4\frac{8^3{\left(\frac{b-a}{4}\right)}^{N_x+1}{\left(\frac{d-c}{4}\right)}^{N_y+1}{\left(\frac{f-e}{4}\right)}^{N_z+1}}{(N_x+1)!(N_y+1)!(N_z+1)!}.\hfill \end{array}$$

(37)

Proof of Theorem 2.

Adopting the relationship given in [37], Equation (33) becomes

$$\begin{array}{ccc}& & L_{N_x+1}\left(\overline{x}\right)=(1-{\overline{x}}^2)T_{N_x}^{{}^{\prime }}\left(\overline{x}\right)=-N_x\overline{x}{T}_{N_x}\left(\overline{x}\right)+N_x{T}_{N_x-1}\left(\overline{x}\right).\hfill \end{array}$$

(38)

Applying the triangular inequality and noting that $|T_{N_x}\left(\overline{x}\right)|\le 1,\forall \widehat{x}\in [-1,1],$ we get

$$\begin{array}{ccc}& & L_{N_x+1}\left(\overline{x}\right)=|-N_x\overline{x}{T}_{N_x}\left(\overline{x}\right)+N_x{T}_{N_x-1}\left(\overline{x}\right)|\le |-N_x\overline{x}{T}_{N_x}\left(\overline{x}\right)|+|N_x{T}_{N_x-1}\left(\overline{x}\right)|\le 2N_x.\hfill \end{array}$$

(39)

The leading coefficient of $L_{N_x+1}\left(\widehat{x}\right)$ is $2^{N_x-1}{N}_x,$ where $2^{N_x-1}$ stem from the leading coefficient of $T_{N_x}\left(\widehat{x}\right)$ and $N_x$ is derived from differentiating $T_{N_x}$ in $N_x^{th}$ times. The product factor in the first term of the error bound formula in Equation (35) can be regarded as the factorized form of monic polynomial $\frac{L_{N_x+1}\left(\overline{x}\right)}{2^{N_x-1}{N}_x}.$ We have

$$\begin{array}{ccc}& & \prod _{i=0}^{N_x}(\overline{x}-{\overline{x}}_i)=\frac{L_{N_x+1}\left(\overline{x}\right)}{2^{N_x-1}{N}_x},\overline{x}\in [-1,1].\hfill \end{array}$$

(40)

Utilizing Equation (39), it can be shown that the monic polynomial (40) is bounded by

$$\begin{array}{ccc}& & \left|\prod _{i=0}^{N_x}(x-{\overline{x}}_i)\right|=\left|\frac{L_{N_x+1}\left(\overline{x}\right)}{2^{N_x-1}{N}_x}\right|\le \frac{2N_x}{2^{N_x-1}{N}_x}=\frac{4}{2^{N_x}}.\hfill \end{array}$$

(41)

Considering the finite domain of interest $x\in [a,b],$ it can be shown that the first product factor in Equation (35) is bounded by

$$\begin{array}{ccc}& & \underset{a\le x\le b}{\max }\left|\prod _{i=0}^{N_x}(x-x_i)\right|=\underset{-1\le \widehat{x}\le 1}{\max }\left|\prod _{i=0}^{N_x}\frac{(b-a)}{2}(\overline{x}-{\overline{x}}_i)\right|={\left(\frac{b-a}{2}\right)}^{N_x+1}\underset{-1\le \overline{x}\le 1}{\max }\left|\prod _{i=0}^{N_x}(\overline{x}-{\overline{x}}_i)\right|\hfill \\ & & ={\left(\frac{b-a}{2}\right)}^{N_x+1}\underset{-1\le \overline{x}\le 1}{\max }\left|\frac{L_{N_x+1}\left(\overline{x}\right)}{2^{N_x-1}{N}_x}\right|\le \frac{4{\left(\frac{b-a}{2}\right)}^{N_x+1}}{2^{N_x}}=\frac{4{\left(\frac{b-a}{2}\right)}^{N_x+1}{2}^{N_x+1}}{2^{N_x}{2}^{N_x+1}}=8{\left(\frac{b-a}{4}\right)}^{N_x+1}\hfill \end{array}$$

(42)

Likewise, the second and third product factor are, respectively, bounded by

$$\begin{array}{ccc}& & \underset{c\le y\le d}{\max }\left|\prod _{j=0}^{N_y}(y-y_j)\right|={\left(\frac{d-c}{2}\right)}^{N_y+1}\underset{-1\le \overline{y}\le 1}{\max }\left|\frac{L_{N_y+1}\left(\overline{y}\right)}{2^{N_y-1}{N}_y}\right|\le \frac{4{\left(\frac{d-c}{2}\right)}^{N_y+1}}{2^{N_y}}\hfill \\ & & =\frac{4{\left(\frac{d-c}{2}\right)}^{N_y+1}{2}^{N_y+1}}{2^{N_y}{2}^{N_y+1}}=8{\left(\frac{d-c}{4}\right)}^{N_y+1},\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & \underset{e\le z\le f}{\max }\left|\prod _{k=0}^{N_z}(z-z_k)\right|={\left(\frac{f-e}{2}\right)}^{N_z+1}\underset{-1\le \overline{z}\le 1}{\max }\left|\frac{L_{N_z+1}\left(\overline{z}\right)}{2^{N_z-1}{N}_z}\right|\le \frac{4{\left(\frac{f-e}{2}\right)}^{N_z+1}}{2^{N_z}}\hfill \\ & & =\frac{4{\left(\frac{f-e}{2}\right)}^{N_z+1}{2}^{N_z+1}}{2^{N_z}{2}^{N_z+1}}=8{\left(\frac{f-e}{4}\right)}^{N_z+1},\hfill \end{array}$$

(44)

Adopting Equation (42) up to (44) and Equation (36) into (35) ends the proof. □

We take note that the error bound via CGL nodes is a bit greater than the optimal Chebyshev nodes [39]. Nevertheless, the CGL nodes are still favored as discretization nodes when compared to the Chebyshev nodes due to the fact that the CGL points are comprised of the boundary point, which is appropriate when dealing with boundary conditions of the equations.

4. Numerical Illustrations

In this section, we provide the various linear and nonlinear elliptic PDEs to be utilized in testing the applicability and accuracy of the numerical scheme. All the examples are available in the literature with exact solutions. The examples are solved subject to the given Dirichlet boundary conditions in various regions $(x,y,z)$$\in [a,b]\times [c,d]\times [e,f].$ The entire computational work was performed on MATLAB 2015a and run on a computer with 4 GB RAM and Intel Core i-3 1.7 GHz CPU.

4.1. Single PDEs

Example 1.

First, the method was tested on 3D Poisson’s equation with singular behaviour [21,40]

$$\begin{array}{ccc}& & \frac{{\partial }^{2}u}{\partial x^2}+\frac{1}{x^2}\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}+\frac{1}{x}\frac{\partial u}{\partial x}=f(x,y,z),a\le x\le b,c\le y\le d,e\le z\le f,\hfill \end{array}$$

(45)

subject to the boundary conditions

$$\begin{array}{ccc}& & u(a,y,z)=a^{2}cos\left(\pi y\right)cos\left(\pi z\right),u(b,y,z)=b^{2}cos\left(\pi y\right)cos\left(\pi z\right),u(x,c,z)=x^{2}cos\left(\pi c\right)cos\left(\pi z\right),\hfill \\ & & u(x,d,z)=x^{2}cos\left(\pi d\right)cos\left(\pi z\right),u(x,y,e)=x^{2}cos\left(\pi y\right)cos\left(\pi e\right),u(x,y,f)=x^{2}cos\left(\pi y\right)cos\left(\pi f\right),\hfill \end{array}$$

(46)

where the function $f(x,y,z)$ was obtained as $f(x,y,z)=(4-{\pi }^2-{\pi }^2{x}^2)cos\left(\pi y\right)cos\left(\pi z\right)$ using the exact solution $u(x,y,z)=x^{2}cos\left(\pi y\right)cos\left(\pi z\right).$ It is worth noting that the solution procedure of the proposed method involves performing the collocation process at the interior points. Thus, the numerical scheme is directly applicable to solve this elliptic PDE without any special modification at the singular point.

Example 2.

Next, we solve a nonlinear convection PDE given by [21,40,41,42,43]

$$\begin{array}{ccc}& & \frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}-u\left(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}\right)=f(x,y,z),a\le x\le b,c\le y\le d,e\le z\le f,\hfill \end{array}$$

(47)

where the function $f(x,y,z)$ was evaluated as

$$\begin{array}{ccc}& & f(x,y,z)=\left(1-\frac{{\pi }^2}{2}\right)e^{x}sin\left(\frac{\pi y}{2}\right)sin\left(\frac{\pi z}{2}\right)-e^{2x}{sin}^2\left(\frac{\pi y}{2}\right){sin}^2\left(\frac{\pi z}{2}\right)\hfill \\ & & -\frac{\pi }{2}{e}^{2x}{sin}^2\left(\frac{\pi y}{2}\right)sin\left(\frac{\pi z}{2}\right)cos\left(\frac{\pi z}{2}\right)-\frac{\pi }{2}{e}^{2x}{sin}^2\left(\frac{\pi z}{2}\right)sin\left(\frac{\pi y}{2}\right)cos\left(\frac{\pi y}{2}\right),\hfill \end{array}$$

(48)

using the exact solution $u(x,y,z)=e^{x}sin\left(\frac{\pi y}{2}\right)sin\left(\frac{\pi z}{2}\right).$ The associated boundary conditions are considered as

$$\begin{array}{ccc}& & u(a,y,z)=e^{a}sin\left(\frac{\pi y}{2}\right)sin\left(\frac{\pi z}{2}\right),u(b,y,z)=e^{b}sin\left(\frac{\pi y}{2}\right)sin\left(\frac{\pi z}{2}\right),\hfill \\ & & u(x,c,z)=e^{x}sin\left(\frac{\pi c}{2}\right)sin\left(\frac{\pi z}{2}\right),u(x,d,z)=e^{x}sin\left(\frac{\pi d}{2}\right)sin\left(\frac{\pi z}{2}\right),\hfill \\ & & u(x,y,e)=e^{x}sin\left(\frac{\pi y}{2}\right)sin\left(\frac{\pi e}{2}\right),u(x,y,f)=e^{x}sin\left(\frac{\pi y}{2}\right)sin\left(\frac{\pi f}{2}\right).\hfill \end{array}$$

(49)

4.2. System of Elliptic PDEs

Example 3.

Next, we take into consideration a system of two nonlinear coupled elliptic PDEs [21,44]

$$\begin{array}{ccc}& & 2\frac{{\partial }^{2}u}{\partial x^2}+3\frac{{\partial }^{2}u}{\partial y^2}+4\frac{{\partial }^{2}u}{\partial z^2}-\alpha uv\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}-\frac{\partial v}{\partial y}-\frac{\partial u}{\partial z}-\frac{\partial v}{\partial z}\right)=f(x,y,z),\hfill \\ & & 5\frac{{\partial }^{2}v}{\partial x^2}+3\frac{{\partial }^{2}v}{\partial y^2}+4\frac{{\partial }^{2}v}{\partial z^2}-\beta uv\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}-\frac{\partial v}{\partial y}-\frac{\partial u}{\partial z}-\frac{\partial v}{\partial z}\right)=g(x,y,z),\hfill \end{array}$$

(50)

where $a\le x\le b,c\le y\le d,e\le z\le f.$ Making use of the exact solution $u(x,y,z)=e^{x}cos\left(y\right)cos\left(z\right)$ and $v(x,y,z)=e^{x}sin\left(y\right)sin\left(z\right),$ the functions $f(x,y,z)$ and $g(x,y,z)$ are given by

$$\begin{array}{ccc}& & f(x,y,z)=e^{x}cos\left(y\right)cos\left(z\right)\left[-5-\alpha e^{2x}sin\left(y\right)sin\left(z\right)\left(cos\left(y\right)cos\left(z\right)+sin\left(y\right)sin\left(z\right)\right)\right],\hfill \\ & & g(x,y,z)=e^{x}sin\left(y\right)sin\left(z\right)\left[-2-\beta e^{2x}cos\left(y\right)cos\left(z\right)\left(cos\left(y\right)cos\left(z\right)+sin\left(y\right)sin\left(z\right)\right)\right],\hfill \end{array}$$

(51)

and the related boundary conditions are given by

$$\begin{array}{ccc}& & u(a,y,z)=e^{a}cos\left(y\right)cos\left(z\right),u(b,y,z)=e^{b}cos\left(y\right)cos\left(z\right),u(x,c,z)=e^{x}cos\left(c\right)cos\left(z\right),\hfill \\ & & u(x,d,z)=e^{x}cos\left(d\right)cos\left(z\right),u(x,y,e)=e^{x}cos\left(y\right)cos\left(e\right),u(x,y,f)=e^{x}cos\left(y\right)cos\left(f\right),\hfill \\ & & v(a,y,z)=e^{a}sin\left(y\right)sin\left(z\right),v(b,y,z)=e^{b}sin\left(y\right)sin\left(z\right),v(x,c,z)=e^{x}sin\left(c\right)sin\left(z\right),\hfill \\ & & v(x,d,z)=e^{x}sin\left(d\right)sinz,v(x,y,e)=e^{x}sin\left(y\right)sin\left(e\right),v(x,y,f)=e^{x}sin\left(y\right)sin\left(f\right).\hfill \end{array}$$

(52)

Example 4.

Lastly, we solve a system of three nonlinear coupled elliptic PDEs modeling Navier–Stokes equations in Cartesian coordinates [21,40,44]

$$\begin{array}{ccc}& & \frac{1}{Re}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}\right)-u\frac{\partial u}{\partial x}-v\frac{\partial u}{\partial y}-w\frac{\partial u}{\partial z}=f(x,y,z),\hfill \\ & & \frac{1}{Re}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}v}{\partial z^2}\right)-u\frac{\partial v}{\partial x}-v\frac{\partial v}{\partial y}-w\frac{\partial v}{\partial z}=g(x,y,z),\hfill \\ & & \frac{1}{Re}\left(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}+\frac{{\partial }^{2}w}{\partial z^2}\right)-u\frac{\partial w}{\partial x}-v\frac{\partial w}{\partial y}-w\frac{\partial w}{\partial z}=h(x,y,z),\hfill \end{array}$$

(53)

where $a\le x\le b,c\le y\le d,e\le z\le f,$$Re>0$ is a Reynolds number and the functions $f(x,y,z),g(x,y,z)$ and $h(x,y,z)$ are evaluated as

$$\begin{array}{ccc}& & f(x,y,z)=-\frac{3{\pi }^2}{Re}sin\left(\pi x\right)cos\left(\pi y\right)cos\left(\pi z\right)+\pi sin\left(\pi x\right)cos\left(\pi x\right){cos}^2\left(\pi y\right)\left[{sin}^2\left(\pi z\right)-{cos}^2\left(\pi z\right)\right]\hfill \\ & & -2\pi sin\left(\pi x\right)cos\left(\pi x\right){sin}^2\left(\pi y\right){cos}^2\left(\pi z\right),\hfill \\ & & g(x,y,z)=\frac{6{\pi }^2}{Re}cos\left(\pi x\right)sin\left(\pi y\right)cos\left(\pi z\right)-2\pi sin\left(\pi y\right)cos\left(\pi y\right){cos}^2\left(\pi z\right)\left[1+{cos}^2\left(\pi x\right)\right]\hfill \\ & & -2\pi sin\left(\pi y\right)cos\left(\pi y\right){cos}^2\left(\pi x\right){sin}^2\left(\pi z\right),\hfill \\ & & h(x,y,z)=-\frac{3{\pi }^2}{Re}cos\left(\pi x\right)cos\left(\pi y\right)sin\left(\pi z\right)-\pi sin\left(\pi z\right)cos\left(\pi z\right){cos}^2\left(\pi x\right)\left[1+{sin}^2\left(\pi y\right)\right]\hfill \\ & & +\pi sin\left(\pi z\right)cos\left(\pi z\right){sin}^2\left(\pi x\right){cos}^2\left(\pi y\right),\hfill \end{array}$$

(54)

using the exact solution $u(x,y,z)=sin\left(\pi x\right)cos\left(\pi y\right)cos\left(\pi z\right),$$v(x,y,z)=-2cos\left(\pi x\right)sin\left(\pi y\right)$$cos\left(\pi z\right)$ and $w(x,y,z)=cos\left(\pi x\right)cos\left(\pi y\right)sin\left(\pi z\right).$ The problem is solved subject to the relevant boundary conditions

$$\begin{array}{ccc}& & u(a,y,z)=sin\left(\pi a\right)cos\left(\pi y\right)cos\left(\pi z\right),u(b,y,z)=sin\left(\pi b\right)cos\left(\pi y\right)cos\left(\pi z\right),\hfill \\ & & u(x,c,z)=sin\left(\pi x\right)cos\left(\pi c\right)cos\left(\pi z\right),u(x,d,z)=sin\left(\pi x\right)cos\left(\pi d\right)cos\left(\pi z\right),\hfill \\ & & u(x,y,e)=sin\left(\pi x\right)cos\left(\pi y\right)cos\left(\pi e\right),u(x,y,f)=sin\left(\pi x\right)cos\left(\pi y\right)cos\left(\pi f\right),\hfill \\ & & v(a,y,z)=-2cos\left(\pi a\right)sin\left(\pi y\right)cos\left(\pi z\right),v(b,y,z)=-2cos\left(\pi b\right)sin\left(\pi y\right)cos\left(\pi z\right),\hfill \\ & & v(x,c,z)=-2cos\left(\pi x\right)sin\left(\pi c\right)cos\left(\pi z\right),v(x,d,z)=-2cos\left(\pi x\right)sin\left(\pi d\right)cos\left(\pi z\right),\hfill \\ & & v(x,y,e)=-2cos\left(\pi x\right)sin\left(\pi y\right)cos\left(\pi e\right),v(x,y,f)=-2cos\left(\pi x\right)sin\left(\pi y\right)cos\left(\pi f\right),\hfill \\ & & w(a,y,z)=cos\left(\pi a\right)cos\left(\pi y\right)sin\left(\pi z\right),w(b,y,z)=cos\left(\pi b\right)cos\left(\pi y\right)sin\left(\pi z\right),\hfill \\ & & w(x,c,z)=cos\left(\pi x\right)cos\left(\pi c\right)sin\left(\pi z\right),w(x,d,z)=cos\left(\pi x\right)cos\left(\pi d\right)sin\left(\pi z\right),\hfill \\ & & w(x,y,e)=cos\left(\pi x\right)cos\left(\pi y\right)sin\left(\pi e\right),w(x,y,f)=cos\left(\pi x\right)cos\left(\pi y\right)sin\left(\pi f\right).\hfill \end{array}$$

5. Results and Discussion

In this section, the TSQLM results are provided and debated by reporting the error estimation, condition numbers and computational time. In the numerical computation, we have considered $(x,y,z)\in {[0,1]}^3$ as a domain of interest unless it has been stated. To validate the efficiency and reliability of the aforesaid numerical approach, the approximate solutions are compared with the exact solutions. The accuracy of the method determined by computing the maximum absolute error over the entire grid points is defined by

$$L_{\infty }=\left|\right|u^n-u^e{\left|\right|}_{\infty }\simeq \underset{i,j,k}{max}\left\{\left|u^a(x_i,y_j,z_k)-u^e(x_i,y_j,z_k)\right|,:0\le i\le N_x,0\le j\le N_y,0\le k\le N_z\right\},$$

(55)

and calculating the values of absolute error at chosen grid points, which is given by the formulae

$$\begin{array}{ccc}& & E_{i,j,k}=|u^n(x_i,y_j,z_k)-u^e(x_i,y_j,z_k)|,\hfill \end{array}$$

(56)

where $u^n$ signifies the numerical solution and $u^e$ is the exact solution at the uniform grid points $(x_i,y_j,z_k).$ The order of convergence for the aforesaid numerical scheme is computed using the formulae

$$\begin{array}{ccc}& & \kappa =ln\left|\frac{u_{\iota +2}^n-u_{\iota +1}^n}{u_{\iota +1}^n-u_{\iota }^n}\right|.\hfill \end{array}$$

(57)

Another quantity that has some influence in the numerical solution of a linear system in the form $Au=b$ is the condition number relative to the norm $\left|\right|.\left|\right|.$ The condition number of a matrix, say $A,$ is defined as

$$\begin{array}{ccc}& & \kappa \left(A\right)=\left|\right|A\left|\right|\left|\right|A^{-1}\left|\right|.\hfill \end{array}$$

(58)

If $\kappa \left(A\right)$ is close to unity $\left(\kappa \right(A)\approx 1),$ then the matrix A is well conditioned, and if $\kappa \left(A\right)$ is significantly greater than 1 ($\kappa \left(A\right)\to \infty$), A becomes more ill-conditioned. Conditioning in this context refers to the relative security that a small residual vector implies a corresponding accurate numerical solution. Since all the computation work was conducted in MATLAB, the command $\mathrm{cond}\left(A\right)$ was used for $\kappa \left(A\right).$

5.1. Absolute Error Estimation, Computational Speed and Stability Analysis

Table 1. Absolute error values calculated at selected grid points in $x,y$ and z, CPU time and condition numbers for Example 1 over the interval ${[0,1]}^3$.

${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{y}}={\mathit{N}}_{\mathit{z}}$		5		10		15
$(\mathit{x},\mathit{y},\mathit{z})$	Exact Solution	TSQLM Solution	Error	TSQLM Solution	Error	TSQLM Solution	Error
(0.9045, 09045, 09045)	0.746688167118849	0.746705378717428	1.721 × 10${}^{-5}$	0.746688166650629	4.682 × 10${}^{-10}$	0.746688167118846	2.331 × 10${}^{-15}$
(0.6545, 0.6545, 0.6545)	0.093251152532562	0.093307715584223	5.656 × 10${}^{-5}$	0.093251152476087	5.648 × 10${}^{-11}$	0.093251152532562	2.914 × 10${}^{-16}$
(0.3455, 0.3455, 0.3455)	0.025983543105084	0.025999430835633	1.589 × 10${}^{-5}$	0.025983543095338	9.746 × 10${}^{-12}$	0.025983543105084	1.006 × 10${}^{-16}$
(0.0955, 0.0955, 0.0955)	0.008322301075952	0.008323357188132	1.056 × 10${}^{-6}$	0.008322301075219	7.333 × 10${}^{-13}$	0.008322301075953	5.725 × 10${}^{-17}$
Cond (A)		351.1623		659.2167		1.6140 × 10${}^3$
CPU Time (s)		0.008614		0.117700		3.434236

,

Table 2. Absolute error values calculated at selected grid points in $x,y$ and z, CPU time and condition numbers for Example 2 over the interval ${[0,1.2]}^3$.

${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{y}}={\mathit{N}}_{\mathit{z}}$		4		8		12
$(\mathit{x},\mathit{y},\mathit{z})$	Exact Solution	TSQLM Solution	Error	TSQLM Solution	Error	TSQLM Solution	Error
(1.0243, 1.0243, 1.0243)	2.781001311184647	2.780055371655581	9.459 × 10${}^{-4}$	2.781001318489854	7.305 × 10${}^{-9}$	2.781001311183933	7.141 × 10${}^{-13}$
(0.6000, 0.6000, 0.6000)	1.192592237740634	1.193313943450853	7.217 × 10${}^{-4}$	1.192592242229544	4.489 × 10${}^{-9}$	1.192592237740563	7.150 × 10${}^{-14}$
(0.1757, 0.1757, 0.1757)	0.088556949386077	0.088653933543552	9.698 × 10${}^{-5}$	0.088556949359024	2.705 × 10${}^{-11}$	0.088556949386076	4.996 × 10${}^{-16}$
Cond (A)		7.4544		81.8051		389.9620
CPU Time (s)		0.002897		0.028628		0.773108

,

Table 3. Absolute error values calculated at selected grid points in $x,y$ and z, CPU time and condition numbers for Example 3 over the interval ${[0,1.6]}^3$ when $\alpha =\beta =1$.

${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{y}}={\mathit{N}}_{\mathit{z}}$		4		8		12
$(\mathit{x},\mathit{y},\mathit{z})$	Exact Solution	TSQLM Solution	Error	TSQLM Solution	Error	TSQLM Solution	Error
$\mathbf{Absolute}\mathbf{error}\left(u\right)$
(1.3657, 1.3657, 1.3657)	0.162550474907984	0.160548370516225	2.002 × 10${}^{-3}$	0.162550451027631	2.388 × 10${}^{-8}$	0.162550474907977	6.939 × 10${}^{-15}$
(0.8000, 0.8000, 0.8000)	1.080278098259261	1.073054459894421	7.224 × 10${}^{-3}$	1.080278098384820	1.256 × 10${}^{-10}$	1.080278098259262	1.776 × 10${}^{-15}$
(0.2343, 0.2343, 0.2343)	1.195902797070561	1.197858394193033	1.956 × 10${}^{-3}$	1.195902797008142	6.242 × 10${}^{-11}$	1.195902797070568	6.883 × 10${}^{-15}$
$\mathbf{Absolute}\mathbf{error}\left(v\right)$
(1.3657, 1.3657, 1.3657)	3.755857431386828	3.754795696039452	1.062 $\times {10}^{-3}$	3.755857412741173	1.865 × 10${}^{-8}$	3.755857431386825	3.553 × 10${}^{-15}$
(0.8000, 0.8000, 0.8000)	1.145262830233207	1.138448902139527	6.814 × 10${}^{-3}$	1.145262830131156	1.021 × 10${}^{-10}$	1.145262830233213	5.551 × 10${}^{-15}$
(0.2343, 0.2343, 0.2343)	0.068139268697581	0.069119178461196	9.799 × 10${}^{-4}$	0.068139268673337	2.424 × 10${}^{-11}$	0.068139268697582	3.608 × 10${}^{-16}$
Cond (C)		11.4521		115.2798		548.4108
CPU Time (s)		0.011463		0.149994		3.780974

and

Table 4. Absolute error values calculated at selected grid points in $x,y$ and z, CPU time and condition numbers for Example 4 over the interval ${[0,1.1]}^3$ when $Re=1$.

${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{y}}={\mathit{N}}_{\mathit{z}}$		4		8		12
$(\mathit{x},\mathit{y},\mathit{z})$	Exact Solution	TSQLM Solution	Error	TSQLM Solution	Error	TSQLM Solution	Error
$\mathbf{Absolute}\mathbf{error}\left(u\right)$
(0.9389, 0.9389, 0.9389)	0.183807501770063	0.182138370703660	1.669 × 10${}^{-3}$	0.183807590422178	8.865 × 10${}^{-8}$	0.183807501767250	2.813 × 10${}^{-12}$
(0.5500, 0.5500, 0.5500)	0.024170454101693	0.024246294103530	7.584 × 10${}^{-5}$	0.024170458865723	4.764 × 10${}^{-9}$	0.024170454101784	9.178 × 10${}^{-14}$
(0.1611, 0.1611, 0.1611)	0.370843708401198	0.367758740938404	3.085 × 10${}^{-3}$	0.370843887767984	1.794 × 10${}^{-7}$	0.370843708393093	8.105 × 10${}^{-12}$
$\mathbf{Absolute}\mathbf{error}\left(v\right)$
(0.9389, 0.9389, 0.9389)	−0.367615003540126	−0.364673539152125	2.941 × 10${}^{-3}$	−0.367615175615953	1.721 × 10${}^{-7}$	−0.367615003535049	5.077 × ${10}^{-12}$
(0.5500, 0.5500, 0.5500)	−0.048340908203385	−0.049463116182947	1.122 × 10${}^{-3}$	−0.048340967794351	5.959 × ${10}^{-8}$	−0.048340908204152	7.671 × 10${}^{-13}$
(0.1611, 0.1611, 0.1611)	−0.741687416802395	−0.735526034375114	6.161 × 10${}^{-3}$	−0.741687681901548	2.651 × 10${}^{-7}$	−0.741687416786753	1.564 × 10${}^{-11}$
$\mathbf{Absolute}\mathbf{error}\left(w\right)$
(0.9389, 0.9389, 0.9389)	0.183807501770063	0.182138370703660	1.669 × 10${}^{-3}$	0.183807590422178	8.865 × 10${}^{-8}$	0.183807501767250	2.813 × 10${}^{-12}$
(0.5500, 0.5500, 0.5500)	0.024170454101693	0.024246294103530	7.584 × 10${}^{-5}$	0.024170458865723	4.764 × ${10}^{-9}$	0.024170454101784	9.184 × 10${}^{-14}$
(0.1611, 0.1611, 0.1611)	0.370843708401197	0.367758740938404	3.085 × 10${}^{-3}$	0.370843887767984	1.794 × 10${}^{-7}$	0.370843708393092	8.105 × 10${}^{-12}$
Cond (C)		8.3394		90.4278		431.2593
CPU Time (s)		0.007821		0.182926		4.158837

display the absolute error values, and exact and approximate solutions for the numerical examples recorded at selected grid points for various numbers of collocation points. It is evident from the tables that the absolute error values decrease with an increase in the number of collocation points. This confirms the merit of any spectral collocation-based method in terms of accuracy improvement as the number of collocation points proliferates. Such accuracy improvement is achieved via small grid sizes. Table 1 depicts that the absolute error values drop drastically from a level of about ${10}^{-6}$ to a level of about ${10}^{-17}$ with the use of grid sizes $N_x\times N_y\times N_z=5\times 5\times 5$ and $N_x\times N_y\times N_z=15\times 15\times 15,$ respectively. Table 2 and Table 3 indicate that considering mesh sizes of $N_x\times N_y\times N_z=4\times 4\times 4$ and $N_x\times N_y\times N_z=12\times 12\times 12$ produces minimum absolute error values of up to approximately ${10}^{-5}$ and ${10}^{-16},$ respectively. On the other hand, in Table 4, the absolute errors reach a value of approximately ${10}^{-5}$ and ${10}^{-14}$ for $N_x=N_y=N_z=4$ and $N_x=N_y=N_z=12$ collocation points, respectively. The grid sizes $N_x\times N_y\times N_z=12\times 12\times 12$ and $N_x\times N_y\times N_z=15\times 15\times 15$ are substantially lower for maximum accuracy to be achieved on course grids. From the tables, we also note that when using these few grid points, the convergence of the numerical scheme to the exact solution is guaranteed in less computational time. For example, in Table 2, supreme accuracy is achieved in less than a second.

The attained good accuracy is also linked to the small condition numbers of the coefficient matrices illustrated at the bottom of each table. These small condition numbers close to unity signify the well-conditioned nature of the resulting coefficient matrices A and C. The well-conditioned nature of a coefficient matrix implies stability in the system of linear algebraic equations being solved. Small condition numbers are attributed to the use of differentiation matrices, which are made smaller in size by excluding entries corresponding to the given boundary conditions. Consequently, the resulting coefficient matrix becomes less dense and easier to invert. Carrying out numerical calculations with smaller-sized and well-conditioned matrices requires a small number of grid points and less computer memory for storage. The use of less number of grid points for discretization minimizes the impact of round-off errors emanating from the functions being approximated with interpolating polynomials having the highest order. Hence, outstandingly accurate and stable results are guaranteed with a limited number of collocation nodes and cheap computational cost.

Another curious observation from the tables is that the absolute error values for those grid points selected quite far from the left boundary are larger than those errors for grid points selected closer to the left boundary. This suggests some loss of accuracy as you move away from the left boundary. This observation is explained by the error bound theorem given in Equation (37). From Equation (37), we note that when $b,d$ and f become large, and $a,c$ and e are fixed, the error increases. Nevertheless, we can still declare that the proposed scheme provides good accuracy as long as we consider a small length of the interval. This declaration is supported by the small difference in errors over the domain $(x,y,z)\in {[0,1]}^3.$

5.2. Rate of Convergence at Various Iteration Levels

Table 5. Computed values for rate of convergence at various iteration levels for examples 2, 3 and 4.

Iterations	Example 2	Example 3		Example 4
	$\mathbf{\kappa }\left(\mathit{u}\right)$	$\mathbf{\kappa }\left(\mathit{u}\right)$	$\mathbf{\kappa }\left(\mathit{v}\right)$	$\mathbf{\kappa }\left(\mathit{u}\right)$	$\mathbf{\kappa }\left(\mathit{v}\right)$	$\mathbf{\kappa }\left(\mathit{w}\right)$
1	-	-	-	-	-	-
2	-	-	-	-	-	-
3	1.085	1.843	1.843	1.714	1.714	1.714
4	1.937	1.961	1.961	1.832	1.832	1.832
5	1.988	1.982	1.982	1.894	1.894	1.894
6	2.014	1.993	1.993	1.956	1.956	1.956
7	1.998	2.001	2.001	1.997	1.997	1.997
8	2.001	2.000	2.000	2.002	2.002	2.002
9	2.000	2.001	2.001	2.000	2.000	2.000
10	2.0002	2.000	2.000	2.000	2.000	2.000

elucidates the numerical rates of convergence after each iteration for examples 2, 3 and 4. The order of convergence is evident to be quadratic since it is close to 2. This is to be anticipated owing to the fact that the convergence of the method is rooted on the underlying Newton-based linearization method, which is well-known to be of order 2.

5.3. Matching Exact Solutions against Approximate Solutions

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 display solutions computed using TSQLM and the corresponding exact solutions for the elliptic PDEs in both three-dimensional (3D) and two-dimensional (2D) forms. All figures demonstrate that the exact solutions and numerical solutions are well consistent in both 2D and 3D forms. This implies that the TSQLM scheme converges to the exact solutions of the elliptic PDEs. It can, therefore, be inferred that the accuracy and computational efficaciousness of the numerical scheme are validated.

5.4. Error Norms at Different Values of z

Figure 7 and Figure 8 are plotted to illustrate what happens to accuracy as the values of z increase within the domain $(x,y,z)\in {[0,1]}^3.$ Most numerical methods available in the literature lose accuracy when a variable becomes large within the domain of interest. However, Figure 7 and Figure 8 demonstrate uniformity in accuracy as the values of z become large in small domains. This is evident in the maximum error norms being close to each other in the considered domain.

5.5. Convergence of the Numerical Approach

Figure 9 illustrates the number of iterations required for the numerical scheme to converge to accurate solutions. As expected, the graphs demonstrate a monotonic reduction in error norms as the number of iterations upsurge. Further, the numerical scheme is noted to fully converge below eight iterations, with the average solution errors being approximately ${10}^{-15}.$ This means that the TSQLM approach requires a limited number of iterations to achieve maximum accuracy.

5.6. Comparison of Numerical Results with Literature Results

Table 6. Comparison of error norms for a different number of collocation points for example 2.

${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{y}}={\mathit{N}}_{\mathit{z}}$	No of Iterations	Aziz et al. [21]	TSQLM
4	6	9.2203 × 10${}^{-4}$	2.731 × 10${}^{-4}$
8	7	2.5331 × 10${}^{-4}$	6.520 × 10${}^{-9}$
16	8	6.4487 × 10${}^{-5}$	5.214 × 10${}^{-12}$

,

Table 7. Comparison of error norms for different number of collocation points for example 3.

	Aziz et al. [21]		TSQLM
${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{y}}={\mathit{N}}_{\mathit{z}}$	${\mathit{L}}_{\infty }\left(\mathit{u}\right)$	${\mathit{L}}_{\infty }\left(\mathit{v}\right)$	${\mathit{L}}_{\infty }\left(\mathit{u}\right)$	${\mathit{L}}_{\infty }\left(\mathit{v}\right)$
$\alpha =1,$$\beta =1$
4	$——$	$——$	2.496 × 10${}^{-5}$	2.313 × 10${}^{-5}$
8	$——$	$——$	1.982 × 10${}^{-11}$	6.459 × 10${}^{-11}$
16	$——$	$——$	2.578 × 10${}^{-15}$	6.997 $\times {10}^{-16}$
$\alpha =5,$$\beta =10$
4	6.5799 × 10${}^{-4}$	3.6681 × 10${}^{-4}$	1.351 × 10${}^{-5}$	7.974 × ${10}^{-5}$
8	1.7957 × 10${}^{-4}$	9.9408 × ${10}^{-5}$	1.769 × 10${}^{-10}$	5.794 × 10${}^{-11}$
16	4.5196 × 10${}^{-5}$	2.4388 × 10${}^{-5}$	1.711 × 10${}^{-15}$	4.789 × 10${}^{-16}$

and

Table 8. Comparison of error norms for varying $Re$ and number of collocation points for example 4.

			Aziz et al. [21]		TSQLM
$\mathit{Re}$	${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{y}}={\mathit{N}}_{\mathit{z}}$	${\mathit{L}}_{\infty }\left(\mathit{u}\right)$	${\mathit{L}}_{\infty }\left(\mathit{v}\right)$	${\mathit{L}}_{\infty }\left(\mathit{w}\right)$	${\mathit{L}}_{\infty }\left(\mathit{u}\right)$	${\mathit{L}}_{\infty }\left(\mathit{v}\right)$	${\mathit{L}}_{\infty }\left(\mathit{w}\right)$
1	4	2.8317 × 10${}^{-3}$	6.0643 × 10${}^{-3}$	2.8317 × 10${}^{-3}$	4.091 × 10${}^{-3}$	6.598 × 10${}^{-3}$	4.091 × 10${}^{-3}$
1	8	1.4786 × 10${}^{-3}$	3.0308 × 10${}^{-3}$	1.4786 × 10${}^{-3}$	1.414 × 10${}^{-6}$	6.770 × 10${}^{-7}$	1.524 × 10${}^{-7}$
1	16	3.7790 × 10${}^{-4}$	7.8002 × 10${}^{-4}$	3.7787 × 10${}^{-4}$	9.219 × 10${}^{-15}$	3.237 × ${10}^{-15}$	8.146 × 10${}^{-16}$
2	4	$——$	$——$	$——$	4.568 × 10${}^{-3}$	6.561 × 10${}^{-3}$	4.568 × 10${}^{-3}$
2	8	$——$	$——$	$——$	1.450 × 10${}^{-6}$	7.057 × 10${}^{-7}$	1.707 × 10${}^{-7}$
2	16	$——$	$——$	$——$	1.094 × 10${}^{-14}$	3.261 × 10${}^{-15}$	1.205× 10${}^{-15}$
6	4	$——$	$——$	$——$	5.986 × 10${}^{-3}$	6.206 × 10${}^{-3}$	5.986 × 10${}^{-3}$
6	8	$——$	$——$	$——$	1.584 × 10${}^{-6}$	9.868 × 10${}^{-7}$	2.965 × 10${}^{-7}$
6	16	$——$	$——$	$——$	1.237 × 10${}^{-14}$	3.679 × 10${}^{-15}$	1.093 × 10${}^{-15}$
8	4	$——$	$——$	$——$	6.389 × 10${}^{-3}$	5.945 × 10${}^{-3}$	6.389 × 10${}^{-3}$
8	8	$——$	$——$	$——$	1.770 × 10${}^{-6}$	1.247 × 10${}^{-6}$	3.429 × 10${}^{-7}$
8	16	$——$	$——$	$——$	1.625 × 10${}^{-13}$	6.781 × 10${}^{-14}$	1.680 × 10${}^{-14}$
10	4	4.5338 × 10${}^{-3}$	1.1035 × 10${}^{-2}$	4.5337 × 10${}^{-3}$	6.603 × 10${}^{-3}$	5.673 × 10${}^{-3}$	6.603 × 10${}^{-3}$
10	8	1.5631 × 10${}^{-3}$	5.1186 × 10${}^{-3}$	1.5631 × 10${}^{-3}$	1.967 × 10${}^{-6}$	1.494 × 10${}^{-6}$	4.015 × 10${}^{-7}$
10	16	4.2250 × 10${}^{-4}$	1.3313 × 10${}^{-3}$	4.2251 × 10${}^{-4}$	8.375 × 10${}^{-12}$	3.599 × 10${}^{-12}$	8.966 × 10${}^{-13}$

are given to show a comparison of the numerical results secured using the TSQLM approach with those obtained by Aziz et al. [21] using the Haar wavelet collocation method. We also study the outcome of increasing the parameter values on the accuracy of the TSQLM scheme. In all tables, both results of Aziz et al. [21] and the TSQLM approach show similar trends in error norms as the number of collocation increases. As discussed earlier, such an observation declares a boost in accuracy by augmenting the number of collocation points. The error norms are noted to be extensively smaller in the TSQLM approach for eight collocation points or more. This means that the TSQLM approach is remarkably more accurate than the Haar wavelet collocation method employed by Aziz et al. [21] to solve the nonlinear elliptic PDEs. For example, when $Re=1$ and $N_x=N_y=N_z=16,$ Aziz et al. [21] reported maximum absolute errors of about ${10}^{-4},$ whereas the TSQLM approach gives maximum absolute errors close to ${10}^{-16}.$ The supremacy of the TSQLM in terms of accuracy can be accredited to the well-conditioned nature of the coefficient matrices and the space variables entirely discretized using spectral collocation and CGL points. When a matrix is well conditioned, then its inverse can be computed with good accuracy. It should be noted that this good accuracy is achieved even when the number of iterations is fixed in both methods, as illustrated in Table 6. With regard to the outcome of increasing parameter values in Table 7 and Table 8, we note that there is no significant change in accuracy as $\alpha$ and $\beta$ are increased in Example 2. This is because the error norms are almost the same even if the parameters are increased. On the other hand, using many grid points along with large values of Reynolds number result in a slight loss in the accuracy of the numerical scheme. This is evident in the difference between error norms obtained using $Re=1$ and $Re=10$ being of order three with a fixed number of collocation points ($N_x=N_y=N_z=16$). We note that making $Re$ very large introduces a very small coefficient of the higher-order differentiation matrices, which eventually makes the entries of the coefficient matrix smaller. Calculations with these small entries and many grid points can give rise to round-off errors that result in a loss of accuracy. Nevertheless, since the error norms for $Re=10$ are small enough to guarantee a reasonable accuracy for practical applications, we can conclude that TSQLM can be preferred for problems with small and large parameter values.

6. Conclusions

In this paper, we examined the adaptability of the novel trivariate spectral collocation method in solving various classes of elliptic PDEs. The reported results indicate that the numerical approach is computationally cost-effective, more accurate and stable when solving three-dimensional elliptic PDEs. With regard to this accuracy and stability, the supremacy of the numerical scheme can be accredited to all spatial variables being discretized using the Chebyshev-spectral collocation approach together with smaller condition numbers of the coefficient matrices. The numerical algorithm is relatively easy to implement and can succeed in overcoming the problem of dealing with singularities at boundaries for elliptic PDEs. The singularity issue is resolved by executing the discretization scheme of the presented collocation-based method only in the interior nodes of the discretized domain. Although the TSQLM possesses many merits, the method has some demerits. The method is sufficiently accurate and efficient in solving PDEs over smaller computational domains. To overcome this limitation, future work will include modifying the TSQLM using domain-decomposition techniques to accommodate problems with stretched domains. Other future works can include extending the method for adaptability in multi-dimensional parabolic and hyperbolic PDEs and time-fractional problems such as Burgers’ equations. The TSQLM approach adds an effective numerical scheme for handling PDEs to the literature and can also be used as a benchmarking tool for existing or newly developed methods.