Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11, 710-725 2025 Publisher: Learning Gate DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate © 2025 by the authors; licensee Learning Gate History: Received: 12 September 2025; Revised: 17 October 2025; Accepted: 22 October 2025; Published: 11 November 2025 * Correspondence: claire_motiun_ds23@iluv.ums.edu.my A bicubic B-spline approximation and application of C-BSEG methods for the solution of 2D elliptic partial differential equations Claire NC Motiun1*, Jumat Sulaiman2, Aini Janteng3, Asep K. Supriatna4 1,2,3Faculty of Science and Technology, Universiti Malaysia Sabah, Malaysia; claire_motiun_ds23@iluv.ums.edu.my (C.N.M.) jumat@ums.edu.my (J.S.) aini_jg@ums.edu.my (A.J.). 4Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang 45363, West Java, Indonesia; ak_supriatna@unpad.ac.id (A.K.S.). Abstract: This paper presents a two-level implicit bicubic B-spline discretization scheme with the collocation approach, particularly known as a bicubic B-spline collocation approach for the solution of two-dimensional elliptic partial differential equations. Then, a system of bicubic B-spline collocation approximation equations generated from the discretization process of the proposed scheme with the collocation approach is normally large-scale, with a sparse matrix. To solve this linear system, a new Bicubic B-spline Explicit Group (C-BSEG) iteration approach has been shown to enhance its convergence rate in solving any linear system. In addition, the capability of the C-BSEG iteration family, such as 2 Point-C-BSEG and 4 Point-C-BSEG methods, has been investigated in solving two- dimensional elliptic partial differential equations. Moreover, the formulation and implementation of both block iterative methods are also presented and used to solve the linear system iteratively. Numerical experiments demonstrate that the 4 Point-C-BSEG iteration combined with the bicubic B-spline collocation approach achieves superior performance compared with existing point and block iterative schemes. Hence, the proposed bicubic B-spline collocation framework provides a reliable and efficient numerical tool with a wide range of applications in fields such as physics, engineering, and applied mathematics. Keywords: Bicubic B-spline block iteration, B-spline collocation, Collocation approach, Two-dimensional Poisson equations. 1. Introduction Numerical methods for solving elliptic partial differential equations (PDEs) have been widely studied by researchers to obtain approximate solutions, which play a fundamental role in addressing various scientific and engineering challenges. For instance, one example of PDEs is the Poisson equation, which is referred to as the generalization of the well-known Laplace's equation. The aforementioned differential equation is commonly employed in theoretical physics and is one example of elliptic PDEs. Based on these equations, there are several mathematical models that can be used to govern numerous scientific and engineering fields, including astronomy, fluid mechanics, electromagnetics, heat transfer, electrostatics, and many more. Aligned with the concept of numerical methods, numerical techniques have been developed to solve two-dimensional (2D) elliptic PDEs. In the late 1960s, the spline interpolation approach was first used to solve differential equations by Bickley [1]. In his work, cubic splines are utilized experimentally to approximate the solution of a basic two-point boundary value problem for a linear ordinary differential equation. Fyfe [2] examined and described the cubic spline method suggested by Bickley and the error predictions of Curtis and Powell [3]. Fyfe [2] concluded that, because the spline can obtain approximate solutions at any point in an interval, the spline method is better than the usual finite difference method (FDM). Due to the effectiveness of this method, numerous authors have been https://orcid.org/0000-0002-9538-6588 https://orcid.org/0000-0003-0129-6344 711 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate interested in solving differential equations using spline approaches. To mention a few, Dağ et al. [4] solved the one-dimensional (1D) Burgers' equation using the cubic B-spline collocation method (CuBsCM) over finite elements in 2005, where Burgers' equation was introduced by Akour et al. [5]. The accuracy of the proposed method was demonstrated through three test problems. According to these problems, CuBsCM is capable of solving Burgers' equation accurately, as evidenced by the comparison of the calculations with the analytical solution. In the study of Demir and Bildik [6], the numerical solution of the heat problem using cubic B-spline (CuBs) was discussed. In this study, the method used to solve the 1D heat problem and the solution were compared with the exact solution. Fourier stability method is considered as an analyzer for the stability of this method, and the result is stable for       = 1, 2 1  . In 2021, Ware and Ashine [7] solved a boundary value problem of an ordinary differential equation by using CuBs and FDM. The resulting system of equations has been solved by a tri-diagonal solver. There are two examples examined, and the results are compared with FDM. The results show that the CuBs method is more efficient and feasible. Beyond that, recent advancements have pushed spline-based methods even further. Du and Sun [8] developed a bicubic B- spline finite element method for solving fourth-order semilinear parabolic optimal control problems in 2D. Their numerical results, which are based on two benchmark problems, demonstrate how effective and practical the suggested strategy is in comparison to classical elements. Likewise, Salama et al. [9] proposed hybrid group iterative methods for solving 2D time-fractional cable equations. From the numerical experiments in this study, these 4-point schemes were proven to be unconditionally stable and outperformed traditional iterative solvers in terms of speed and memory usage. In 2022, Salama et al. [10] further refined their approach by introducing a modified hybrid explicit group (MHEG) method, which improved convergence for 2D diffusion problems. In their study, two benchmark problems were examined to evaluate the proposed method. The results showed that the MHEG iterative method achieved significantly faster simulations without substantially compromising accuracy compared to the hybrid standard point (HSP) iterative method [11]. Besides numerical techniques, the discovery of the iterative method for PDEs modeled issues began in the early 20th century [12]. Through the discretization process of numerical techniques for 2D elliptic PDEs, it can be observed that the coefficient matrix of the resulting linear system was large- scale and sparse. Based on the previous studies by Young [13], Hackbusch [14], and Saad [15], the findings of their studies stated that the iterative methods are the best linear solvers for solving any linear system. Recently, the explicit group (EG) iterative methods, which were introduced by Evans and Abdullah [16] that have been used successfully to solve numerical problems involving parabolic and hyperbolic PDEs [17]. A small group of 2, 4, 9, 16, and 25 points was generated in the iterative processes for solving Laplace's equation using the EG iterative approach [18]. The numerical findings demonstrate that the EG method takes less storage and is easier to implement than the block (line) iterative approaches. This approach, however, was developed only utilizing the conventional standard finite difference discretization, which limits the solutions to certain locations within the solution domain. However, the study by Mohanty [19] already introduced and applied a two-level implicit cubic spline approach together with the cubic spline alternating group explicit (C-SPLAGE) iterative method for solving 1D nonlinear parabolic PDEs. He concluded that the proposed iterative method is superior to the successive over-relaxation (SOR) method. From the development of the spline function and the use of EG and C-SPLAGE iterative methods, as mentioned in previous paragraphs, and dealing with numerical solutions of 2D elliptic PDEs, the application of the concept of the bicubic B-spline collocation approach and a family of block iteration methods has attracted attention for obtaining numerical solutions. Based on Mohanty's [19] findings, the author focused on constructing the spline approximation equation and the AGE method for solving 1D problems. There have been limited studies concerned with a bicubic collocation approach being applied to solve 2D elliptic PDEs via the block iteration family. Therefore, this research intends to 712 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate construct the derivation of the bicubic B-spline collocation approximation equation, in which this approximate equation leads us to construct a system of bicubic B-spline collocation approximation equations. Therefore, the objectives of this research are to investigate the applicability of the C-BSEG iteration family, such as 2 Point-C-BSEG and 4 Point-C-BSEG methods, based on the established bicubic B-spline collocation approximation equation for solving 2D elliptic PDEs. 2. Formulation of Bicubic B-spline Collocation Approximation Equations Before starting to perform the bicubic B-spline collocation discretization process and implementing the proposed iteration family, let us consider the general form of the aforementioned PDEs as Ali et al. [20]. , 2   −= u (1) where 2 2 2 2 2 yx   +   = is the Laplace operator, u is the potential difference,  is the volume charge density, and represents the permittivity of the medium, respectively. Problem (1) can be extended specifically to the Poisson equation over the region, ],[],[ baba = as follows with Dirichlet boundary conditions, ),(),( 0 xfyaU = ),(),( 1 yfybU = ,bya  ),(),( 2 xfaxU = ),(),( 3 xfbxU = .bxa  Now, let us start the process of discretization over the proposed Problem (2) in constructing a bicubic B-spline collocation approximation equation for generating a system of linear equations. To do this, the solution domain ],[],[ baba = needs to be partitioned for x and y directions, and their sub-interval distance for both directions is given as . m ab h − = All node points of Problem (1) are denoted as ),( ji yx with ihaxi += and ,jhay j += mji  ,0 as indicated in Figure 1. Prior to having the 2D mesh network, let B-spline basis functions of order d and degree 1−d denoted as )(, xB di [21] and defined as    + = .,0 ],,[,1 )( 1 ,0 otherwise xxx xB ii d (3) ).()()( 1,1 1 1,, xB xx xx xB xx xx xB di idi di di idi i di −+ ++ + − +             − − + − − = (4) The cubic B-spline is defined as a piecewise function [22], ),,( yxfUU yyxx =+ ],,[, bayx  (2) 713 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate ( ) ( )   ( ) ( ) ( )   ( ) ( ) ( )   ( )   3 1 2 33 2 1 1 1 2 ,4 3 2 33 2 3 3 3 2 3 3 4 3 4 , , 3 3 3 , ,1 6 3 3 3 , , , , i i i i i i i i i i i i i i i i i x x x x x h h x x h x x x x x x x B x h h h x x h x x x x x x x x x x x x + + + + + + + + + + + + +  −    + − + − − −  =   + − + − − −    −  (5) Figure 1. Distribution of uniform node points over the solution domain for m = 8. Clearly, it can be stated that the cubic B-spline basis, )(4, xBi Function (5), can be known as a piecewise polynomial of degree 3. The bicubic B-spline approximation function, derived from the basis function in Equation (5), defines a surface known as the bicubic B-spline surface because it is constructed using two cubic B-spline bases and could be defined as,  = − −= − −= 1 3 1 3 4,4,, ),()(),( m i m j jijiB yBxBCyxS ],,[ 0 mxxx ].,[ 0 myyy (6) where 1,3,, −− mjiC ji are unknown coefficients are to be determined. By applying the simplification of the bicubic B-spline approximation function, ),( yxSB at any arbitrary node point, 714 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate ( ) 3, 1 2, 1 1, 1 3, 2 2, 2 1, 2 3, 3 2, 3 1, 3 4 1 , 4 16 4 36 4 i j i j i j i j i j i j i jB i j i j i j C C C S x y C C C C C C − − − − − − − − − − − − − − − − − −  + +   = + + +     + + +   (7) Then, the first and second derivatives of the bicubic B-spline approximation function, ),( yxSB with respect to x could be simplified at any point as ,44 12 1 ),( 3,13,3 2,12,3 1,11,3           −−−− −−−− −−−− +− +− +− =   jiji jiji jiji jiB CC CC CC h yxS x (8) . 2 484 2 6 1 ),( 3,13,23,3 2,12,22,3 1,11,21,3 22 2           −−−−−− −−−−−− −−−−−− +−+ +−+ +− =   jijiji jijiji jijiji jiB CCC CCC CCC h yxS x (9) Similarly to derive Equations (8) and (9), the first and second derivatives of the bicubic B-spline approximation function, ),( yxSB with respect to y could be simplified at any point as , 4 4 12 1 ),( 3,13,23,3 1,11,21,3       −−−−−− −−−−−− −−− ++ =   jijiji jijiji jiB CCC CCC h yxS y (10) . 4 282 4 6 1 ),( 3,13,23,3 2,12,22,3 1,11,21,3 22 2           −−−−−− −−−−−− −−−−−− +++ −−− ++ =   jijiji jijiji jijiji jiB CCC CCC CCC h yxS y (11) Before getting the approximate solution ),( yxU over Problem (2), the values of 1,3,, −− mjiC ji on the boundary conditions of the domain solution in Problem (2) can be determined by using the approximate solution (6) in the boundary conditions (2) [23]. Based on the bottom boundary condition, )(),( 2 xfaxU = gives ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3, 3 2 0 2 0 2, 3 2 1 1, 3 2 2 2 12, 3 2 21, 3 6 24 2 61 4 1 61 4 1 61 4 1 6 22 4 mm m mm C f x hf x C f x C f x f xC f x hf xC − − − − − − −− − − −    +                   =                     −         (12) The top boundary condition, )(),( 3 xfbxU = yields 715 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3, 1 3 0 3 0 2, 1 3 1 1, 1 3 2 3 12, 1 3 31, 1 6 24 2 61 4 1 61 4 1 61 4 1 6 22 4 m m m mm m m mm m C f x hf x C f x C f x f xC f x hf xC − − − − − − −− − − −    +                   =                     −         (13) The left boundary condition, )(),( 0 yfyaU = leads to ( ) ( ) ( ) ( ) ( ) 3, 2 1 0 3, 3 3, 1 0 1 3,0 0 2 0 13, 1 0 3, 13, 2 64 1 1 4 1 6 1 4 1 6 1 4 1 6 1 4 6 mm m mm C f y C C f y C f y f yC f y CC − − − − − − − −− − − −− −    −                    =                    −       (14) The right boundary condition, )(),( 1 yfybU = leads to ( ) ( ) ( ) ( ) ( ) 1, 2 1 0 1, 3 1, 1 1 1 1,0 1 2 1 11, 1 1 1, 11, 2 64 1 1 4 1 6 1 4 1 6 1 4 1 6 1 4 6 m m m m mm m m m mm m C f y C C f y C f y f yC f y CC − − − − − − − −− − − −− −    −                    =                    −       (15) The tridiagonal system of Equations (12) - (15) was solved by forward and backward substitution. It means that the LU decomposition approach has been performed over the tridiagonal linear systems (12) – (15) to obtain the value of jiC , on the boundary conditions. Next, we attempt to calculate the approximate values of jiC , at all interior points over the solution domain  . To do that, the discretization process over Problem (2) needs to be conducted. Firstly, let jiU , and jif , represent ),( ji yxU and ),( ji yxf respectively. Then Problem (2) becomes ),(),(),( 2 2 2 2 jijiBjiB yxfyxS y yxS x =   +   (16) Then, substitute Equations (9) and (11) into Equation (16). Problem (2) can be approximated and simplified as a bicubic B-spline collocation approximation equation, which is given by 3, 3 2, 3 1, 3 3, 2 2, 2 1, 2 3, 1 2, 1 1, 1 2 , 8 3 i j i j i j i j i j i j i j i j i j i j C C C C C C C C C h f − − − − − − − − − − − − − − − − − −+ + + − + + + + = (17) 716 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate for , 1,2,3, , 1i j m= − . By imposing the bicubic B-spline collocation approximation Equation (17) over all interior node points, ( ), , , 1,2,3, , 1i jx y i j m= − , a large-scale and sparse linear system generated from Equation (17) can be rewritten in a matrix form as .fAC = (18) 3. Formulation of Bicubic B-spline Explicit Group Iteration Family As stated in the linear system (18), the main characteristics of the coefficient matrix, A (18), are that it is large-scale and sparse. In this paper, 2 Point-C-BSEG and 4 Point-C-BSEG iterative methods will be applied to solve linear systems (18) generated from the approximation Equation (16) through the discretization of Problem (2). As mentioned in the second section, this paper aims to demonstrate the effectiveness of the family of iterative methods, specifically the 2-Point and 4-Point C-BSEG methods, for solving Problem (2). These methods are based on bicubic B-spline interpolation in conjunction with the collocation methodology. To establish the formulation of the proposed method, first, let us decompose the coefficient matrix, A, in Equation (18) as A D L V= + + (19) where D is a diagonal matrix, L is a lower triangular and V is the upper triangular parts of A respectively. From the linear system (18), the general scheme for full sweep Gauss-Seidel iterative methods can be written as, ). )( ( 1 )( )1( k VCfLD k C − − += +  (20) 3.1. Formulation of 2-Point Explicit Group Iterative Methods The formulation and implementation of the 2-Point EG method to solve the 2D Poisson equation will be presented in this section. By referring to algebraic Equation (17), consider any point of the two points, jiC , and jiC ,1+ that are used simultaneously to compute the value of )1( +k C . Thus, at a point jiC , , the solution is approximated by ,,1,11,1,1,1,,11,11,1,1 8 jijijijijijijijijiji fCCCCCCCCC =++++−+++ ++++−+−−+−−− (21) where at point 1, +jiC , the solution is given by .1,1,11,1,1,1,,11,11,1,1 8 +++++−+−−+−−− =++++−+++ jijijijijijijijijiji fCCCCCCCCC (22) Now, the Equations (21) and (22) can be written simultaneously in the matrix form as follows,       =               − − + 2 1 1, , 81 18 S S C C ji ji (23) where, 717 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate ,3 , 2 ,1,11,11,1,11,11,11 jijijijijijijiji fhCCCCCCCS −++++++= +−−+−−−+++− .3 1, 2 2,12,2,11,11,1,1,12 +++++−+++−+− −++++++= jijijijijijijiji fhCCCCCCCS The above, Equation (23), can be inverted to result in a two-point explicit form,             −− −− =      + 2 1 1, , 81 18 63 1 S S C C ji ji (24) whose individual explicit equations are given by, ].8[ 63 1 ],8[ 63 1 211, 21, SSC SSC ji ji += += + (25) For the case of an ungrouped (single) point, the following iteration formula will be applied to compute the point, jiC , where ,1,1 −=−= njmi . 38 1 ,,1,11,1 1,1,11,11,1,1 1,1 2         −++ ++++ = +−−+ −−−++++− −− jijijiji jijijijiji nm fhCCC CCCCC C (26) By considering Equations (25) and (26), the algorithm of the 2EG method for both cases—complete grouped (Case 1) and incomplete grouped (with one point ungrouped) (Case 2)—is illustrated in Algorithm 1, respectively. Algorithm 1 : 2EG iteration I. Initialize 0 )0( =C and 10 100.1 − = . II. For ,1,1 −=−= njmi calculate Equation (26). For 3,,5,3,1,1 −=−= mjmi L and ,1,3,,5,3,1 −=−= mjmi L calculate Equation (25). III. If − + || )()1( kk CC is satisfied, then proceed to Step IV. Otherwise, go back to Step II. IV. Calculate and display approximate values of ),( jiB yxS . 3.2. Formulation of 4-Point Explicit Group Iterative Methods From Figure 2, let us consider that the solution at any group of four points on the solution domain can be obtained using Equation (18). This leads to a (4 × 4) linear system as follows,             − − − − − − − − − − − − 8 1 1 1 1 8 1 1 1 1 8 1 1 1 1 8               ++ + + 1,1 1, ,1 , ji ji ji ji C C C C =             4 3 2 1 S S S S (27) 718 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate where, 1,12,22,12,1,2,24 1,2,12,2,11,1,13 ,11,2,21,21,11,2 ,1,1,11,11,1,11 2 2 2 2 3 3 3 3 ++++++++++ +++++−+−− ++++−+−+− +−−−+−−− −++++= −++++= −++++= −++++= jijijijijiji jijijijijiji jijijijijiji jijijijijiji fhCCCCCS fhCCCCCS fhCCCCCS fhCCCCCS (28) Figure 2. Computational Molecule Equation (27). The above Equation (27) can be inverted to result in a 4-Point equation,                         =               ++ + + 4 3 2 1 1,1 1, ,1 , 486 81 81 81 81 486 81 81 81 81 486 81 81 81 81 486 3645 1 S S S S C C C C ji ji ji ji (29) whose individual explicit equations are given by, 719 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate        441,1 331, 22,1 11, 486 3645 1 486 3645 1 486 3645 1 486 3645 1 STC STC STC TSC ji ji ji ji += += += += ++ + + (30) where, )(81 )(81 )(81 )(81 3214 4213 4312 4321 SSST SSST SSST SSST ++= ++= ++= ++= (31) Algorithm 2: 4EG iteration I. Initialize 0 )0( =C and 10 100.1 − = . II. For 3,,5,3,1 −= mi L and ,3,,5,3,1 −= mj L calculate Equation (30). For 3,,5,3,1,1 −=−= mjmi L and ,1,3,,5,3,1 −=−= mjmi L calculate Equation (25). III. If − + || )()1( kk CC is satisfied, then proceed to Step IV. Otherwise, go back to Step II. IV. Calculate and display approximate values of ),( jiB yxS . 4. Numerical Experiments To investigate the performance of the cubic B-spline Gauss-Seidel (C-BSGS), 2 Point-C-BSEG, and 4 Point-C-BSEG iterative methods, we evaluated three examples of the 2D Poisson equation. The goal was to validate the efficiency of both iterative approaches based on the number of iterations ),(W execution time in seconds )(t and maximum errors ).norm( −L Throughout the implementation of the point iterations, a convergence test was performed by considering a tolerance error, .100.1 10− = This ensured that the iterative methods continued until the desired level of accuracy was achieved. Example 1: Elsherbeny, et al. [24] ),,( yxfyyUxxU =+ ]1,0[, yx (32) with )sin()sin(),( yxyxf = .Then, the analytical solution of Problem (32) is obtained as follows )),sin()(sin( 2 2 1 ),( yxyxU   −= ].1,0[, yx (33) Example 2: Roslan and Hoe [25] 720 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate ( )  , , , 0,1xx yyU U f x y x y+ =  (34) with ).)(362()1(2))(1)(45(),( 2222 yxyx eyyxxyeyxxyxf −−−− +−−+−+−= Then, the analytical solution of Problem (34) is given as follows ( ) ( ) ( )  2, 1 1 , , 0,1x yU x y e x x y y x y− −= − −  (35) Example 3: Stonko, et al. [26] ),( yxfyyUxxU =+ , ]1,0[, yx (36) with )2cos()(sin2)(sin)2cos(2),( 2222 yxyxyxf  −−= . Then, the analytical solution of Problem (36) is stated as follows, )(sin)(sin),( 22 yxyxU = , ]1,0[, yx . (37) 5. Discussion The numerical results presented in Table 1 indicate that the 4-C-BSEG iterative method achieves a reduction in the number of iterations by approximately 29.73% to 33.23% compared to the C-BSGS method. Furthermore, the 4-C-BSEG method demonstrates faster computational performance, reducing execution time by 25.00% to 44.66%. This implies that the 4-C-BSEG iterative method requires fewer iterations and is more efficient in terms of computational time than both the C-BSGS and 2-C-BSEG iterative methods. Similarly, as shown in Table 2, it can be concluded that the 4-C-BSEG iterative method has fewer iterations by 34.17% - 35.18% compared to the C-BSGS method. Additionally, in terms of computational time, the 4-C-BSEG iterative method is faster than the C-BSGS iterative method, with a range of 36.84% - 51.21%. This indicates that the 4-C-BSEG iterative method is significantly more efficient than both the C-BSGS and 2-C-BSEG iterative methods for solving the second problem of 2D Poisson equations. From the numerical results recorded in Table 3, it can be observed that the 4-C-BSEG iterative method has a lesser number of iterations by 33.58% - 34.49% compared to C-BSGS. In terms of computational time, the implementation of the 4-C-BSEG iterative method is faster by 25.00% - 47.02% than the C-BSGS iterative method. This indicates that the 4-C-BSEG iterative method requires fewer iterations and is more efficient in computational time than both the C-BSGS and 2-C-BSEG iterative methods. 721 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate Table 1. Comparison of the number of iterations (W), execution time in seconds (t) and maximum errors (L∞-norm) using Example 1 m Method Number of iterations (W) Execution time (t) Maximum Errors ( L -norm) 16 C-BSGS 207 0.14 2.656698e-05 2-C-BSEG 184 0.12 2.656718e-05 4-C-BSEG 139 0.09 2.709432e-05 32 C-BSGS 626 0.32 2.354121e-05 2-C-BSEG 557 0.28 2.354195e-05 4-C-BSEG 418 0.24 2.379541e-05 64 C-BSGS 1943 1.03 2.203216e-05 2-C-BSEG 1734 0.77 2.203508e-05 4-C-BSEG 1306 0.57 2.216327e-05 128 C-BSGS 5942 3.03 2.122272e-05 2-C-BSEG 5333 2.73 2.123435e-05 4-C-BSEG 4057 2.00 2.131874e-05 256 C-BSGS 17152 26.85 2.058207e-05 2-C-BSEG 15533 24.32 2.062849e-05 4-C-BSEG 12052 16.49 2.075144e-05 Figure 3. The number of iterations and execution time in seconds for different mesh sizes (m) by C-BSGS, 2-C-BSEG, and 4-C-BSEG for Example 1. 722 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate Table 2. Comparison of the number of iterations (W), execution time in seconds (t), and maximum errors (L∞-norm) using Example 2 m Method Number of iterations (W) Execution time (t) Maximum Errors ( L -norm) 16 C-BSGS 313 0.19 3.432516e-04 2-C-BSEG 277 0.14 3.432516e-04 4-C-BSEG 203 0.12 3.432517e-04 32 C-BSGS 1012 0.61 1.938521e-04 2-C-BSEG 894 0.45 1.938522e-04 4-C-BSEG 656 0.32 1.938523e-04 64 C-BSGS 3407 2.07 1.052995e-04 2-C-BSEG 3016 1.34 1.052995e-04 4-C-BSEG 2217 1.01 1.052996e-04 128 C-BSGS 11646 7.10 5.576506e-05 2-C-BSEG 10323 5.58 5.576514e-05 4-C-BSEG 7615 3.98 5.576532e-05 256 C-BSGS 39653 65.21 2.901369e-05 2-C-BSEG 35221 57.65 2.901381e-05 4-C-BSEG 26105 37.89 2.901405e-05 Figure 4. The number of iterations and execution time in seconds for different mesh sizes (m) by C-BSGS, 2-C-BSEG, and 4-C-BSEG for Example 2. 723 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate Table 3. Comparison of the number of iterations (W), execution time in seconds (t), and maximum errors (L∞-norm) using Example 3. m Method Number of iterations (W) Execution time (t) Maximum Errors ( L -norm) 16 C-BSGS 290 0.16 1.105950e-03 2-C-BSEG 257 0.14 1.105950e-03 4-C-BSEG 192 0.12 1.125276e-03 32 C-BSGS 922 0.44 9.689167e-04 2-C-BSEG 816 0.43 9.689175e-04 4-C-BSEG 604 0.29 9.786855e-04 64 C-BSGS 3059 1.68 9.040932e-04 2-C-BSEG 2711 1.21 9.040964e-04 4-C-BSEG 2005 0.89 9.088428e-04 128 C-BSGS 10269 5.34 8.723361e-04 2-C-BSEG 9119 4.60 8.723487e-04 4-C-BSEG 6765 3.53 8.747496e-04 256 C-BSGS 34187 53.12 8.563620e-04 2-C-BSEG 30439 45.97 8.564117e-04 4-C-BSEG 22707 31.78 8.577004e-04 Figure 5. The number of iterations and execution time in seconds for different mesh sizes (m) by C-BSGS, 2-C-BSEG, and 4-C-BSEG for Example 3. 6. Conclusion A comparison of the numerical method using two methods is considered with all measured parameters for five different mesh sizes, as in Tables 1, 2, and 3, respectively. The numerical results for three examples indicate that the 4-C-BSEG iterative method outperforms the C-BSGS and 2-C-BSEG iterative methods, which require significantly fewer iterations to converge compared to the C-BSGS and 2-C-BSEG iterative methods. This demonstrates that the proposed iterative method exhibits a faster convergence rate, particularly for larger grids. As the grid size increases, the execution time to approximate the exact solution for each example was shorter than with the C-BSGS and 2-C-BSEG methods. The observations in Figures 3, 4, and 5 illustrate how the 4-C-BSEG iteration method can be used to demonstrate the effectiveness of the bicubic B-spline collocation strategy in solving 2D Poisson equations. It can be inferred that their bicubic B-spline collocation, using both iterative approaches, may converge to their known exact solution effectively in terms of the infinity norm across various mesh sizes. For future work, this study will continue to investigate the capability of the explicit group (EG) iteration family in conjunction with the Successive Over-Relaxation (SOR) iteration approach, as introduced by Young [13] and Young 724 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 11: 710-725, 2025 DOI: 10.55214/2576-8484.v9i11.10962 © 2025 by the authors; licensee Learning Gate [27]. This method belongs to the family of point-iterative techniques. The motivation for using SOR in this study comes from the work of Mohammed and Rivaie [28], who found that among the three indirect methods, namely Jacobi-Davidson, GS, and SOR, the SOR method is the most efficient and performs best. 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