A bicubic B-spline approximation and application of C-BSEG methods for the solution of 2D elliptic partial differential equations

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.