Bi-Conjugate Gradient Stabilized and Multigrid Approaches Based on Nonstandard Finite Differences for the Solution of Elliptic Partial Differential Equations
Sweilam, N., Hanafy, I., Ramadan, M., Eseily, A. (2025). Bi-Conjugate Gradient Stabilized and Multigrid Approaches Based on Nonstandard Finite Differences for the Solution of Elliptic Partial Differential Equations. EKB Journal Management System, 6(3), 304-320. doi: 10.21608/ajbas.2025.385670.1255
N. H. Sweilam; Ibrahim M. Hanafy; Moutaz Ramadan; Azza Eseily. "Bi-Conjugate Gradient Stabilized and Multigrid Approaches Based on Nonstandard Finite Differences for the Solution of Elliptic Partial Differential Equations". EKB Journal Management System, 6, 3, 2025, 304-320. doi: 10.21608/ajbas.2025.385670.1255
Sweilam, N., Hanafy, I., Ramadan, M., Eseily, A. (2025). 'Bi-Conjugate Gradient Stabilized and Multigrid Approaches Based on Nonstandard Finite Differences for the Solution of Elliptic Partial Differential Equations', EKB Journal Management System, 6(3), pp. 304-320. doi: 10.21608/ajbas.2025.385670.1255
Sweilam, N., Hanafy, I., Ramadan, M., Eseily, A. Bi-Conjugate Gradient Stabilized and Multigrid Approaches Based on Nonstandard Finite Differences for the Solution of Elliptic Partial Differential Equations. EKB Journal Management System, 2025; 6(3): 304-320. doi: 10.21608/ajbas.2025.385670.1255

Bi-Conjugate Gradient Stabilized and Multigrid Approaches Based on Nonstandard Finite Differences for the Solution of Elliptic Partial Differential Equations

Alfarama Journal of Basic & Applied Sciences
Volume 6, Issue 3, July 2025, Page 304-320  XML PDF (1.11 MB)
Document Type: Original Article
DOI: 10.21608/ajbas.2025.385670.1255
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Authors
N. H. Sweilam1; Ibrahim M. Hanafy2; Moutaz Ramadanorcid 2, 3; Azza Eseily email 2
1Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt.
2Department of Mathematics, Faculty of Science, Port Said University, Port Said, Egypt.
3Department of Mathematics, College of Sciences, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.
Abstract
This study introduces a robust and highly accurate numerical scheme for the solution of two-dimensional elliptic partial differential equations subject to Dirichlet boundary conditions. The proposed method is based on the Nonstandard Finite Difference technique, which is designed to achieve improved accuracy and stability compared to the classical Standard Finite Difference approach. To solve the resulting large, sparse linear systems, we employ two iterative solvers: the Bi-Conjugate Gradient Stabilized method and a Multigrid method. Within the multigrid framework, the Generalized Minimal Residual algorithm is utilized as a smoothing strategy to enhance convergence behavior. A comprehensive set of numerical experiments is carried out to assess and compare the performance of these approaches in terms of convergence rate, computational time, and number of iterations. The results, obtained for a range of grid resolutions and problem configurations, demonstrate the superior performance and efficiency of the proposed NSFD-based scheme, particularly on finer grids, confirming its effectiveness and reliability for solving elliptic problems.
Keywords
Iterative solvers; elliptic PDEs; Dirichlet boundary conditions
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