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

Sweilam, N. H.; Hanafy, Ibrahim M.; Ramadan, Moutaz; Eseily, Azza

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

Document Type : Original Article

Authors

¹ Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt.

² Department of Mathematics, Faculty of Science, Port Said University, Port Said, Egypt.

³ Department of Mathematics, College of Sciences, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.

10.21608/ajbas.2025.385670.1255

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.

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