Malaysian Journal of Mathematical Sciences, June 2026, Vol. 20, No. 2
Multigrid and Bi-Conjugate Gradient Stabilized Methods for the Ill-Posed Complex Helmholtz Equation: Comparative Studies
Sweilam, N. H., Ibrahim, M. H., Ramadan, M., Eseily, A.
Corresponding Email: nsweilam@sci.cu.edu.eg
Received date: 27 May 2025
Accepted date: 16 October 2025
Abstract:
In this study, we develop and analyze efficient numerical schemes for solving the ill-posed complex Helmholtz equation in both two and three spatial dimensions under Dirichlet boundary conditions. The spatial discretization is carried out using the classical finite difference method and the nonstandard finite difference method, with the latter offering enhanced stability. The resulting linear systems are large, sparse, and highly sensitive to perturbations, reflecting the ill-posed nature of the problem. To address these challenges, two advanced iterative solvers are employed: the multigrid method, with the generalized minimal residual used as a smoother, and the bi-conjugate gradient stabilized method. Numerical experiments compare the computed solutions with solutions to verify accuracy, and condition numbers are evaluated to assess ill-posedness and stability. The results demonstrate that multigrid achieves robust and stable convergence, particularly in cases where the bi-conjugate gradient stabilized method fails, while the nonstandard finite difference method consistently improves numerical stability compared with the classical finite difference method.
Keywords: complex Helmholtz equation; nonstandard finite difference method; finite difference method; iterative solvers; ill-posed systems of equations.
