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Year 2025
 
Issue 6 [12 - 2025], Year XXXIV

Finite Element and Fictitious Component Methods for Solvingthe Poisson Equation in a Rectangular Domain

https://mij.hoimovietnam.vn/en/archives?article=25066

  • Affiliations:

    Long An University of Economics and Industry, Khanh Hau W., Tay Ninh, Vietnam

  • *Corresponding:
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  • Received: 18th-Sept-2025
  • Revised: 25th-Oct-2025
  • Accepted: 31st-Oct-2025
  • Online: 31st-Dec-2025
Pages: 24 - 35
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Abstract:

This study presents a novel hybrid numerical framework for solving the Poisson equation ∇²u = f in a unit square domain Ω, discretized into a uniform 14×14 coarse grid, with the computational domain encompassing a triangular subregion defined by vertices (0,0), (0,1), and (1,1). The methodology integrates the Finite Element Method (FEM) for triangulation—refining each grid cell into two linear triangular elements along a diagonal—with the Fictitious Component Method (FCM) to embed the irregular domain into a fictitious square, enhancing iterative efficiency. A stiffness matrix is assembled via FEM weak formulation, modified by FCM penalty terms, and solved using Richardson iteration with an optimal parameter ω ≈ 2/(λ_min + λ_max), where λ_min and λ_max are extreme non-zero eigenvalues of the system matrix. Numerical simulations across mesh refinements (3,249 to 201,601 nodes) demonstrate rapid convergence in 39 iterations to ||r|| < 〖10〗^({-6}), achieving O(h²) accuracy (L_error^2< 0.00015) and linear time scaling (~1.2 ms/node). Compared to pure FEM, the hybrid reduces iterations by 54% and computation time by 40% on fine meshes, underscoring its novelty in preconditioning ill-conditioned systems for irregular geometries. This framework offers significant potential in geomechanics and mining engineering, enabling efficient modeling of stress-strain fields and rock deformation around tunnels, with broader applicability to nonlinear and 3D problems.

How to Cite
Nguyen, D.Huu 2025. Finite Element and Fictitious Component Methods for Solvingthe Poisson Equation in a Rectangular Domain (in Vietnamese). Mining Industry Journal. XXXIV, 6 (Dec, 2025), 24-35. .
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