Finite element and fictitious component methods for solving the poisson equation in a rectangular domain

Tóm tắt

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/(  + ), where  and  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|| < , achieving O(h²) accuracy ( < 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.