A finite difference approach to solving boundary value problems for Poisson’s equation

This study investigates the numerical solution of the two-dimensional Poisson’s equation using the Finite Difference Method (FDM) with a five-point stencil discretization under Dirichlet boundary conditions. The equation is solved on a structured square domain, and the resulting linear system is computed using a direct solver in Mathematica. Numerical experiments were conducted for various grid sizes (n = 5, 10, 15, 20), and the solutions were compared with the exact analytical solution. Results indicate that finer grids significantly improve accuracy, as demonstrated by decreasing absolute error and smoother surface plots. The findings confirm the second-order accuracy and convergence of the method. While direct solvers are effective for small to mediumsized problems, future work is recommended to explore more scalable iterative methods and extensions to complex geometries or nonlinear systems for broader applicability.