Solving Poisson’s equation using the Finite Difference Method

This paper discusses the numerical solution of Poisson’s and Laplace equations using the Finite Difference Method (FDM). The two equations were solved twodimensionally in a rectangular domain under Dirichlet boundary conditions. A fivepoint stencil scheme was used to discretize the equations, and the linear systems obtained were solved using the Mathematica function ‘LinearSolve’. The primary objective was to test the method with various grid sizes (n = 5, 10, 15, 20) and compare the precision and reliability of the two problems—one with a source term (Poisson) and one without (Laplace). The findings revealed that the method produced accurate and consistent solutions, with error decreasing as the grid size increased. Surface plots verified the smoothness of the solution on finer grids, and error analysis confirmed the convergence behaviour. This project shows that FDM is a stable and viable technique for solving boundary value problems and is useful to students and researchers working in numerical analysis and partial differential equations.