Numerical Solution of Classical Poisson Equation using MATLAB

The classical Poisson equation is a two-dimensional partial differential equation with applications across mathematics, physics, and engineering. To solve this equation, MATLAB numerical solution program code can be employed. The program typically implements either finite difference method (FDM) or finite element method (FEM) for numerical approximation. In the finite difference approach, the continuous Poisson equation is discretized into a grid format using centered difference approximations for Laplace operator. This transforms the PDE into a system of linear equations Ax = b, where matrix A represents the discretized Laplace operator and vector b contains source terms. MATLAB's built-in functions like `poisolv` or sparse matrix solvers (`\` operator) can efficiently solve this system. The finite element method decomposes the solution domain into smaller subdomains (elements) and constructs piecewise polynomial approximations. Key implementation steps include mesh generation using `pdegeom` and `initmesh`, defining boundary conditions with `pdebound`, and solving using PDE toolbox functions like `assempde`. The method yields solution values at nodal points through variational formulation and assembly of stiffness matrices. Using MATLAB's numerical solution code provides better understanding of the Poisson equation solving process while achieving accurate results through proper discretization techniques and efficient linear algebra computations. The code typically includes visualization components using `pdeplot` or `contourf` to display potential distribution and field patterns.