Solving Poisson's Equation Using Finite Difference Method

In the field of numerical analysis, the finite difference method is a common approach for solving partial differential equations. Among these, Poisson's equation represents an important class of partial differential equations. In this example, we will implement the finite difference method to solve Poisson's equation. The implementation involves discretizing the domain using a grid system, approximating derivatives using central difference schemes, and setting up a system of linear equations that can be solved using matrix operations. Key algorithmic steps include boundary condition handling, coefficient matrix construction, and solving the resulting linear system using efficient numerical methods like Gaussian elimination or iterative solvers. We will compare our numerical solution with the analytical solution to evaluate the accuracy of our implementation. This comparison will involve calculating error metrics such as root mean square error and maximum absolute error. Through this example, we aim to provide readers with a deeper understanding of finite difference method principles and demonstrate how to apply this technique to broader numerical problems. The code implementation will showcase practical aspects including mesh generation, matrix assembly, and solution visualization techniques.