Numerical Solution of Classical Poisson Equation Using MATLAB

As a typical representative of elliptic partial differential equations, the Poisson equation widely appears in physical problems such as electromagnetic fields, fluid dynamics, and heat conduction. In the MATLAB environment, we can efficiently solve it numerically using the finite difference method.

The core solution approach mainly consists of three steps: Grid discretization - Dividing the continuous solution domain into a uniform two-dimensional grid, replacing continuous space with discrete points. Equal spacing is typically adopted to facilitate the construction of difference schemes. In MATLAB implementation, this can be achieved using meshgrid() function to create coordinate matrices. Difference scheme construction - Approximating second-order derivative terms using central differences to convert the differential equation into a system of linear algebraic equations. Boundary conditions (such as Dirichlet boundaries) require special treatment through matrix modifications. The Laplacian operator can be discretized using the 5-point stencil method. Matrix solution - The resulting sparse matrix system can be solved directly using the backslash operator (\), or iterative methods (like the conjugate gradient method) can be employed to improve computational efficiency for large-scale problems. MATLAB's built-in pcg() function is particularly suitable for symmetric positive definite systems.

Special attention is required when the source term exhibits singularity or when dealing with complex boundary geometries, necessitating adjustments in grid generation strategies and boundary treatment approaches. For periodic boundary conditions, the Fast Fourier Transform (FFT) method can be considered to accelerate the solution process using MATLAB's fft2() and ifft2() functions.

In practical applications, this method can be extended to solve nonlinear Poisson equations or coupled with other physical equations. By adjusting discretization accuracy and implementing adaptive mesh techniques using MATLAB's PDE Toolbox functions, one can effectively balance computational precision and efficiency.