Fast Poisson Equation Solver - MATLAB Implementation for Square Domains

This project focuses on developing an optimized computational method for solving the Poisson equation within square domains using MATLAB. Our implementation leverages discrete Fourier transforms (DFT) and spectral decomposition techniques to achieve superior computational efficiency compared to conventional approaches.

The core algorithm employs matrix diagonalization through fast Fourier transforms (FFT), where we discretize the Laplacian operator using second-order finite differences. The MATLAB implementation utilizes built-in FFT functions (fft2/ifft2) for rapid transformation between spatial and frequency domains, while implementing eigenvalue decomposition for the discrete Laplacian matrix. Key functions include boundary condition handling through zero-padding and spectral coefficient calculation for the right-hand side source term.

Validation involves comparative analysis with standard methods including finite difference schemes (using sparse matrices with backslash operator) and spectral methods. Performance benchmarking evaluates computational complexity O(n log n) versus traditional O(n³) methods, with metrics assessing solution accuracy through L²-norm residuals and convergence rates.

This optimized solver demonstrates significant potential for applications in computational fluid dynamics, electromagnetic field simulation, and image processing, where rapid Poisson solutions are critical for large-scale scientific computing.