Fast Poisson Equation Solver in a Square Domain - MATLAB Implementation

Fast Poisson equation solving is a fundamental problem in scientific computing, particularly when dealing with Poisson's equation on two-dimensional square domains. MATLAB provides highly efficient numerical methods for this purpose. The Poisson equation is commonly used to describe potential field distributions, heat conduction, and similar phenomena, with its mathematical form being ∇²u = f, where u is the unknown potential function and f is the known source term.

For square domains, fast Poisson solvers typically employ the Finite Difference Method (FDM) for discretization, transforming the differential equation into a system of linear algebraic equations. MATLAB's fast Poisson solver leverages Discrete Sine Transform (DST) or Discrete Fourier Transform (DFT) to accelerate computations, as these transforms can diagonalize the discrete Laplacian operator, enabling efficient equation solving in the frequency domain.

The specific solution procedure involves: Grid Discretization: Divide the square domain into a uniform grid using the five-point stencil scheme to approximate the Laplacian operator. Source Term Handling: Adjust the discrete values of the right-hand side f to accommodate boundary conditions. Transform-based Solution: Utilize Fast Fourier Transform (FFT) or Fast Sine Transform (FST) to convert the problem to the frequency domain and solve the diagonalized linear system. Inverse Transformation: Convert the frequency-domain solution back to physical space to obtain the final numerical solution.

This approach avoids direct solving of large sparse matrices, reducing computational complexity from O(n³) to O(n log n), significantly improving computational efficiency. Both MATLAB's built-in `poisolv` function and custom FFT/DST implementations can achieve this objective. In code implementation, this typically involves using MATLAB's `fft2` and `ifft2` functions for periodic boundaries, or `dst` and `idst` for Dirichlet boundaries.

In practical applications, this method is suitable for fixed boundary conditions (such as Dirichlet boundaries), while other boundary conditions may require adjustments to the transformation method or additional preprocessing steps. Furthermore, MATLAB's optimized matrix operations ensure high computational efficiency, making it a widely used tool in engineering and scientific computing. The algorithm's implementation can be verified through convergence tests and comparison with analytical solutions where available.