MATLAB Implementation for Solving the Two-Dimensional Poisson Equation

To solve the two-dimensional Poisson equation using MATLAB, you need to follow a systematic computational workflow. The process begins with data importation using functions like `load` or `readmatrix` to input the equation parameters and domain specifications. Next, you must define the computational mesh grid using MATLAB's `meshgrid` function, which generates a structured grid of points where the solution will be calculated. For example: `[X,Y] = meshgrid(xmin:h:xmax, ymin:h:ymax)` where h represents the grid spacing. The critical step involves specifying boundary conditions (Dirichlet, Neumann, or mixed) using conditional statements or dedicated boundary arrays. MATLAB's partial differential equation toolbox offers `pdepe` for simple cases, while for custom implementations, finite difference methods can be coded using sparse matrices (`spdiags`) to build the discretized Laplacian operator. The core solving phase typically utilizes linear algebra solvers like the backslash operator (`\`) for direct methods or iterative solvers (`pcg`) for large sparse systems. For Poisson's equation ∇²u = f, this involves solving AU = F where A is the discretized Laplacian matrix. Post-solving analysis includes visualization through `surf` or `contour` plots to examine the solution landscape. Validation can be performed by comparing with analytical solutions using `norm` for error calculation, or by conducting parameter studies through loop structures that modify boundary conditions and source terms. Key functions to implement include: - `meshgrid` for domain discretization - `delsq` for discrete Laplacian generation - Matrix solvers for linear systems - Visualization functions like `pcolor` and `quiver` for result analysis In summary, the MATLAB implementation requires: data importation, mesh generation, boundary condition specification, numerical solution using appropriate algorithms, and comprehensive result validation through both quantitative and visual methods. The entire process leverages MATLAB's strengths in matrix operations and scientific visualization for accurate PDE solutions.