MATLAB Source Code for Solving 2D Poisson's Equation with Finite Difference Implementation

In numerical analysis, solving the two-dimensional Poisson's equation represents a classical problem typically addressed through finite difference discretization methods. The core methodology involves transforming continuous partial differential equations into discrete algebraic systems using grid-based approximations to approach the true solution. Fundamental Approach Problem Formulation The standard form of the 2D Poisson's equation is ∇²u = f(x,y), where u represents the unknown function and f denotes the known source term. Boundary conditions may include Dirichlet, Neumann, or mixed types. Discretization Process On a uniform grid, second-order derivatives are approximated using central difference schemes, converting the partial differential equation into a linear system. The five-point stencil method, for instance, connects each interior point's solution with its four adjacent neighbors through discrete Laplacian operators. Matrix Construction The discretized system typically manifests as a sparse matrix structure, suitable for iterative methods (like Gauss-Seidel) or direct solvers. For Dirichlet boundary conditions, boundary values are directly incorporated into the equations, while other boundary types require additional handling through ghost points or modified stencils. Parameter Scanning When treating the z-axis as a parametric degree of freedom (e.g., varying with time or another variable), iterative loop structures can solve the equation for different parameter values sequentially. Results can be stored in three-dimensional arrays for subsequent visualization or analysis, implemented through nested for-loops with matrix preallocation. Implementation Considerations Grid generation requires balancing precision against computational efficiency, with mesh density determined by error tolerance requirements. Sparse matrix storage (using MATLAB's spalloc or sparse functions) significantly reduces memory consumption for large-scale problems. Proper handling of boundary conditions proves crucial for solution accuracy, requiring careful implementation of boundary value assignments and coefficient modifications. This methodology can be extended to non-uniform grids or more complex boundary conditions, finding applications in physical modeling scenarios such as electromagnetic field simulations and heat conduction analysis. The implementation typically involves meshgrid creation, coefficient matrix assembly using diagonal matrices, and solver selection based on problem size and conditioning.