MATLAB Finite Element Program for 2D Elliptic Equations with Rectangular Meshing

This MATLAB program implements a finite element solution for two-dimensional elliptic equations using rectangular mesh discretization, designed with excellent extensibility. Although 2D finite element source codes are considerably more complex than their 1D counterparts, this program features a clear structure that facilitates comprehension and modification. The implementation employs Galerkin's method with bilinear basis functions for rectangular elements, incorporating proper boundary condition handling and linear system solvers. The code has been thoroughly validated and includes visualization capabilities that plot both exact and numerical solutions, enabling users to intuitively assess computational results through comparative graphs. Beyond solving elliptic equations, the program serves as a valuable reference implementation for other partial differential equations. With minor adaptations to the weak formulation and boundary conditions, it can be readily extended to handle various PDE types. The program's modular design separates mesh generation, matrix assembly, and solution modules, making it practically valuable as both a computational tool and educational resource for finite element methodology.