MATLAB Code for Solving Poisson's Equation Using Finite Element Method

In the following paragraphs, I will provide MATLAB code for solving Poisson's equation using the finite element method to help you better understand this process. The implementation begins with defining boundary conditions for the problem domain, followed by mesh generation to discretize the domain into smaller elements. The code utilizes linear triangular elements and assembles the global stiffness matrix through element-wise integration. Key MATLAB functions involved include mesh generation using pdetool or custom functions, assembly of finite element matrices, and handling of boundary conditions through penalty methods or direct imposition.

The implementation proceeds by applying the finite element formulation to solve the variational form of Poisson's equation, where we compute element stiffness matrices and load vectors. The code demonstrates how to handle Dirichlet boundary conditions by modifying the system matrices and solving the resulting linear system using MATLAB's built-in solvers like backslash operator or iterative methods. The solution process includes post-processing steps for result visualization using contour plots or surface plots to analyze the potential distribution.

Finally, we interpret and analyze the solution results by examining field distributions, calculating derived quantities like gradients, and verifying solution accuracy through convergence studies. The discussion covers practical applications in engineering fields such as electrostatics, heat transfer, and fluid mechanics, where Poisson's equation models various physical phenomena. This implementation serves as an educational foundation for understanding finite element method principles and their application to partial differential equations.