A Finite Element Method Implementation for 1D Heat Conduction Equation with MATLAB Code Examples

This document presents 1DHeatEquFEM, a MATLAB example program implementing the Finite Element Method (FEM) for solving one-dimensional heat conduction equations. FEM is a numerical computation technique that breaks down complex problems into simpler sub-problems (elements), then reconstructs the global solution from elemental solutions. The program demonstrates key implementation aspects including: - Domain discretization using linear Lagrange elements - Assembly of global stiffness and mass matrices through element-wise contributions - Implementation of boundary conditions (Dirichlet/Neumann) via matrix modifications - Time integration using finite difference schemes (e.g., backward Euler method) The code illustrates essential MATLAB programming techniques such as sparse matrix operations for efficient computation, vectorized loops for element matrix assembly, and conditional statements for boundary handling. Key functions include mesh generation (linspace), matrix assembly (sparse), and linear system solvers (backslash operator). Through this example, learners gain practical understanding of FEM principles and MATLAB's computational capabilities for engineering simulations, providing a foundation for extending these methods to broader applications like structural analysis or fluid dynamics.