Practical Introduction to FEM with GMSH: A MATLAB/Octave Perspective

TL;DR

This paper presents an educational yet efficient 3D FEM implementation in MATLAB/Octave integrated with GMSH for mesh generation, facilitating understanding and customization of PDE solutions.

Contribution

It introduces a novel FEM implementation in MATLAB/Octave that seamlessly integrates GMSH, enhancing educational value and flexibility for complex PDE problems.

Findings

Efficient 3D FEM implementation in MATLAB/Octave.

Seamless integration with GMSH for mesh handling.

Flexible foundation for advanced PDE problems.

Abstract

The Finite Element Method (FEM) is a powerful computational tool for solving partial differential equations (PDEs). Although commercial and open-source FEM software packages are widely available, an independent implementation of FEM provides significant educational value, provides a deeper understanding of the method, and enables the development of custom solutions tailored to specialized applications or integration with other solvers. This work introduces a 3D $\mathbb{P}_m$-element FEM implementation in MATLAB/Octave that is designed to balance educational clarity with computational efficiency. A key feature is its integration with GMSH, an open-source 3D mesh generator with CAD capabilities that streamlines mesh generation for complex geometries. By leveraging GMSH data structures, we provide a seamless connection between geometric modeling and numerical simulation. The implementation focuses on solving the general convection-diffusion-advection equation and serves as a flexible foundation for addressing advanced problems, including elasticity, mixed formulations, and integration with other numerical methods.