Introduction to the Finite Element Method

These lecture notes accompany the course Introduction to the Finite Element Method. They cover the mathematical foundations, 1D and multi-dimensional finite element formulations, implementation details, and error analysis.

The notes are designed so that students can follow along during lectures without needing to copy derivations — instead, focus on understanding the concepts and annotating where needed.

0.1 How to Use These Notes

The chapters follow the lecture sequence. Each chapter contains:

• Mathematical derivations presented step by step
• Python code demonstrating the concepts computationally (click to expand code blocks)
• Key definitions and results highlighted in callout boxes

0.2 Companion Resources

These lecture notes are part of a three-part site:

• FEM Companion Notebooks — hands-on Jupyter notebooks that let you work through the key ideas symbolically and numerically using the symbolic_fem_workbench package. Relevant notebooks are linked at the top of each chapter.
• Package Documentation — API reference and getting-started guide for the symbolic_fem_workbench Python package that powers all of the code examples.
• Source Repository — issues, source for the book, notebooks and package, and edit-on-GitHub links on every chapter.

A single-file PDF of the entire lecture book is also available (rebuilt on every push to main), and the sidebar carries a Download PDF shortcut.

0.3 Prerequisites

Students should be familiar with:

• Linear algebra (matrix operations, eigenvalues)
• Calculus (integration by parts, partial derivatives)
• Basic Python programming
• Introductory continuum mechanics