Advanced Finite Element Methods

This repository contains an open graduate-level book and teaching materials for ME/BME 736: Advanced Finite Element Methods at South Dakota Mines. The material develops the mathematical formulation of finite element methods and connects it to explicit numerical implementation and FEniCSx computations. Heat conduction and solid mechanics are used as recurring model problems.

• Read the book online
• Download the book as a PDF
• See the current chapter status

Book

The Quarto sources are in book/. The current complete chapters cover:

1. introduction to the finite element method;
2. mathematical foundations;
3. partial differential equations and variational formulations;
4. model problems in heat transfer and solid mechanics;
5. the finite element method in one dimension; and
6. finite element approximation in multiple dimensions.

The book also includes appendices on real-analysis essentials and tensor notation and index calculus. Later chapters will address error estimation, adaptivity, constraints, mixed methods, nonlinear and transient problems, and PDE-constrained optimization. The development status of each chapter is maintained in book/status.qmd.

To render both the HTML and PDF versions from the repository root, use:

quarto render book

The rendered files are written to book/_book/.

Assignments and computational notebooks

The assignment sources are in assignments/:

• Assignments 1 and 2: mathematical foundations;
• Assignment 3: variational formulations and energy principles; and
• Assignment 4: finite element computation in one and two dimensions.

Assignment 4 is supported by two executable notebooks:

• notebooks/assembly/assembly_demo.ipynb develops one-dimensional assembly explicitly with NumPy; and
• notebooks/diffusion/diffusion_2d_demo.ipynb solves a two-dimensional diffusion problem with FEniCSx and examines mesh refinement and error.

See assignments/README.md for assignment rendering and notebook-submission instructions.

Lecture materials and handouts

The topic-based lecture materials in lectures/slides/ currently include:

• one-dimensional finite element assembly, boundary conditions, and reactions; and
• multidimensional finite elements, linear triangles, basis-function support, matrix sparsity, and a two-dimensional diffusion example.

Each topic directory contains the LaTeX source, figures, and final PDF. Build instructions are given in lectures/slides/README.md.

Focused notes in handouts/ cover one-dimensional finite elements, local-to-global assembly, and examples of weak derivatives.

Course documents

The course/ directory contains the Fall 2026 ME/BME 736 syllabus and course flyer.

Software environment

The book, notebooks, and assignments use a single Conda environment containing Quarto, Python, Jupyter, FEniCSx, MPI, PETSc, and the supporting scientific Python packages. Create it with:

conda env create -f environment/environment.yml
conda activate finite-elements

Complete setup, TinyTeX, Jupyter-kernel, MPI, and verification instructions are provided in environment/README.md.

Copyright and license

Copyright (C) 2026 Prashant K. Jha.

This work is free: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. It is distributed without any warranty; see LICENSE for the full terms.