ME/BME 736 — Advanced Finite Element Methods

A graduate course on the analysis, implementation and verification of finite element methods, taught alongside an open textbook and code repository that are developed in public as the course runs.

The course follows a continuous sequence from variational formulation to discretization, implementation and verification, with convergence studies used as evidence for the numerical result. Students build solvers rather than only use them.

Sequence

Foundations and approximation theory; a priori and a posteriori error estimation and adaptivity; constraints, mixed and nonlinear formulations; transient problems and PDE-constrained optimization. The final computational project is designed to integrate all four, formulation, implementation, verification and interpretation, in one piece of work.

Scaffolding

Entering students come from uneven mathematical backgrounds, so the assignments supply intermediate steps rather than assuming them. The second assignment works through equality almost everywhere, uniform against L² convergence, extensions of the divergence theorem, the derivative of a nonlinear energy functional, continuity and coercivity of a diffusion form, and a staged contradiction proof that a tent function lies in H¹ but not H²: the machinery the error analysis later depends on.

Assessment is homework 40%, quizzes 10%, midterm 25%, final computational project 25%.

The open book

The accompanying Quarto book and its code are public and versioned. Four of the thirteen chapters are complete — introduction, mathematical foundations, variational formulations, and model problems in heat transfer and solid mechanics — the one-dimensional finite element chapter is in progress, and the rest follow the course as it is delivered. The book carries its own status page saying exactly that. It is an active book, not a finished textbook.