A new graduate course launched this fall: ME/BME 736, Advanced Finite Element Methods, at South Dakota Mines.
It runs in four parts. Review and foundations — function and Sobolev spaces, variational formulations, energy minimization, and finite element approximation on heat conduction and elasticity. Error analysis and adaptivity — a priori estimation and Galerkin orthogonality, then residual-based a posteriori estimators, local indicators, marking, and adaptive refinement. Advanced formulations — penalty and Lagrange-multiplier constraints, mixed displacement–pressure methods for nearly incompressible elasticity, consistent linearization and Newton methods for hyperelasticity, and transient problems. PDE-constrained optimization — state and design variables, direct sensitivities, continuous and discrete adjoints, gradient verification.
The arc through all four is the same: analysis, then formulation, then FEniCSx, then verification, then optimization. Computational assignments pair selected methods with implementations, convergence checks and interpretation of the numerical results.
What makes it unusual is that the textbook is being written in public alongside it. The open Quarto book and its code repository — FEniCSx examples, notebooks, assignments and data — are versioned and free to read; chapters appear as the course reaches them. It is an active book, not a finished one, and that is the point: students see the material being built rather than handed down.
