Finite element methods
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
Variational foundations
- Weak formulations of elliptic boundary-value problems
- Lax–Milgram theorem and well-posedness
- Galerkin and Petrov–Galerkin approximations
- Energy norms and best approximation in Hilbert spaces
- Abstract error estimates (Céa's lemma)
Finite element spaces
- Piecewise polynomial elements on triangulations and meshes
- Reference elements, shape functions, and assembly
- Interpolation error and approximation theory
- h-refinement vs p-refinement (introduction)
- Mixed finite elements (introduction)
Time-dependent and hyperbolic problems
- Method of lines for parabolic problems
- Stability of time discretizations (theta methods overview)
- A posteriori error estimation (introduction)
- Adaptive mesh refinement driven by error indicators
- Introduction to discontinuous Galerkin methods
STEM / applied
Implementation and software
- Assembly algorithms and sparse linear algebra
- Mesh generation and quality metrics
- Solving large systems: direct vs iterative solvers
- Commercial and open-source FEM workflows (FEniCS, deal.II overview)
- Verification and validation in engineering simulation
Applications
- Linear elasticity and structural analysis
- Steady and transient heat conduction
- Incompressible flow: Stokes problem (introduction)
- Electromagnetics: Maxwell in frequency domain (introduction)
- Multiphysics coupling strategies (overview)
Notes
Topics reflect common graduate finite element method syllabi at US universities, often cross-listed with computational science and engineering programs.