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Finite element methods · Graduate · Math
Learning objectives from the Finite element methods syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Finite Element Methods — scope drawn from open FEM notes and typical US graduate finite element syllabi.
Variational foundations
- Weak formulations of elliptic boundary-value problems — Weak formulations of elliptic boundary-value problems
- Lax–Milgram theorem and well-posedness — Lax–Milgram theorem and well-posedness
- Galerkin and Petrov–Galerkin approximations — Galerkin and Petrov–Galerkin approximations
- Energy norms — Energy norms
- Best approximation in Hilbert spaces — Best approximation in Hilbert spaces
- Abstract error estimates (Céa's lemma) — Abstract error estimates (Céa's lemma)
Finite element spaces
- Piecewise polynomial elements on triangulations and meshes — Piecewise polynomial elements on triangulations and meshes
- Reference elements, shape functions, and assembly — Reference elements, shape functions, and assembly
- Interpolation error — Interpolation error
- Approximation theory — Approximation theory
- h-refinement vs p-refinement (introduction) — h-refinement vs p-refinement (introduction)
- Mixed finite elements (introduction) — Mixed finite elements (introduction)
Time-dependent and hyperbolic problems
- Method of lines for parabolic problems — Method of lines for parabolic problems
- Stability of time discretizations (theta methods overview) — Stability of time discretizations (theta methods overview)
- A posteriori error estimation (introduction) — A posteriori error estimation (introduction)
- Adaptive mesh refinement driven by error indicators — Adaptive mesh refinement driven by error indicators
- Introduction to discontinuous Galerkin methods — Introduction to discontinuous Galerkin methods
Implementation and software
- Assembly algorithms — Assembly algorithms
- Sparse linear algebra — Sparse linear algebra
- Mesh generation and quality metrics — Mesh generation and quality metrics
- Solving large systems: direct vs iterative solvers — Solving large systems: direct vs iterative solvers
- Commercial — Commercial
- Open-source FEM workflows (FEniCS, deal.II overview) — Open-source FEM workflows (FEniCS, deal.II overview)
- Verification — Verification
- Validation in engineering simulation — Validation in engineering simulation
Applications
- Linear elasticity — Linear elasticity
- Structural analysis — Structural analysis
- Steady and transient heat conduction — Steady and transient heat conduction
- Incompressible flow: Stokes problem (introduction) — Incompressible flow: Stokes problem (introduction)
- Electromagnetics: Maxwell in frequency domain (introduction) — Electromagnetics: Maxwell in frequency domain (introduction)
- Multiphysics coupling strategies (overview) — Multiphysics coupling strategies (overview)
Learning objectives
Click an objective for study materials.
Variational foundations
- Weak formulations of elliptic boundary-value problems — Weak formulations of elliptic boundary-value problems
- Lax–Milgram theorem and well-posedness — Lax–Milgram theorem and well-posedness
- Galerkin and Petrov–Galerkin approximations — Galerkin and Petrov–Galerkin approximations
- Energy norms — Energy norms
- Best approximation in Hilbert spaces — Best approximation in Hilbert spaces
- Abstract error estimates (Céa's lemma) — Abstract error estimates (Céa's lemma)
Finite element spaces
- Piecewise polynomial elements on triangulations and meshes — Piecewise polynomial elements on triangulations and meshes
- Reference elements, shape functions, and assembly — Reference elements, shape functions, and assembly
- Interpolation error — Interpolation error
- Approximation theory — Approximation theory
- h-refinement vs p-refinement (introduction) — h-refinement vs p-refinement (introduction)
- Mixed finite elements (introduction) — Mixed finite elements (introduction)
Time-dependent and hyperbolic problems
- Method of lines for parabolic problems — Method of lines for parabolic problems
- Stability of time discretizations (theta methods overview) — Stability of time discretizations (theta methods overview)
- A posteriori error estimation (introduction) — A posteriori error estimation (introduction)
- Adaptive mesh refinement driven by error indicators — Adaptive mesh refinement driven by error indicators
- Introduction to discontinuous Galerkin methods — Introduction to discontinuous Galerkin methods
Implementation and software
- Assembly algorithms — Assembly algorithms
- Sparse linear algebra — Sparse linear algebra
- Mesh generation and quality metrics — Mesh generation and quality metrics
- Solving large systems: direct vs iterative solvers — Solving large systems: direct vs iterative solvers
- Commercial — Commercial
- Open-source FEM workflows (FEniCS, deal.II overview) — Open-source FEM workflows (FEniCS, deal.II overview)
- Verification — Verification
- Validation in engineering simulation — Validation in engineering simulation
Applications
- Linear elasticity — Linear elasticity
- Structural analysis — Structural analysis
- Steady and transient heat conduction — Steady and transient heat conduction
- Incompressible flow: Stokes problem (introduction) — Incompressible flow: Stokes problem (introduction)
- Electromagnetics: Maxwell in frequency domain (introduction) — Electromagnetics: Maxwell in frequency domain (introduction)
- Multiphysics coupling strategies (overview) — Multiphysics coupling strategies (overview)
Definitions and structure
- Weak formulations of elliptic boundary-value problems — Weak formulations of elliptic boundary-value problems
- Lax–Milgram theorem and well-posedness — Lax–Milgram theorem and well-posedness
- Galerkin and Petrov–Galerkin approximations — Galerkin and Petrov–Galerkin approximations
- Energy norms — Energy norms
- Best approximation in Hilbert spaces — Best approximation in Hilbert spaces
Proofs and reasoning
- Abstract error estimates (Céa's lemma) — Abstract error estimates (Céa's lemma)
- Piecewise polynomial elements on triangulations and meshes — Piecewise polynomial elements on triangulations and meshes
- Reference elements, shape functions, and assembly — Reference elements, shape functions, and assembly
- Interpolation error — Interpolation error
- Approximation theory — Approximation theory
Abstraction and generalization
- h-refinement vs p-refinement (introduction) — h-refinement vs p-refinement (introduction)
- Mixed finite elements (introduction) — Mixed finite elements (introduction)
- Method of lines for parabolic problems — Method of lines for parabolic problems
- Stability of time discretizations (theta methods overview) — Stability of time discretizations (theta methods overview)
- A posteriori error estimation (introduction) — A posteriori error estimation (introduction)
Modeling and computation
- Apply weak formulations of elliptic boundary-value problems in engine... — Apply weak formulations of elliptic boundary-value problems in engineering contexts
- Apply lax–milgram theorem and well-posedness in engineering contexts — Apply lax–milgram theorem and well-posedness in engineering contexts
- Apply galerkin and petrov–galerkin approximations in engineering cont... — Apply galerkin and petrov–galerkin approximations in engineering contexts
- Apply energy norms in engineering contexts — Apply energy norms in engineering contexts
- Apply best approximation in hilbert spaces in engineering contexts — Apply best approximation in hilbert spaces in engineering contexts
Data and technology
- Use software to explore finite element methods problems numerically — Use software to explore finite element methods problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Abstract error estimates (Céa's lemma) — Abstract error estimates (Céa's lemma)
- Piecewise polynomial elements on triangulations and meshes — Piecewise polynomial elements on triangulations and meshes
- Reference elements, shape functions, and assembly — Reference elements, shape functions, and assembly
- Interpolation error — Interpolation error
- Approximation theory — Approximation theory
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Variational foundations
Weak formulations of elliptic boundary-value problems
Coming soon - Finite element spaces
Piecewise polynomial elements on triangulations and meshes
Coming soon - Time-dependent and hyperbolic problems
Method of lines for parabolic problems
Coming soon - Implementation and software
Assembly algorithms
Coming soon - Applications
Linear elasticity
Coming soon
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$1,162 · Finite element methods · 18 tutoring hrs
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