• This is an advanced course on finite element methods, focusing on time-dependent Partial Differential Equations (PDEs). It covers the spatial and time discretization of parabolic and hyperbolic equations using finite element methods and mixed finite element methods. Both conforming and non-conforming finite elements are considered.
• Prerequisites: This course is designed for graduate students in mathematics, engineering, or related fields who have a basic knowledge of analysis, linear algebra, and numerical methods. Additionally, a foundational understanding of evolution problems and basic finite element methods is required.

The following topics will be covered

• Non conforming Finite Element methods: Variational crimes, Non conforming Lagrange finite elements on triangles, Weak enforcement of Dirichlet boundary conditions, Discontinuous Galerkin methods.
• Parabolic Problems: Heat equation, Abstract parabolic problems, Weak formulation and well-posedness, Semi-discretization in space, discretization in time (Explicit/Implicit Euler and Runge–Kutta schemes), Application to the finite element approximation of the Heat equation.
• Second-Order Evolution Problems in Time: Wave equation, Abstract second-order problems in Time, Semi-discretization in space, discretization in time (Crank–Nicolson scheme), Application to the finite element approximation of the Wave equation.
• Time-dependent First-order PDEs: Friedrichs’ systems, space semi-discretization, time discretization, Finite Element Method for Conservation Equations.
• Time-dependent Maxwell’s Equations: Variational formulation for Maxwell equations, Construction of the finite element spaces, Exact sequences of finite element spaces, Time discretization (Leap Frog scheme.

• Lecture notes will be available from the instructor.
• Recommended Textbooks:
  • 1. A. Ern, J-L. Guermond, Finite Elements III, Springer, 2021.
  • A. Ratnani and E. Sonnendrücker, Advanced Finite Element Methods.
  • D. N. Arnold, Finite Element Exterior Calculus, CBMS-NSF regional conference series in applied mathematics, SIAM, 2018.

1. Homeworks/Projects: 40 %
2. Final Exam : 60 %