• This course provides an introduction to the theory and numerical methods for Partial Differential Equations (PDEs) using variational techniques. It covers a range of topics from the fundamentals of weak formulation to the numerical approximation of PDEs using finite element methods and mixed finite element methods. Both coercive and non-coercive problems for scalar and vector-valued PDEs 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 method

The following topics will be covered

• Mathematical Preliminaries: Distributions, Sobolev spaces, Traces, Green’s formulas, Introduction to the concept of weak formulation, Well-Posedness Results (Lax–Milgram and Banach–Nečas–Babuška theorems), Study of model problems (Laplace, Stokes…).
• Galerkin approximation in Banach spaces: Galerkin and Petrov-Galerkin methods, Discrete Well-Posedness Results (Discrete Lax–Milgram and Banach–Nečas–Babuška theorems), Error analysis (Céa lemma, and Babuška lemma)
• Coercive problems: Study of scalar elliptic problems (theory and approximation, coercivity loss), Solution of the Laplace problem in 1-d and 2-d with Lagrange finite elements, Solution of the Laplace problem in 1-d with spectral methods and comparison with Lagrange finite elements, Vector-valued coercive PDEs (defined Maxwell’s equations).
• Mixed finite element approximation: Study of saddle point problems, Babuška-Brezzi theorem, Solution of the scalar and vector Poisson problem in mixed formulation and comparison with a classical formulation, Solution of the Stokes problem in mixed formulation.
• Non-coercive PDEs: The Helmholtz problem, First-order PDEs (Scalar first-order PDEs, First-order PDEs in L2, Galerkin/least squares approximation).

Lecture notes based mainly on the lectures by A. Ratnani and E. Sonnendrücker, "Advanced Finite Element Methods," will be available from the instructor.

Recommended Textbooks:

1. A. Ern, J-L. Guermond, Finite Elements I & II, Texts in Applied Mathematics, Springer, 2021.
2. D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications, Springer, 201

1. Homeworks/Projects: 25 %
2. Final Exam : 75 %