Summary

The aim of the course is to give a theoretical and practical knowledge of the finite element method for saddle point problems, such as the ones of fluid dynamics, elasticity and electromagnetic problems.

Content

Keywords

Partial differential equations,  saddle point problems, finite element method, Galerkin approximation, stability and convergence analysis.

Learning Prerequisites

Required courses

Analysis I II III IV, Numerical Analysis,  Numerical Approximations of PDEs

Recommended courses

Sobolev spaces and elliptic equations,

Important concepts to start the course

• Basic knowledge of functional analysis: Banach and Hilbert spaces, L^p spaces.
• Some knowledge on theory of PDEs: classical and weak solutions, existence and uniqueness.
• Basic concepts in numerical analysis: stability, convergence, condition number, solution of linear systems, quadrature formulae, finite difference formulae, polynomial interpolation.
• Basic information on finite element theory for elliptic problems 

Learning Outcomes

By the end of the course, the student must be able to:

• Identify features of a PDE relevant for the selection and performance of a numerical algorithm.
• Assess / Evaluate numerical methods in light of the theoretical results.
• Implement numerical methods for saddle point problems
• Choose an appropriate method to solve a given differential problem
• Prove convergence of a discretisation scheme

Transversal skills

• Write a scientific or technical report.

Teaching methods

Ex cathedra lectures, exercises in the classroom and computer lab sessions.

Expected student activities

• Attendance of lectures.
• Completing exercises.
• Solving problems with an academic software as Free FEM ++

Assessment methods

Oral exams and evaluation of the report of a mini-project.

Supervision

Resources

Bibliography

• S.C. Brenner, L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer 2007.
• A. Ern, J-L. Guermond, Theory and Practice of Finite Elements. Springer 2004.
• D. Boffi, F. Brezzi, M. Fortin Mixed Finite elements and Applications, Springer Verlag. 2013. 

Ressources en bibliothèque

• The Mathematical Theory of Finite Element Methods / S.C. Brenner & L.R. Scott
• Theory and Practice of Finite Elements / A. Ern & J-L. Guermond
• Mixed Finite elements and Applications / D. Boffi, F. Brezzi & M. Fortin

Notes/Handbook

Notes for each lectures will be provided every week.