# Fiches de cours

## Numerical methods for saddle point problems

Buffa Annalisa

English

#### 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 fluid dynamics, elasticity and electromagnetic problems.

#### Content

- Minimization of convex functionals (energies) under linear constraints and their interpretation as saddle point problems. Wellposedness and inf-sup conditions.

- Finite element approximation of saddle point problems, discrete inf-sup conditions, stability and approximation estimates

- Finite elements for Stokes flows, (quasi-)incompressible linear elasticity, and Darcy flows

- Compatible discretisations of differential forms and of Maxwell equations

#### Keywords

Finite element methods, Galerkin approximation, mixed finite elements, Darcy flows, incompressible fluids and linear elasticity, Maxwell equations, discrete differential forms.

#### Learning Prerequisites

##### Required courses

Analysis I II III IV, Numerical Analysis, Advanced numerical analysis, Sobolev spaces and elliptic equations, Numerical Approximations of PDEs I

##### Recommended courses

Functional analysis I, measure and integration, Programming

##### Important concepts to start the course

- Basic knowledge of functional analysis, Banach and Hilbert spaces, L^p spaces

- Some knowledge on the theory of elliptic PDEs, weak solutions, existence and uniqueness

- Basic concepts in numerical analysis: stability, convergence, condition number, solution of linear systems, quadrature formulae, polynomial interpolation.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Choose an appropriate discretisation scheme to solve a specific PDEs
• Analyse numerical errors
• Interpret results of a computation in light of theory
• Prove theoretical properties of discretisation schemes
• Propose a theoretical and numerical solution to a mini-project on a topic going beyond the material of the course
• Formalise the solution of a mini-project in a scientific report

#### Transversal skills

• Use a work methodology appropriate to the task.
• Write a scientific or technical report.
• Use both general and domain specific IT resources and tools

#### Teaching methods

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

#### Expected student activities

- Attendance of lectures

- Completing exercises

- Solving problems on the computer

- Work out a small project and write a technical report

#### Assessment methods

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

#### Supervision

 Office hours No Assistants Yes

#### Resources

##### Bibliography

- D. Boffi, F. Brezzi, M. Fortin Mixed Finite Element Methods and Applications, Springer Series in Computatioanl mathematics, 2013.

- P. Monk, Finite Element Methods for Maxwell Equations, Oxford University press, 2003

- A. Ern, J-L. Guermond, Theory and Practise of Finite Elements, Springer 2004.

##### Notes/Handbook

Notes for each lectures will be provided every week.

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand