# Advanced PDE models - 4MMMVAM6

• #### Number of hours

• Lectures 16.5
• Projects -
• Tutorials 16.5
• Internship -
• Laboratory works -
• Written tests -

ECTS 3.0

## Goal(s)

To deepen knowledges on mathematical modeling with PDEs and their numerical resolution. We present mainly finite element methods whose theoretical bases, numerical schemes and programming aspects are studied.

Responsible(s)

Clement JOURDANA

## Content(s)

I - Introduction to modeling through some examples: Heat transfer (1D/2D, Steady/Transcient), transport, elasticity (Lamé), fluid (Stokes), fluid-structure coupling (flow around an elastic obstacle). Comments on specific mathematical caveats of above problems.
II - Boundary value problems 1D. Weak forms.
III - Steady-state models / elliptic equations
Variationnal context. Symmetrical case and minization. Green formulaes.
IV - Finite elements method: basis functions, algorithms, implementation, a-priori estimates. Transport term, stabilization. Non linear case : linearization.
III - Unsteady models / Parabolic equations
Time scheme, splitting methods. FD-FE schemes.
IV - Possible extensions: ALE methods for fluid-structure models, models reduction,
Semi-lagrangian approach (characteristics), A-posteriori estimates, mesh refinement
Discontinuous-Galerkin methods. Some of these extensions could be part of the practical homework.

Prerequisites

2nd year: Models of PDEs or Advacnce numerical methods; 1st year: numerical methods, mathematical analysis .

Test

Written exam (2 h) + practical homework

N1=(2*E1+P)/3
N2=max(N1,(2*E2+P)/3)

The exam is given in english only

Calendar

The course exists in the following branches:

• Curriculum - Math. Modelling, Image & Simulation - Semester 8 (this course is given in english only )
see the course schedule for 2023-2024

Course ID : 4MMMVAM6
Course language(s):

The course is attached to the following structures:

You can find this course among all other courses.

Bibliography

G. ALLAIRE : Analyse numerique et optimisation . Edts de l’école polytechnique. Version PDF disponible sur la page de l'auteur.
A. QUARTERONI and A. VALLI : « Numerical approximation of PDEs », Springer.
A. Ern, J.-L. Guermond, Eléments finis : théorie, applications, mise en œuvre, Springer.
P.-A. RAVIART et J.-M. THOMAS : Introduction à l'analyse numérique des équations aux dérivées partielles, Coll. Mathématiques appliquées pour la Maîtrise, Dunod