Ensimag Rubrique Formation 2022

Advanced PDE models - 4MMMVAM6

  • Number of hours

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


    ECTS 3.0


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.




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.


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


Written exam (2 h) + practical homework


The exam is given in english only FR


The course exists in the following branches:

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

Additional Information

Course ID : 4MMMVAM6
Course language(s): FR

The course is attached to the following structures:

You can find this course among all other courses.


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