ADVANCED PDE MODELS

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 30) + practical homework

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

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