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Finite Element Methods & Model Reductions


Program (detailed contents) :


  • *Analysis of elliptic PDEs : weak solution vs classical solution, Sobolev spaces, Lax-Milgram theory, a-priori estimations. Boundary conditions. Energy minimization.
  • Modelling with a FE software (eg FreeFEM++) : classical models, 1 real-like problem.
  • Finite Element Method (FEM) principles: discretisation, approximation, data structures, implementation. Error analysis (a-priori). Convergence curves.


Advective term: stabilisation (eg SUPG).


  • Practical, Programming (Python): assembly algorithm ; code assessments, scheme accuracy, error estimator.
  • FEM extensions:

Unsteady models: semi-discretisation.

Non-linear models: linearization.

A-posteriori estimations, mesh refinement.


  • Model Reduction: offline-online strategy.

POD basis reduction.


  • Domain Decomposition Method (DDM) : method(s), practical (FreeFEM++).


Organisation :


18 classes + 12 tutorials-exercises + 14 lab. Tutorials (Python-FEniCS, FreeFEM++).



Main difficulties for students :


Understand the large range of required knowledge: from the physical modeling to computational aspects via the mathematical & numerical analysis.


At the end of this module, the student will have understood and be able to explain (main concepts) :


-          How to write the weak (variational) form of the classical PDE models (with the corresponding energy minimization, symmetric case).


-          Write and implement a finite element scheme (for linear and non-linear models).


-          Employ Finite Element libraries (eg. FEniCS - Python and FreeFem++).


-          Set up an offline – online strategy for real time computaions (POD reduction for linear PDEs)


-          Decompose large problems on multiple processors using domain decomposition methods (with preconditionners).


-          Simulate classical phenomena (diffusive, convective, linear – non linear etc).

Needed prerequisite

Fundamentals of PDE equations I4MMNP71

Numerical analysis I3MIMT11

Matrix computation I3MIMT31

Basic numerical methods-nbalaysis.

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...