 # Finite Element Methods & Model Reductions

## Presentation

 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.

## Objectives

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...