Program (detailed contents)
Finite Elements Methods
FE BASES
* Recalls. Analysis of elliptic PDEs : weak solution vs classical solution, Sobolev spaces, Lax-Milgram theory, a-priori estimations. Boundary conditions. Energy minimization. (2 Lectures, 1Tutorial)
* Modelling with a FE software (FreeFEM++): classical models, one real-like problem.
1 Lab Work with FreeFEM++
* Finite Element Method (FEM) principles in multi-D: discretisation, approximation, Error analysis (a-priori). (1Lecture, 1Tutorial).
* In practice: data structures, implementation (assembly algorithm). Convergence curves (code validation from exact solutions). (2 Lectures, 1 Tutorial, 2 Lab Works with Python)
DISCONTINUOUS GALERKIN (DG) METHODS
* Discontinuous Galerkin Method for Diffusive and transport problem: Broken spaces, jumps and interior penalty techniques, stability, introduction to non-conforming analysis.
(3 Lectures, 1 Tutorial, 1 Lab Work)
ADVANCED FE METHODS
* The advection-diffusion equation:
– Analysis and naïve discretization. (1 Lecture, 1 Tutorial)
– Transport term: FE stabilisation (SD, SUPG). (1 Lecture, 1 Tutorial)
* Non-linear models: linearization (1 Lecture, 2 Tutorials, 3 Lab Works)
(Including: differential calculus / Riez-Frechet theorem)+ help to the marked practical).
* Unsteady models: semi-discretisation. 1 Lecture, 1 Lab Work
PDE MODEL REDUCTIONS
* Model Reduction for parametrized PDEs, offline-online strategies.
Linear PDEs: POD method. (1,5 Lecture, 1 Lab Work)
Non-linear PDEs: Hybrid approaches POD – Neural Networks. (1.5 Lecture, 1 Lab Work)
