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This course provides a hands-on introduction to the foundations and implementation of numerical algorithms to solve various PDE problems in 1D/2D/3D domains with the finite element method. Each session contains a short theoretical presentation of the numerical method, directly followed by its implementation in the form of computer programs. The free software FreeFem++ (www.freefem.org) offers an ideal framework to start a scientific computing activity: all the technicalities of the finite-element implementation are hidden from the user and the syntax is very close to mathematical formulations. For advanced users, FreeFem++ allows to focus on numerical algorithms to solve complex problems. A graduate-level presentation of theoretical and technical aspects will enable a fast progress in mastering numerical methods and the software: from linear to non-linear PDEs, from 2D to 3D problems, from sequential to parallel computing. A large variety of PDEs will be addressed: heat and wave equation, problems arising in linear elasticity, Stokes and Navier-Stokes-Boussinesq systems, etc.

The sessions of the second week will be followed by short technical presentations of complex problems solved with FreeFem++: Navier-Stokes-Boussinesq equations for phase-change materials, Schrodinger equation for Bose-Einstein condensates, fluid-structure interaction.

Instructors: Prof I. Danaila (University of Rouen, France), Prof F. Hecht (University Pierre et Marie Curie, France)

Features: Theory and Examples, Computer Implementation. 

Prerequisites: basic programming skills, elementary calculus. Participants are advised to bring their laptops.

Week 1: Introduction and basic notions: solving 2D linear and non-linear PDEs

Week 2: Advanced notions: non-linear PDEs, 3D domains, parallel computing