École Doctorale Mathématiques et STIC

Cours Ludovic Chamoin du 27 novembre au 11 décembre 2014

ERROR ESTIMATION AND ADAPTIVE TECHNIQUES IN NUMERICAL SIMULATIONS

Ludovic Chamoin, LMT ENS Cachan

  • Jeudi 27 novembre 2014 et jeudi 11 décembre 2014
  • De 14h00 à 16h00, Salle B211, Aile Becquerel, Bâtiment Coriolis, Cité Descartes

Résumé :

A large number of environmental and physical phenomena is described by PDEs. Unfortunately, it is usually not possible to find the analytical exact solutions of these equations, so that numerical methods are employed as simulation tools. As these only deliver approximate solutions which are defined in some finite-dimensional spaces, there are two extremely important questions to adress :

  • How large is the overall error between exact and approximate solutions ?
  • Where is this error localized ?

Answers to these two questions may be crucial in many engineering activities (construction of bridges or planes, weather forecast, advanced health care techniques, financial predictions,...) as a decision is often taken on the basis of the numerical simulation result. Taking this reflection one step further, the ultimate goal in scientific computing is to design algorithms such that a given precision is attained at the end of the simulation white minimizing the amount of computational work. This objective of the course is therefore to explain basic methods that have been designed since the 80s in order to perform reliable numerical simulations. Focusing on the Finite Element Method (FEM) context, we review the construction of error estimators that enable :

  • (I) to give a fully computable upper bound on the overall error (error control)
  • (II) to distinguish between the different error sources (error localization)
  • (III) to adjust optimally the computation parameters during the simulation (adaptivity). We also present state-of-the-art research developments such as goaloriented error estimation, applicability to complex problems, and modeling error estimation.