Université Paris 13
MACS 2 - Advanced Numerical Analysis - 2

Professors : Caroline Japhet and Michel Kern

In scientific computing, complex physical phenomena can be formulated as systems of partial differential equations (PDE), for example the airflow around an aircraft, the evolution of a glacier or the temperature in your room. Most of these equations however can not be solved exactly by analytical tools. Their solution has to be approximated using numerical methods, involving the solving of linear systems of large size, that cannot be solved on one computer.
Krylov methods (e.g. GMRES), Preconditioning techniques, and Domain Decomposition Methods (DDM) are iterative methods commonly used for solving these large problems (with sparse matrices). The principle of DDM is to transform the problem (the PDE or the linear system) into a series of decoupled subproblems, of smaller size, which can be solved in parallel on several processors. The saving of time is then considerable. These methods are also widely used to couple different models (e.g. ocean and atmosphere models to predict cyclones).
The goal of this course is to study :
    - Krylov methods (Conjugate Gradient, GMRES) and Preconditioners
    - a class of DDM : the Schwarz and Optimized Schwarz methods.

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Handouts

Cours

Diapositives du cours, devoir et TD/TP corrigés : voir le site où a lieu le cours

References

  • M. Kern. Cours d’Analyse Numérique Avancée 1, MACS 2, 2020-2021.
  • F. Cuvelier. Cours d’Analyse Numérique Elémentaire, MACS 1, 2020.
  • M.J. Gander and L. Halpern. Méthodes de décomposition de domaines - Notions de base. In Encyclopédie
    des techniques de l’ingénieur, méthodes numériques. AF137, 2012.
  • P.-L. Lions. On the Schwarz alternating method. I. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Pé-
    riaux, editors, First International Symposium on Domain Decomposition Methods for Partial Differential
    Equations, pages 1–42. Philadelphia, PA, SIAM, 1988.
  • H. A. Schwarz. Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Natur-
    forschenden Gesellschaft in Zürich, 15 :272–286, May 1870.
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