Introduction to finite elements

General information:     See the syllabus
                                                  The course will follow the book by S.C. Brenner and L.R. Scott : The Mathematical theory of
finite element methods, springer-verlag 1994.
                                                  The scilab software can be downloaded here 
                                                  Classes start Feb. 9, 2009

Schedule :
Course : Friday 8-10, Lab. Dieudonné; P-E Jabin
Exercises : Monday 15:15-17:15, EPU; A. Sangam
                   Friday    10:15-12:15, Lab. Dieudonné; V. Dolean


Office hours :     P-E Jabin : Thursday   15:15-17:15

Homework : 4 pages max. on chapt. 1 and chapt. 2 (up to 2.4 included). Due friday 13th march.

Exercises :  1st class and correction,  2nd class and correction, 3rd class and correction

Introduction to scilab  and computer implementation :  Introduction      1st class

Short Program : it may be slighty modified/updated along the course. The section and chapter numbers are always given
with respect to Brenner-Scott.

1st class :
0. Quick presentation of the course.
1. Derivation of diffusion or heat equation from discrete jump processes : first the simple brownian case and then non homogeneous diffusion with sources.
2. Elliptic problems introduced as stationary solutions or asymptotic in time of  diffusion problems. Formal proof based on the energy dissipation for the asymptotic in time.
3. Other examples without full derivation : Black and Scholes for finance, fluid mechanics (Navier-Stokes only).

2nd Class : Chapter 0 of Brenner-Scott, Introduction to finite elements
1. section 0.1 : Weak formulation
2. section 0.2 : Ritz-Galerkin approximation
3. section 0.4 : The finite element method
4. section 0.6 : Computer implementation

3rd class : End of chapter 0, Error Estimates
1. section 0.6 : the sequel of the 2nd class
1. section 0.5 : Comparison with other methods
2. section 0.3 and 0.7 : Error estimates

4th class : Chapter 3, Construction of a finite element space
1. section 0.8 : Weighted norm estimates
2. section 3.1 : definition of a finite element
3. section 3.2 : triangular finite elements


5th class : End of chapter 3
1. section 3.3 : the interpolant
2. section 3.5 : rectangular elements and some words of 3.6.
3. Crash course on numerical analysis for matrices : 1st part.

6th class
1. Crash course on numerical analysis for matrices : 2nd part.
2. Summary of chapt. 1

7th class :  
1.  Summary of standard Hilbert theory 2.1 to 2.4.
2. Chapter 2, from 2.5.

8th class :
1.  End of chapter 2.
2. Summary of 4.1, 4.2
3. Section 4.3
4. Test

9th class:
1. End of section 4.3
2. Part of 4.4
3. Section 5.1

10th class:
Chapter 5 at least  the end of 5.1,  and then 5.3, 5.4 solving Poisson equation and if there is more time : 5.2 Neumann problem, 5.5 regularity estimates for Poisson.