March 2008
if you send an email, please put MA691 or CC60X in the subject |

Lecture: | Mon | 14:30 - 17:45 | room: B213 (2nd floor) |

Computer exps: | Fri | 16:15 - 17:45 |

#### Part I: Mathematical requisites

- The concept of distributions
- Definitions
- Main properties
- Classical approach
- The concept of quasi derivative
- Hilbert and Sobolev spaces
- Hilbert spaces
- Hilbert and boundary-value problems
- Sobolev spaces
- Partial differential equations
- Partial differential equations
- Analysis of PDEs
- Fundamental examples
- General properties
- Weak formulation of elliptic problems
- Dirichlet problem
- Neumann problem
- Abstract variational problems
- Variational approximations of elliptic problems
- Application to boundary value problems in R
- Numerical linear algebra
#### Part II: Numerical methods

- The finite difference method
- Finite difference approximations
- Finite difference formulation for a 1d problem
- The finite element approximation
- General principles
- The one dimensional case
- Triangular finite elements in higher dimensions
- The finite element method for the Stokes problem
- A posteriori error estimate
- Motivations
- A Laplacian model
- The Stokes problem
- Numerical considerations
- Mesh generation and approximation
- Terminology and definitions
- Mesh generation
- Mesh adaptation
#### Part III: Post processing techniques

- An introduction to scientific visualization
- Elements of analytical and projective geometry
- Illumination models
- Vector fields
#### Appendices

- D. Braess,
*Finite elements*, Cambridge University Press, (1997). - H. Brezis,
*Analyse fonctionnelle: théorie et applications*, Dunod, (2005). - P.G. Ciarlet,
*The Finite Element Method*, North Holland, (1978). - A. Ern and J.L. Guermond,
*Theory and practice of finite elements*, vol. 159 of Applied Mathematical Series, Springer-Verlag, New York, (2004). - L.C. Evans,
*Partial differential equations*, AMS, (2002). - P. Frey and P.L. George,
*Mesh generation: application to finite elements*, 1st edition, Hermès Science, (2000). - V. Girault and P.A. Raviart,
*Finite element methods for Navier-Stokes equations*, Springer Series in Computational Mathematics, 5, Springer, (1986). - G.H. Golub and F. Van Loan,
*Matrix computations*, 3rd edition, The Johns Hopkins University Press, (1993). - R.J. LeVeque,
*Numerical methods for conservation laws*, Birkhauser, Basel, (1992). - K.W. Morton and D. Mayers,
*Numerical solution of partial differential equations*, 2nd edition, Cambridge University Press, (2005). - J.T. Oden and J.N. Reddy,
*An introduction to the mathematical theory of finite element methods*n vol 2, Handbook of numerical analysis, North Holland, (1991). - A. Quarteroni, R. Sacco, F. Saleri,
*Numerical Mathematics*, Springer, (2000). - W. Rudin,
*Functional analysis*, 2nd edition, Mc Graw Hill, (1991). - P. Solin,
*Partial differential equations and the finite element method*, Wiley-Interscience, (2005). - K. Yosida,
*Functional analysis*, 6th edition, Springer-Verlag, (1980).

Tentative grading scheme:

where

- Group assignments

Research papers will be assigned to small teams to make presentations to the class. The performance will be factored into the homework/midterm exam grades. - Research papers
- V. Girault, H. Lopez and B. Maury, Energy balance of a 2d model for lubricated oil transportation in a pipe,
*Divulgaciones Matematicas*,**16**, 87-105, (2008). - J.A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Natl., Acad., Sci., USA,
**93**, 1591-1595, (1996). - B. Bourdin, Image segmentation with a finite element method, M2AN,
**33**(2), 229-244, (1999) - P.O. Persson and G. Strang, A simple mesh generator in Matlab, SIAM Review,
**46**(2), 329-345, (2004).

- V. Girault, H. Lopez and B. Maury, Energy balance of a 2d model for lubricated oil transportation in a pipe,
- Homework assignments
- Around the Stokes problem : hw1.pdf

- Numerical experiments in finite differences are conducted in Scilab or Matlab or Octave. Finite element simulations will be implemented using FreeFEM++. The documentation on FreeFEM++ can be found at freefem++doc.pdf. The project will be implemented in C language.
- Project reports

The project will involve small written project reports. Each report must contain equations, tables and graphics. We suggest you use Latex to typeset it.

This "Simplified Introduction" is a helpful reference guide. - Numerical experiments
- week 1: TP0.pdf (introduction to Scilab)
- week 2: TP1.pdf (finite difference method I, Scilab)
- week 3: TP2.pdf (finite difference method II, Scilab)
- weeks 4+5: TP3.pdf (finite element method I, Scilab)
- weeks 6+7: TP4.pdf (finite element method II, FreeFem++); the program Poisson.edp
- week 8: research papers presentation
- weeks 9-11: research project (writing a finite element code in C)

**solving a linear elasticity problem in two dimensions.**

- load the source files (contain mesh examples)
- read the project description document
- write the matrix assembly routine
- compile the source code on your machine
- visualize the results using medit.

- week 12: presentation of the research projects.

Updated 2008-04-03 12:11 CLT