Simulación numérica de problemas de EDP sobre dominios complejos

Curso electivo y Seminario avanzado de Matematicas I: MA691 (DIM) - CC60X (DCC)

March 2008 Course instructors: Pascal Frey Maya de Buhan Office: 622, CMM-DIM Office hours: MWF at 2pm or by appointment Phone: 978 4802 (office) if you send an email, please put MA691 or CC60X in the subject

0. Schedule

1. Syllabus

• Part I: Mathematical requisites

1. The concept of distributions
• Definitions
• Main properties
• Classical approach
• The concept of quasi derivative
2. Hilbert and Sobolev spaces
• Hilbert spaces
• Hilbert and boundary-value problems
• Sobolev spaces
3. Partial differential equations
• Partial differential equations
• Analysis of PDEs
• Fundamental examples
• General properties
4. Weak formulation of elliptic problems
• Dirichlet problem
• Neumann problem
• Abstract variational problems
• Variational approximations of elliptic problems
• Application to boundary value problems in R
5. Numerical linear algebra

• Part II: Numerical methods

1. The finite difference method
• Finite difference approximations
• Finite difference formulation for a 1d problem
2. The finite element approximation
• General principles
• The one dimensional case
• Triangular finite elements in higher dimensions
• The finite element method for the Stokes problem
3. A posteriori error estimate
• Motivations
• A Laplacian model
• The Stokes problem
• Numerical considerations
4. Mesh generation and approximation
• Terminology and definitions
• Mesh generation
• Mesh adaptation

• Part III: Post processing techniques

1. An introduction to scientific visualization
• Elements of analytical and projective geometry
• Illumination models
• Vector fields

• Appendices

  1. Metric and Banach spaces
  2. Measure theory and Lebesgue integration

2. References

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

3. Grading policy

4. Homework

• 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
  1. 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).
  2. J.A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Natl., Acad., Sci., USA, 93, 1591-1595, (1996).
  3. B. Bourdin, Image segmentation with a finite element method, M2AN, 33(2), 229-244, (1999)
  4. P.O. Persson and G. Strang, A simple mesh generator in Matlab, SIAM Review, 46(2), 329-345, (2004).
• Homework assignments
  1. Around the Stokes problem : hw1.pdf

4. Projects

• 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
1. week 1: TP0.pdf (introduction to Scilab)
2. week 2: TP1.pdf (finite difference method I, Scilab)
3. week 3: TP2.pdf (finite difference method II, Scilab)
4. weeks 4+5: TP3.pdf (finite element method I, Scilab)
5. weeks 6+7: TP4.pdf (finite element method II, FreeFem++); the program Poisson.edp
6. week 8: research papers presentation
7. 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.
8. week 12: presentation of the research projects.