Analisis Numerico de Ecuaciones en Derivadas parciales

Curso MA53L-1 (DIM)

Semestre de Primavera 2008 Course instructors: Axel Osses 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 MA53L-1 in the subject

0. Schedule

1. Syllabus

• Part I: Finite differences method

1. Problem model: advection-diffusion
• Notion of numerical scheme, principle of maximum, existence and uniqueness
• Approximation error estimate, variational formulation of the continuous scheme, Lax-Milgram lemma, consistency error
2. Problem model: transport equation
• Analysis of various numerical schemes, consistency and approximation error
• Stability and convergence
3. Problem model: 2d diffusion
• Finite difference discretization
• Existence and uniqueness
• Convergence

• Part II: Finite elements method

1. Problem model: non homogeneous diffusion
• Functional framework
• Céa's Lemma
2. Convergence of the finite elements method
• Generalization of Céa's lemma
• Deny-Lions lemma
• Error estimate in H norms

• Part III: Finite volumes method

1. Discrete space and discrete problem
2. Uniqueness, convergence and error estimate

2. References

1. G. Allaire, Analyse numérique et optimisation, Editions de l’Ecole Polytechnique, Paris, 2006.
2. I. Babuska, A. K. Aziz, Foundations of the finite element method, In A. K. Aziz, editor, The Mathematics Foundations of the Finite Element Method with Applications to Partial Differential Equations, 3–362, Academic Press, New York, 1972.
3. C. Bernardi, Y. Maday, F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques et Applications, vol. 45, Springer, Paris, 2004.
4. S. Brenner, L.-R. Scott, The mathematical theory of finite element method, Springer, 2000.
5. J.-H. Bramble, S.-R. Hilbert, Estimations of linear functionals on Sobolev spaces with application to Fourier transform and spline interpolation, SIAM J. Numer. Anal., 25(6), 1237–1271, 1970.
6. H. Brezis, Analyse fonctionnelle: théorie et applications, Dunod, (2005).
7. Z. Cai, On the finite volume element method, Numer. Math. 58, 713–735, 1991.
8. P.G. Ciarlet, The Finite Element Method, North Holland, (1978).
9. R. Dautray, J.-L. Lions, Analyse mathématique et calcul numérique, Tomes 3, 4, Masson, 1987.
10. A. Ern and J.L. Guermond, Theory and practice of finite elements, vol. 159 of Applied Mathematical Series, Springer-Verlag, New York, (2004).
11. L.C. Evans, Partial differential equations, AMS, (2002).
12. R. Eymard, T. Gallouet, R. Herbin, The Finite Volume Method, Handbook of Numerical Analysis, Ph. Ciarlet J.L. Lions eds, North Holland, 2000, 715-1022
13. M. Krizek, P. Neittaanm¨aki, Finite element approximation of Variational Problems and Applications, London, Longman, 1990.
14. J. Necas, Les méthodes directes dans la théorie des équations elliptiques, Academia, Prague, 1967.
15. K.W. Morton and D. Mayers, Numerical solution of partial differential equations, 2nd edition, Cambridge University Press, (2005).
16. A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer, (2000).
17. P.-A. Raviart, J.-M. Thomas, Introduction à l’analyse numérique des équations aux derivées partielles, Masson, 1998.
18. P. Solin, Partial differential equations and the finite element method, Wiley-Interscience, (2005).
19. A. Tveito and R. Winther, Introduction to partial differential equations: a computational approach, Texts in Applied Mathematics, 29, Springer, 1998.
20. O. Zienkiewicz, J.-Z. Zhu, R-L. Taylor, The Finite Element Method: Its Basis and Fundamentals, Elsevier, Paris, 2005.

3. Grading policy

4. Homework

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.