## Analisis Numerico de Ecuaciones en Derivadas parciales

### Curso MA53L-1 (DIM)

 Semestre de Primavera 2008 Course instructors: 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

 Lecture: Tue-Thu 10:15 - 11:45 room: F9 Computer exps: Fri 14:30 - 17:45 room: B214

### 1. Syllabus

MA53L-1 is an introduction to mathematical and numerical methods for the resolution of partial differential equations: finite differences, finite elements and finite volume methods. We will cover basic theoretical results, and we will apply these results in numerical analysis projects.
Details of the Syllabus (PDF file)
• #### Part I: Finite differences method

• 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

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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).
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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.