1. Introduction: Types of PDE. Elliptic equation. Boundary conditions.
Heat equation, wave equation, convection-diffusion equation.
2. Fundamental notions: Convergence. Consistency. Stability. Lax’s theorem.
3. Finite difference schema for equations of evolution: Parabolic equation. Explicit and implicit methods.
Hyperbolic equation. Ecuatii parabolice. Explicit and implicit methods. Stability and convergence. Applications.
4. Finite element method: Lagrange finite element. Mesh of a regular domain. Families of triangulations.
Approximation of the solutions of elliptic equation.
5. Convergence of the finite element method: Cea’s Lemma. Condtions of convergence.
Interpolation Lagrange in Sobolev spaces. Error evaluation in Lagrange interpolation.
Error evaluations for finite element method.
6. Parabolic equation: Semi-discretization and total-discretization. Trapezoidal method.
Stability. Convergece.
7. Hyperbolic equation: Semi-discretization and total-discretization. Newmark method. Stability. Convergece.
Bibliography
1. H. Brezis: Analyse fonctionelle: Théorie et applications, Masson, Paris, 1983.
2. P.G. Ciarlet: Introduction á l’anayse numérique matricielle et à l’optimisation, Masson, Paris, 1988.
3. P.G. Ciarlet: The finite element method for eliptic problems, North-Holland, Amsterdam, 1978.
4. K. Eriksson, D. Estep, P. Hansbo si C. Johnson: Computational differential Equations,
Studentlitteratur, Lund, 1996.
5. P.A. Raviart and J.M. Thomas: Introduction à l’analyse numérique des équations aux dérives partielles,
Masson, Paris, 1983.
6. P. Rabier and J.M. Thomas: Exercices d’analyse numérique des équations aux dérivées partielles,
Masson, Paris, 1985.
7. J. Strickwerda: Finite difference schemes and partial differential equations, Pacific Grove, California, 1989.