MATH-451 / 5 crédits

Enseignant: Buffa Annalisa

Langue: Anglais

## Summary

The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions.

## Content

• Linear elliptic problems: weak form, well-posedness, Galerkin approximation.
• Finite element approximation: stability, convergence, a priori error estimates in different norms, implementation aspects.
• Extensions to parabolic and hyperbolic problems

The numerical methods proposed along the class will be implemented during the exercise sessions. The implementation may be in MatLab or python.

## Keywords

Partial differential equations, finite element method, Galerkin approximation, stability and convergence analysis.

## Required courses

Analysis I-II-III-IV, Numerical analysis.

## Recommended courses

Functional analysis I, Measure and integration, Espaces de Sobolev et équations elliptiques, Advanced numerical analysis, Programming.

## Important concepts to start the course

• Basic knowledge of functional analysis: Banach and Hilbert spaces, L^p spaces.
• Some knowledge on theory of PDEs: classical and weak solutions, existence and uniqueness.
• Basic concepts in numerical analysis: stability, convergence, condition number, solution of linear systems, quadrature formulae, finite difference formulae, polynomial interpolation.

## Learning Outcomes

By the end of the course, the student must be able to:

• Identify features of a PDE relevant for the selection and performance of a numerical algorithm.
• Assess / Evaluate numerical methods in light of the theoretical results.
• Implement fundamental numerical methods for the solution of PDEs.
• Choose an appropriate discretization scheme to solve a specific PDE.
• Analyze numerical errors and stability properties.
• Interpret results of a computation in the light of theory.
• Prove theoretical properties of discretization schemes.
• State theoretical properties of PDEs and corresponding discretization schemes.
• Analyze numerical errors and stability properties.

## Transversal skills

• Use a work methodology appropriate to the task.
• Write a scientific or technical report.
• Use both general and domain specific IT resources and tools

## Teaching methods

Ex cathedra lectures, exercises in the classroom and computer lab sessions.

## Expected student activities

• Attendance of lectures.
• Completing exercises.
• Solving problems on the computer.

## Assessment methods

85% Written exam. The exam may involve the use of a computer.

15% Project involving both computer simulation and theoretical developements.

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

## Supervision

 Office hours Yes Assistants Yes Forum No

No

## Prerequisite for

Numerical approximation of PDEs II, Numerical methods for conservation laws, Numerical methods for fluids, structures & electromagnetics

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Numerical approximation of PDEs
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel

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