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# Coursebooks 2017-2018

## Numerical approximation of PDE's I

#### MATH-451

#### Lecturer(s) :

Nobile Fabio#### Language:

English

#### Summary

The aim of the course is to give a theoretical and practical knowledge of finite difference and finite element methods for the numerical approximation of partial differential equations in one or more dimensions.#### Content

- Finite difference methods for elliptic, parabolic and hyperbolic equations; stability and convergence analysis
- Linear elliptic problems: weak form, well-posedness, Galerkin approximation
- Finite element approximation in two and three dimensions: stability, convergence, a-priori error estimates in different norms, implementation aspects
- Transport dominated problems and stabilization techniques

#### Keywords

Partial Differential Equations, Finite difference method, Finite element method, Galerkin approximation, convergence analysis.

#### Learning Prerequisites

##### Required courses

Analysis I-II-III-IV, Numerical analysis

##### Recommended courses

Functional Analysis I, Introduction aux équations aux dérivées partielles, Measure and Integration, Programming

##### Important concepts to start the course

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

#### Learning Outcomes

By the end of the course, the student must be able to:- Choose an appropriate discretization scheme to solve a specific PDE
- Analyze numerical errors
- Interpret results of a computation in the light of theory
- Prove theoretical properties of discretization schemes
- Solve a PDE using available software
- State theoretical properties of PDEs and corresponding discretization schemes
- Describe discretization methods for PDEs

#### Transversal skills

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

#### Teaching methods

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

#### Expected student activities

- Attendance of lectures
- Completing exercicies
- Solving simple problems on the computer

#### Assessment methods

written exam. The exam may involve the use of a computer.

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 |

#### Resources

##### Virtual desktop infrastructure (VDI)

Yes

##### Bibliography

- A.Quarteroni, Numerical Models for Differential Problems, Springer, 2009
- S.C. Brenner, L.R. Scott The Mathematical Theory of Finite Element Methods, Springer, 3rd ed, 2007
- A. Ern, J-L. Guermond, Theory and Practice of Finite Elements, Springer, 2004
- Lecture notes by the teacher

##### Ressources en bibliothèque

- The Mathematical Theory of Finite Element Methods / Brenner
- Numerical Models for Differential Problems / Quarteroni
- Theory and Practice of Finite Elements / Ern

##### Moodle Link

#### Prerequisite for

Numerical Approximation of Partial Differential Equations II

### In the programs

**Semester**Spring**Exam form**Written**Credits**

5**Subject examined**

Numerical approximation of PDE's I**Lecture**

2 Hour(s) per week x 14 weeks**Exercises**

2 Hour(s) per week x 14 weeks

**Semester**Spring**Exam form**Written**Credits**

5**Subject examined**

Numerical approximation of PDE's I**Lecture**

2 Hour(s) per week x 14 weeks**Exercises**

2 Hour(s) per week x 14 weeks

### Reference week

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21-22 |

Mo | Tu | We | Th | Fr | |
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21-22 |

Under construction

Lecture

Exercise, TP

Project, other

### legend

- Autumn semester
- Winter sessions
- Spring semester
- Summer sessions

- Lecture in French
- Lecture in English
- Lecture in German