# Fiches de cours 2018-2019

## Numerical approximation of PDE's I

Nobile Fabio

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; implementation aspects
• Linear elliptic problems: weak form, well-posedness, Galerkin approximation
• Finite element approximation in one, two and three dimensions: stability, convergence, a-priori error estimates in different norms, implementation aspects

#### 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, 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:
• 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

No

##### Bibliography

• D.F. Griffiths, J.W. Dold, D.J. Silvester, Essential partial differential equations. Springer 2015.
• S. Larsson, V. Thomée, Partial differential equations with numerical methods (Vol. 45). Springer Science & Business Media, 2008
• 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

#### Prerequisite for

Numerical Approximation of Partial Differential Equations II, Numerical methods for conservation laws, Numerical methods for saddle point problems

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand