# Coursebooks

## Numerical approximation of PDEs

Buffa Annalisa

English

#### Summary

The course pertains to the derivation, theoretical analysis and implementation 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: stability, convergence, a priori error estimates in different norms, implementation aspects.

#### Keywords

Partial differential equations, finite difference method, finite element method, Galerkin approximation, stability and 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:
• 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.

#### 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 simple 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

### Reference week

MoTuWeThFr
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
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