Coursebooks

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

In the programs

• Financial engineering, 2018-2019, Master semester 2
• 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
• Financial engineering, 2018-2019, Master semester 4
• 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
• Mathematics, 2018-2019, Bachelor semester 6
• 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
• Computational science and Engineering, 2018-2019, Master semester 2
• 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
• Computational science and Engineering, 2018-2019, Master semester 4
• 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

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

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German