Numerics III

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General Information

Description

Mathematical modelling of spatial or spatial/temporal phenomena such as porous medium flow, solidification of melts, weather prediction, etc. typically leads to partial differential equations (PDEs). After some remarks on the modelling with and classification of PDEs, the course will concentrate on elliptic problems. Starting with a brief introduction to the classical theory (existence and uniqueness of solutions, Green's functions) and assiciated difference methods we will mainly focus on weak solutions and their approximation by finite element methods. Adaptivity and multigrid methods will be also discussed.

Target Audience

Students of Bachelor and Master courses in Mathematics and of BMS.

Prerequisites

Basic knowledge in calculus (Analysis I-III) and Numerical Analysis (Numerik I). Some knowledge in Functional Analysis will help but is not necessary.

Registration

All participants should register at the KVV, so that we know who is participating. The overall number of participants is also necessary to justify the equipment of the course. In addition, depending on your program of study, you have to register in the Campus Management.

Exercises and Criteria for a Certificate

Tutorial & Exercises

• The tutorial offers the possibility to discuss and better understand the presented material and exercises. Furthermore solutions of exercises are presented by the students.
• Each week a sheet with exercises will be made available electronically on this web page (see below).
• The exercises are intended to be solved by teams .
• The exercises consist of theoretical and numerical problems that should be solved by the teams of 3-4 people.
• Programming exercises should be solved using Matlab. Octave can also be used if Matlab is not available. Both types of exercises are rated separately.
• The solutions of the numerical problems and of theoretical problems should be delivered by email to xingjianz@gmail.com. Please use LateX/Word/Scans(Smartphone apps) for theoretical problems. A complete solution for a numerical experiment (programming exercises) consists of a running Matlab/Octave code, a program executing the required test runs, and protocols of the execution of these test runs. Delivering a correct and running code without knowing what is going on in the code will be regarded and rated as attempt of deception.

Individual Exam

• There will be either a written or an oral video exam during July 6-10. The decision will be taken in due course depending on the given conditions.

Criteria for the Certificate

Necessary and sufficient for a certificate are:

• exam: passing the exam
• active participation: presentation of solutions in the tutorial, at least 60 % of both, theoretical and programming points.
• bullet participation: (virtual or physical) presence in lectures and tutorials (not officially monitored)

The grade of final certificate is the grade of the individual exam.

Exercises

• Exercise 1
• Exercise 2
• Exercise 3
• Exercise 4
• Exercise 5
• Exercise 6
• Exercise 7
• Exercise 8
• Exercise 9
• Exercise 10 (provided sources: basis.m, quadrature.m, uniform_grid.m, generate_grid.m, circle.mat, assemble_mass.m, assemble_stiff.m)

Lecture Notes

The lecture notes are available here.

Accompanying Material

Jinchao Xu, Iterative methods by space decomposition and subspace correction

Harry Yserentant, Old and New Convergence Proofs for Multigrid Methods

A short survey about the Dirichlet Problem can be found here.

Notes from June 22 can be found here.

A survey about hierarchical error estimates can be found here.

Matlab

Here you can find an introduction to Matlab (in German, sorry).

Literature

• D. Braess: Finite Elemente. Springer, 3rd edition (2002)
• P. Knabner, L. Angermann: Numerik partieller Differentialgleichungen. Springer (2000)
• P. Deuflhard, M. Weiser: Numerische Mathematik 3. De Gruyter (2011)
• J. Wloka: Partielle Differentialgleichungen. Teubner (1982)
• D. Werner: Funktionalanalysis. Springer, Berlin (2000)
• H. Alt: Lineare Funktionalanalysis. Sprinter, 6th edition (2012)
• W. Rudin: Functional Analysis. McGraw-Hill, 2nd edition (1991)
• L. Evans: Partial Differential Equations. AMS, 19th volume (1998)
• F. John: Partial Differential Equations. Springer (1982)
• M. Renardy, R. C. Rogers: An introduction to partial differential equations. Springer, 2nd volume (2004)
• A. Quarteroni, R. Sacco, F. Saleri: Numerische Mathematik 2. Springer (2002)
• P. A. Raviart, J. M. Thomas: Introduction à l'analyse numérique des équations aux dérivées partielles. Dunod (1998)

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