Numerics III

News

• The results of the second exam can be found here.
• For your preparation for the second exam, the exercise sheet of the first exam is avaliable here.
• The post-exam review (Klausureinsicht) was on Tuesday, September 13th from 12:00 to 13:00 in the seminar room 126 in Arnimallee 6. Please bring an ID card with your picture on it.
• The results of the first exam can be found here.
• As an example for the strcuture of the upcoming exam you can have a look at the NumericsII exams from last semester.

Dates

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 a lot 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 of three members.
• The exercises consist of theoretical and numerical problems that should be solved by the teams.
• Programming exercises should be solved using Matlab (available at the students computer pool at the institute). Both types of exercises are rated separately.
• The solutions have to be finished before the tutorial two weeks after they were handed out.
• The solutions of the numerical problems should be delivered by email to tobias.kies@fu-berlin.de. Note that a complete solution for a numerical experiment consists of a running Matlab 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.

Exam

• There will be an exam at the end of the semester.

Criteria for the Certificate

Necessary and sufficient for a certificate are:

tba

Certificates are graded according to the result of the exam.

Exercises

• Exercise 1
• Exercise 2
• Exercise 3
• Exercise 4
• Exercise 5
• Exercise 6 (provided sources: createFDGrid.m)
• Exercise 7 (provided sources: mesh.mat)
• Exercise 8
• Exercise 9 (provided sources: basis.m, quadrature.m, uniform_grid.m, generate_grid.m, circle.mat, assemble_mass.m, assemble_stiff.m)
• Exercise 10 (provided sources: hierarchic_basis.m, refine_grid.m, assemble_P1operator.m)

Accompanying Material

Lecture Notes

The lecture notes are available 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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