Numerical Methods for Partial Differential Equations

Ferienpräsenz: we offer consultation hours during the semester break where you can ask question about the lecture material. For the dates see  this document (in German). It also lists the dates for the Prüfungseinsicht.

Lecturer: Ralf Hiptmair

Assistants: Eivind Fonn, Laura Scarabosio, Cecilia Pagliantini, Fabian Müller

Lecture hours:

• Monday, 15-17, HG E 7
• Friday, 08-10, HG E 3

Exercise classes:

• CSE: Wednesday, 08-10, ML F 40 (Eivind Fonn, Laura Scarabosio)
• INFK: Wednesday, 13-15, HG F 26.5 (Fabian Lukas Müller)
• INFK: Thursday, 13-15, HG E 41 (Cecilia Pagliantini)

Consultation hours:

They will be established the first exercise class.

Please choose an exercise class here. (Will be online in the first week of the semester.)

Assignments

There will be 12 weekly assignments handed out every Monday (first time on February 25th). They should be handed in the following week before midnight on Monday in the labeled box at HG G 53.X.

Exercises are marked either as core or non-core problems. Core problems (about 2 per sheet) are supposed to be essential for following the course.

Correction of homework problems will be done on request (to be written on the problem sheet you hand in) for at most two problems per sheet.

Testat requirements: NONE.

For submitting your MATLAB code, please use the online submission interface and choose the course n.7.

Assignments, lecture notes and other handouts

Examination

3 hour computer-based examination, with both programming and theoretical exercises. More info to come as the date draws nearer.

There will be two quick assessments during the semester:

• Mid-term assessment: April 19, 2013
• End-term assessment: May 31, 2013

The points awarded in the quick assessments will be taken into account for the grade of the final examination with a weight of roughly 20%.

Aims of the course

Introduce students of Applied Mathematics, Computational Science and Engineering and Computer Science to the most widely used numerical solution methods for ordinary and partial differential equations, their mathematical properties, and their computer implementation.

Students should be able to:

• implement advanced numerical methods for the solution of partial differential equations efficiently.
• modify and adapt numerical algorithms guided by awareness of their mathematical foundations.
• select and assess numerical methods in light of the predictions of theory.
• identify features of a PDE (= partial differential equation) based model
  that are relevant for the selection and performance of a numerical
  algorithm.
• understand research publications on theoretical and practical aspects of numerical methods for partial differential equations.

Programming Language: All implementations will be done in MATLAB.

Content of the course

• The finite element method (FEM)
• Special elliptic boundary value problems
• Parabolic boundary value problems
• Numerical methods for conservation laws

See a more comprehensive list on the VVZ.

Matlab links

Students of ETH can download Matlab via Stud-IDES for free (product name 'Matlab free').

• Matlab online documentation from MathWorks
• Matlab primer

Literature

This list should be interpreted as supplementary reading beyond the lecture notes, and is neither important nor required for following the course.

• W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2003.
• D. Braess, Finite Elements, 2nd Edition, Cambridge University Press, 2001.
• S. Brenner and R. Scott, Mathematical theory of finite element methods, Texts in Applied Mathematics. Springer–Verlag, New York, 1994.
• A. Ern and J.-L. Guermond Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Sciences. Springer, New York, 2004.
• Ch. Großmann and H.-G. Roos, Numerik partieller Differentialgleichungen, Teubner Studienbücher Mathematik. Teubner, Stuttgart, 1992.
• W. Hackbusch, Elliptic Differential Equations. Theory and Numerical Treatment, volume 18 of Springer Series in Computational Mathematics. Springer, Berlin, 1992.
• T. Hughes, The Finite Element Method, Dover Publications, 2000.
• C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987.
• P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.
• S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.
• R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
• A. Quarteroni, R. Sacco und F. Salieri, Numerische Mathematik, Springer, 2002.

Further literature

These are beyond the scope of this course, and are listed here for the particularly interested.

• C. Canuto, M.Y. Hussaini, A. Quarteroni und T.A. Zang, Spectral Methods, Springer, 2006.
• M. Feistauer, J. Felcman, and I. Straskraba, Mathematical and Computational Methods for Compressible Flow, Oxford: Clarendon Press, 2003.
• A. Tveito und R. Winther, Einführung in Partielle Differentialgleichungen, Springer, 2002.