Numerical Methods for Elliptic and Parabolic PDEs

Lecture

Siehe auch:
SAM Colloquia

• Tue, 15:15-17:00, HG E 21
• Thu, 13:15-15:00, HG E 21
• Lecturer: P. Grohs
• Catalogue info: vvz
• First lecture: Tue, Sep 17, 2013

Lecture notes

Will be published here.

Tutorial

Siehe auch:
HA Stellen
BSc, MSc, ETc. thesis topics

• Mon, 16:15-17:00, ML H 34.3
• First tutorial: Mon, Sep 23
• TA: Fabian and Markus
• We will be happy to help in case of questions concerning the exercises or the lecture
• Office hours preferably by appointment (e-mail)
• Submission of practical Matlab exercises: Hand in here.
• Testate condition: As testate conditions are not in place anymore, it is not compulsory, yet recommended, to hand in the exercises for correction.
• Hand in your handwritten solutions in the exercise class or deposit them in the marked box in front of HG G 53.x. If possible, please also provide a printout of your Matlab code

Assignments

LehrFEM

Programming exercises are based on the LehrFEM software written in MATLAB. During the course we will release more and more parts of the Software. Here, you can find a (reduced) package fitted to the contents of the lecture.

You will find an introduction to the most important files in this manual.

Exam

Please find all necessary information here.

Aims of the course

The course gives a comprehensive introduction into the numerical treatment of linear and non-linear elliptic boundary value problems and related eigenvalue problems and parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises involve Matlab implementation of finite element methods.

Participants of the course will become familiar with:

• concepts underlying the discretization of elliptic and parabolic boundary value problems
• analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
• methods for the efficient solution of discretized boundary value problems
• implementational aspects of the finite element method

Content of the course (cf. vvz)

core:

• Elliptic boundary value problems
• Galerkin discretization of variational problems
• The primal finite element method
• Finite difference and finite volume methods
• Mixed finite element methods

optional:

• Discretization of parabolic initial boundary value problems
• Discontinuous Galerkin methods
• Sparse grids

Literature

recommended (but not mandatory):

• D. Braess, Finite Elements, 3rd Ed., Cambridge University Press, 2007 (or German original, 4th Ed.)
• V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 2nd Ed., Springer Verlag, 2006 (available online)

more useful books:

• S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3rd Ed., Springer, 2008 (available online)
• D. A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, Springer, 2012 (available online)
• Ch. Grossmann and H.-G. Roos, Numerik partieller Differentialgleichungen, 2nd Ed., Teubner, 1994 (see also homepage of H.-G. Roos)

yet more:

• K. Atkinson and W. Han, Theoretical numerical analysis, 3rd Ed., Springer, 2009 (available online)
• H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010: online. Cf. the French original
• L. C. Evans, Partial differential equations, 2nd Ed., AMS, 2010 (unfortunately, not online)
• A. Ern and J.-L. Guermond, Theory and practice of finite elements, Springer, 2004
• S. Sauter and Ch. Schwab, Randelementmethoden, Teubner, 2004 (available online)
• P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer, 2003

Note: "available online" applies to users in the ETH domain (student computers / ETH WiFi / VPN)

Matlab links

ETH students can download Matlab with a free network license from Stud-IDES

Matlab tutorials, etc.

• Matlab Online Documentation von MathWorks
• Matlab Primer
• Matlab Summary and Tutorial
• Matlab Online Documentation from the MathWorks
• Matlab Guide, D.J. Higham and N.J. Higham, SIAM 2000, ISBN 0-89871-516-4
• Matlab Einführungskurs, P. Arbenz