Numerical Analysis for Elliptic and Parabolic Partial Differential Equations Autumn Semester 2020
- Lecturer
- Prof. Dr. Christoph Schwab
- Assistants
- Marcello Longo
Lectures
- Tuesday - 10:15-12:00 - HG D 5.2
- Wednesday- 12:15-14:00 - HG D 5.2
- First Lecture: Tuesday 15th of September, 2020
- Lectures will be moved to online format starting: Tuesday 3rd of November, 2020
Zoom ID: 999 2174 6569 (nethz account required), Password communicated by email to registered students
Exercise Classes
- Wednesday - 09:15-10:00 - ML F 40
- First Exercise Class: Wednesday 23rd of September, 2020
- Exercise classes will be moved to online format starting: Wednesday 28th of October, 2020
Zoom ID: 998 8821 3741 (nethz account required), Password communicated by email to registered students
Contents
The course will address the mathematical analysis of numerical solution methods
for linear and nonlinear elliptic and parabolic partial differential equations.
Functional analytic and algebraic (De Rham complex) tools will be provided.
Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed.
Particular attention will be placed on developing mathematical foundations
(Regularity, Approximation theory) for a-priori convergence rate analysis.
A-posteriori error analysis and mathematical proofs of adaptivity and optimality
will be covered.
Implementations for model problems in MATLAB will illustrate the
theory.
A selection of the following topics will be covered:
- Elliptic boundary value problems
- Galerkin discretization of linear variational problems
- The primal finite element method
- Mixed finite element methods
- Discontinuous Galerkin Methods
- Boundary element methods
- Spectral methods
- Adaptive finite element schemes
- Singularly perturbed problems
- Sparse grids
- Galerkin discretization of elliptic eigenproblems
- Non-linear elliptic boundary value problems
- Discretization of parabolic initial boundary value problems
For more details, please check the
ETH Course Catalogue
Videos
- Video recordings of the lectures, starting 03 Nov 2020, will be published on the Moodle page .
- Students registered for the course can access these for online streaming using nethz username and password.
- Disclaimer: Videos are not intended to replace lecture notes.
Relevant for the exam is the content of the lecture notes (excluding sections marked with "*"), as well as exercises and their solutions.
Exercises
The new exercises will be posted here every week by Tuesday.
We expect you to look at the problems beforehand and to prepare
questions for the exercise class on Wednesday.
Submission of Matlab Codes:
- First establish a VPN connection, see
VPN
- The online submission is then done via this Link
- More details can be found
here
Please
upload a scan/high-quality image of your handwritten solutions together with your Matlab codes
before the next exercise session.
Your solutions will be corrected and
returned via SAMup before the following exercise class.
Literature
Lecture Notes:
Chapter 1 ,
Chapter 2 ,
Chapter 3 ,
Chapter 4 ,
Chapter 5 ,
Chapter 6 ,
Chapter 7 ,
Chapter 8 ,
Bibliography
Main literature
- S.C. Brenner and L. Ridgway Scott, The mathematical theory of Finite Element Methods, New York, Berlin, Springer, cop.1994. (online PDF)
- A. Ern and J.L. Guermond, Theory and Practice of Finite Element Methods, Springer Applied Mathematical Sciences Vol. 159, Springer, 1st Ed. 2004. (online PDF)
- R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013. (Chapter 1)
- H. Li and V. Nistor, LNG−FEM: graded meshes on domains of polygonal structure. Recent advances in scientific computing and applications, 239–246, Contemp. Math., 586, Amer. Math. Soc., 2013.
Additional literature
- D. Braess, Finite Elements, Cambridge Univ. Press, 3rd Ed. 2007. (Also available in German)
- D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Springer, vol. 69 SMAI Mathématiques et Applications, 2012. (online PDF)
- V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Verlag, 2nd Ed. 2006. (online PDF)
- A. Quarteroni, A. Manzoni and F. Negri, Reduced basis methods for partial differential equations: an introduction. Vol. 92. Springer, 2015. (online PDF)
- D. N. Arnold, Finite element exterior calculus. CBMS-NSF Regional Conference Series in Applied Mathematics, 93. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018. xi+120 pp.
- H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Springer Science & Business Media, 2010. (online PDF)
Note: "online PDF" applies to users in the ETH domain (student computers / ETH WiFi / VPN)
Matlab Resources
Matlab links
ETH students can download Matlab with a free network license from the IT-Shop .
Matlab tutorials
