Numerical Methods for Elliptic and Parabolic Partial Differential Equations Autumn Semester 2019
- Lecturer
- Prof. Dr. Christoph Schwab
- Assistants
- Fernando Henriquez
- Maksim Rakhuba
Lectures
- Tuesday - 10:15-12:00 - HG E 21
- Thursday- 08:15-10:00 - HG E 1.2
- First Lecture: Tuesday 17th of September, 2019
Exercise Classes
- Wednesday - 09:15-10:00 - HG E 1.2
- First Exercise Class: Wednesday 25th of September, 2019
Presence Hours
From 10:00 - 12:00 in front of
HG G 53.2 .
- Tuesday, 7.01.2020
- Thursday, 9.01.2020
- Monday, 13.01.2020
- Wednesday, 15.01.2020
- Friday, 17.01.2020
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 and python 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
Exercises
The new exercises will be posted here on Thursday.
We expect you to look at the problems over the weekend and to prepare
questions for the exercise class on Thursday.
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 hand in your solutions exercise class or in the marked box in front of
HG G 53.2 before the next exercise session (if possible, please also provide
a printout of your code)
Your solutions will be corrected and
returned in the following exercise class or, if not collected, returned
to the box in HG G 53.2.
Literature
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)
- A. Quarteroni, A. Manzoni and F. Negri, Reduced basis methods for partial differential equations: an introduction. Vol. 92. Springer, 2015. (online PDF)
- 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.
- 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)
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)
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
