401365100L Numerical Analysis for Elliptic and Parabolic Partial Differential Equations
Semester  Autumn Semester 2020 
Lecturers  C. Schwab 
Periodicity  yearly recurring course 
Language of instruction  English 
Comment  Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students. Other ETHstudents are advised to attend the course "Numerical Methods for Partial Differential Equations" (401067400L) in the CSE curriculum during the spring semester. 
Courses
Number  Title  Hours  Lecturers  

401365100 V  Numerical Analysis for Elliptic and Parabolic Partial Differential Equations  4 hrs 
 C. Schwab  
401365100 U  Numerical Analysis for Elliptic and Parabolic Partial Differential Equations  1 hrs 
 C. Schwab 
Catalogue data
Abstract  This course gives a comprehensive introduction into the numerical treatment of linear and nonlinear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods. 
Objective  Participants of the course should 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 discrete boundary value problems * implementational aspects of the finite element method 
Content  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 apriori convergence rate analysis. Aposteriori 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 * Nonlinear elliptic boundary value problems * Discretization of parabolic initial boundary value problems 
Literature  SUPPLEMENTARY Literature (core material will be in lecture notes) Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp. A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods, Springer Applied Mathematical Sciences Vol. 159, Springer, 1st Ed. 2004, 2nd Ed. 2015. R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013 Additional Literature: D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007). (Also available in German.) Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp. D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications, Springer, 2012 [DOI: 10.1007/9783642229800] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006). 
Prerequisites / Notice  Practical exercises based on MATLAB Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations 
Performance assessment
Performance assessment information (valid until the course unit is held again)  
Performance assessment as a semester course  
ECTS credits  10 credits 
Examiners  C. Schwab 
Type  session examination 
Language of examination  English 
Repetition  The performance assessment is offered every session. Repetition possible without reenrolling for the course unit. 
Mode of examination  oral 30 minutes 
This information can be updated until the beginning of the semester; information on the examination timetable is binding. 
Learning materials
Main link  Information 
Only public learning materials are listed. 
Groups
401365100 U  Numerical Analysis for Elliptic and Parabolic Partial Differential Equations  
Group  G01 

Restrictions
There are no additional restrictions for the registration. 
Offered in
Programme  Section  Type  

Doctoral Department of Mathematics  Graduate School  W  
Mathematics Master  Core Courses: Applied Mathematics and Further Appl.Oriented Fields  W 