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Numerical Methods for Elliptic and Parabolic Partial Differential Equations
Last Updated: 2026-06-03 00:07:54
Abstract
This course gives a comprehensive introduction into the numerical treatment of elliptic boundary value problems and parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods.Practical exercises include MATLAB and python implementations of finite difference methods, finite element methods and time integration schemes.
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 elliptic and parabolic partial differential equations. Functional analytic tools (Sobolev spaces) and the basic analysis of partial differential equations is discussed in the course. Particular attention will be placed on developing mathematical foundations for a-priori convergence rate analysis. The basic ideas behind a-posteriori error analysis and adaptivity are covered. Implementations for model problems in MATLAB and python will illustrate the theory.
Resources
Literature
Lecture Notes will be provided. Some excerpts from the first chapter of the following reference may be used as well. Bartels, Sören: Numerical approximation of partial differential equations. Texts in Applied Mathematics, 64. Springer, New York, 2016 Additional Literature: 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 D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007). (Also available in German.) Haim Brezis: 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: 10.1007/978-3-642-22980-0] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006).
General Information
- Language
- English
- Levels
- BSC , MSC
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- oral 30 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Numerical Methods for Elliptic and Parabolic Partial Differential Equations | No time listed | 4 h weekly |
| exercise | Numerical Methods for Elliptic and Parabolic Partial Differential Equations | No time listed | 1 h weekly |
Offered In
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Core Courses (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 14 of the required 26 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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