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401-0674-00L 10 Credits BSC , MSC D-BSSE , D-INFK , D-MATH , D-PHYS , D-ITET
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Numerical Methods for Partial Differential Equations

Lecturers & Examiners: Prof. Dr. Ralf Hiptmair
Not meant for BSc/MSc students of mathematics.
VVZ CR n/a

Last Updated: 2026-02-05 16:23:12

Abstract

Derivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations: convection-diffusion, heat equation, wave equation, conservation laws. Implementation in C++ based on a finite element library.

Objective

Main skills to be acquired in this course: * Ability to implement fundamental numerical methods for the solution of partial differential equations efficiently. * Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations. * Ability to select and assess numerical methods in light of the predictions of theory * Ability to identify features of a PDE (= partial differential equation) based model that are relevant for the selection and performance of a numerical algorithm. * Ability to understand research publications on theoretical and practical aspects of numerical methods for partial differential equations. * Skills in the efficient implementation of finite element methods on unstructured meshes. This course is neither a course on the mathematical foundations and numerical analysis of methods nor an course that merely teaches recipes and how to apply software packages.

Content

Second-order scalar elliptic boundary value problems Finite-element methods (FEM) FEM: Convergence and Accuracy Non-linear elliptic boundary value problems Second-order linear evolution problems Convection-diffusion problems Numerical methods for conservation laws

Resources

Lecture Notes

The lecture will be taught in flipped classroom format:- Video tutorials for all thematic units will be published online.- Tablet notes accompanying the videos will be made available to the audience as PDF.- A comprehensive lecture document will cover all aspects of the course, seehttps://www.sam.math.ethz.ch/~grsam/NUMPDEFL/NUMPDE.pdf

Literature

Chapters of the following books provide supplementary reading (detailed references in course material): * D. Braess: Finite Elemente, Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, Springer 2007 (available online). * S. Brenner and R. Scott. Mathematical theory of finite element methods, Springer 2008 (available online). * A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Sciences. Springer, New York, 2004. * Ch. Großmann and H.-G. Roos: Numerical Treatment of Partial Differential Equations, Springer 2007. * W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment, volume 18 of Springer Series in Computational Mathematics. Springer, Berlin, 1992. * P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial Differential Equations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003. * S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, Heidelberg, 2003. * R. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, 2002. However, study of supplementary literature is not important for for following the course.

General Information

Language
English
Levels
BSC , MSC
Frequency
Yearly recurring

Examination

Type
session examination
Mode
written 225 minutes
Aids
None
Digital
The exam takes place on devices provided by ETH Zurich.
Three-hour (180 minute) computer based examination involving coding problems beside theoretical questions. Some of the lecture materials and other documents will be made available as PDF or HTML during the examination. The total exam time of 225 minute also include 30-minute reading time in the beginning of the exam.A 30-minute mid-term exam and a 30-minute end term exam (non-mandatory) will be held during the teaching period on dates specified in the beginning of the semester. The grades of these interim examinations will be taken into account through a bonus of up to 30% for the final grade.

Course Components

Type Title Time & Place Hours
lecture with exercise Numerical Methods for Partial Differential Equations
This course is designed in a flipped classroom format based on video tutorials and supplemented by a weekly question-and-answer session, for which attendance is highly recommended.
  • Mon 16:15-18:00 (HG F 1)
2 h weekly
exercise Numerical Methods for Partial Differential Equations
Groups are selected in myStudies.
  • Fri 10:15-12:00 (ETZ E 8)
  • Fri 10:15-12:00 (ETZ H 91)
  • Fri 12:15-14:00 (ETZ E 8)
  • Fri 12:15-14:00 (ETZ H 91)
2 h weekly
practical/laboratory course Numerical Methods for Partial Differential Equations
Homework C++ coding projects for the course "Numerical Methods for Partial Differential Equations"
No time listed 2 h weekly
independent project Numerical Methods for Partial Differential Equations
Video guided self-study or group-study for the course "Numerical Methods for Partial Differential Equations" This course coincides with 401-0674-00 Numerical Methods for Partial Differential Equations, which is taught this Spring Term. All students who have to take this course must also enrol in 401-0674-00 Numerical Methods for Partial Differential Equations.
No time listed 4 h weekly

Offered In