401-0674-00L  Numerical Methods for Partial Differential Equations

SemesterFrühjahrssemester 2021
DozierendeR. Hiptmair
Periodizitätjährlich wiederkehrende Veranstaltung
LehrspracheEnglisch
KommentarNicht für Studierende BSc/MSc Mathematik



Lehrveranstaltungen

NummerTitelUmfangDozierende
401-0674-00 GNumerical 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.
2 Std.
Mo16-18HG F 1 »
R. Hiptmair
401-0674-00 UNumerical Methods for Partial Differential Equations
Gruppeneinteilung erfolgt über myStudies.
2 Std.
Fr10-12ETZ E 8 »
10-12HG D 1.1 »
10-12HG G 3 »
11-13ETZ G 91 »
R. Hiptmair
401-0674-00 PNumerical Methods for Partial Differential Equations
Homework C++ coding projects for the course "Numerical Methods for Partial Differential Equations"
2 Std.R. Hiptmair
401-0674-00 ANumerical Methods for Partial Differential Equations
Video guided self-study or group-study for the course "Numerical Methods for Partial Differential Equations"
4 Std.R. Hiptmair

Katalogdaten

KurzbeschreibungDerivation, 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.
LernzielMain 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.
Inhalt1 Second-Order Scalar Elliptic Boundary Value Problems
1.2 Equilibrium Models: Examples
1.3 Sobolev spaces
1.4 Linear Variational Problems
1.5 Equilibrium Models: Boundary Value Problems
1.6 Diffusion Models (Stationary Heat Conduction)
1.7 Boundary Conditions
1.8 Second-Order Elliptic Variational Problems
1.9 Essential and Natural Boundary Conditions
2 Finite Element Methods (FEM)
2.2 Principles of Galerkin Discretization
2.3 Case Study: Linear FEM for Two-Point Boundary Value Problems
2.4 Case Study: Triangular Linear FEM in Two Dimensions
2.5 Building Blocks of General Finite Element Methods
2.6 Lagrangian Finite Element Methods
2.7 Implementation of Finite Element Methods
2.7.1 Mesh Generation and Mesh File Format
2.7.2 Mesh Information and Mesh Data Structures
2.7.2.1 L EHR FEM++ Mesh: Container Layer
2.7.2.2 L EHR FEM++ Mesh: Topology Layer
2.7.2.3 L EHR FEM++ Mesh: Geometry Layer
2.7.3 Vectors and Matrices
2.7.4 Assembly Algorithms
2.7.4.1 Assembly: Localization
2.7.4.2 Assembly: Index Mappings
2.7.4.3 Distribute Assembly Schemes
2.7.4.4 Assembly: Linear Algebra Perspective
2.7.5 Local Computations
2.7.5.1 Analytic Formulas for Entries of Element Matrices
2.7.5.2 Local Quadrature
2.7.6 Treatment of Essential Boundary Conditions
2.8 Parametric Finite Element Methods
3 FEM: Convergence and Accuracy
3.1 Abstract Galerkin Error Estimates
3.2 Empirical (Asymptotic) Convergence of Lagrangian FEM
3.3 A Priori (Asymptotic) Finite Element Error Estimates
3.4 Elliptic Regularity Theory
3.5 Variational Crimes
3.6 FEM: Duality Techniques for Error Estimation
3.7 Discrete Maximum Principle
3.8 Validation and Debugging of Finite Element Codes
4 Beyond FEM: Alternative Discretizations [dropped]
5 Non-Linear Elliptic Boundary Value Problems [dropped]
6 Second-Order Linear Evolution Problems
6.1 Time-Dependent Boundary Value Problems
6.2 Parabolic Initial-Boundary Value Problems
6.3 Linear Wave Equations
7 Convection-Diffusion Problems [dropped]
8 Numerical Methods for Conservation Laws
8.1 Conservation Laws: Examples
8.2 Scalar Conservation Laws in 1D
8.3 Conservative Finite Volume (FV) Discretization
8.4 Timestepping for Finite-Volume Methods
8.5 Higher-Order Conservative Finite-Volume Schemes
SkriptThe 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.
LiteraturChapters 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.
Voraussetzungen / BesonderesMastery of basic calculus and linear algebra is taken for granted.
Familiarity with fundamental numerical methods (solution methods for linear systems of equations, interpolation, approximation, numerical quadrature, numerical integration of ODEs) is essential.

Important: Coding skills and experience in C++ are essential.

Homework assignments involve substantial coding, partly based on a C++ finite element library. The written examination will be computer based and will comprise coding tasks.

Leistungskontrolle

Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird)
Leistungskontrolle als Semesterkurs
Im Prüfungsblock fürBachelor-Studiengang Rechnergestützte Wissenschaften 2012; Ausgabe 13.12.2016 (Prüfungsblock G3)
Bachelor-Studiengang Rechnergestützte Wissenschaften 2016; Ausgabe 27.03.2018 (Prüfungsblock G3)
Bachelor-Studiengang Rechnergestützte Wissenschaften 2018; Ausgabe 25.02.2020 (Prüfungsblock G3)
ECTS Kreditpunkte10 KP
PrüfendeR. Hiptmair
FormSessionsprüfung
PrüfungsspracheEnglisch
RepetitionDie Leistungskontrolle wird in jeder Session angeboten. Die Repetition ist ohne erneute Belegung der Lerneinheit möglich.
ZulassungsbedingungNone
Prüfungsmodusschriftlich 180 Minuten
Zusatzinformation zum PrüfungsmodusComputer based examination involving coding problems beside theoretical questions. Some of the lecture materials will be made available as PDF during the examination.

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 20% for the final grade.
Hilfsmittel schriftlichNone
Online-PrüfungDie Prüfung kann am Computer stattfinden.
FernprüfungDas Ablegen als Fernprüfung ist nicht möglich.
Falls die Lerneinheit innerhalb eines Prüfungsblockes geprüft wird, werden die Kreditpunkte für den gesamten bestandenen Block erteilt.
Diese Angaben können noch zu Semesterbeginn aktualisiert werden; verbindlich sind die Angaben auf dem Prüfungsplan.

Lernmaterialien

Keine öffentlichen Lernmaterialien verfügbar.
Es werden nur die öffentlichen Lernmaterialien aufgeführt.

Gruppen

401-0674-00 UNumerical Methods for Partial Differential Equations
GruppenG-01
Fr10-12HG D 1.1 »
G-02
Fr10-12HG G 3 »
G-03
Fr10-12ETZ E 8 »
G-04
Fr11-13ETZ G 91 »

Einschränkungen

Keine zusätzlichen Belegungseinschränkungen vorhanden.

Angeboten in

StudiengangBereichTyp
Computational Biology and Bioinformatics MasterTheorieWInformation
Data Science MasterWählbare KernfächerWInformation
Informatik BachelorWahlfächerWInformation
Physik BachelorAuswahl an Lehrveranstaltungen aus höheren SemesternWInformation
Physik MasterAuswahl: MathematikWInformation
Rechnergestützte Wissenschaften BachelorBlock G3OInformation
Rechnergestützte Wissenschaften BachelorBlock G3OInformation