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

SemesterSpring Semester 2021
LecturersR. Hiptmair
Periodicityyearly recurring course
Language of instructionEnglish
CommentNot meant for BSc/MSc students of mathematics.



Courses

NumberTitleHoursLecturers
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 hrs
Mon16-18HG F 1 »
R. Hiptmair
401-0674-00 UNumerical Methods for Partial Differential Equations
Groups are selected in myStudies.
2 hrs
Fri10-12ETZ 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 hrsR. 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 hrsR. Hiptmair

Catalogue data

AbstractDerivation, 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.
ObjectiveMain 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.
Content1 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
Lecture notesThe 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.
LiteratureChapters 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.
Prerequisites / NoticeMastery 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.

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
In examination block forBachelor's Programme in Computational Science and Engineering 2012; Version 13.12.2016 (Examination Block G3)
Bachelor's Programme in Computational Science and Engineering 2016; Version 27.03.2018 (Examination Block G3)
Bachelor's Programme in Computational Science and Engineering 2018; Version 25.02.2020 (Examination Block G3)
ECTS credits10 credits
ExaminersR. Hiptmair
Typesession examination
Language of examinationEnglish
RepetitionThe performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Admission requirementNone
Mode of examinationwritten 180 minutes
Additional information on mode of examinationComputer 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.
Written aidsNone
Online examinationThe examination may take place on the computer.
Distance examinationIt is not possible to take a distance examination.
If the course unit is part of an examination block, the credits are allocated for the successful completion of the whole block.
This information can be updated until the beginning of the semester; information on the examination timetable is binding.

Learning materials

No public learning materials available.
Only public learning materials are listed.

Groups

401-0674-00 UNumerical Methods for Partial Differential Equations
GroupsG-01
Fri10-12HG D 1.1 »
G-02
Fri10-12HG G 3 »
G-03
Fri10-12ETZ E 8 »
G-04
Fri11-13ETZ G 91 »

Restrictions

There are no additional restrictions for the registration.

Offered in

ProgrammeSectionType
Computational Biology and Bioinformatics MasterTheoryWInformation
Data Science MasterCore ElectivesWInformation
Computer Science BachelorElectivesWInformation
Physics BachelorSelection of Higher Semester CoursesWInformation
Physics MasterSelection: MathematicsWInformation
Computational Science and Engineering BachelorBlock G3OInformation
Computational Science and Engineering BachelorBlock G3OInformation