Ralf Hiptmair: Katalogdaten im Frühjahrssemester 2018
|Name||Herr Prof. Dr. Ralf Hiptmair|
Seminar für Angewandte Mathematik
ETH Zürich, HG G 58.2
|Telefon||+41 44 632 34 04|
|Fax||+41 44 632 11 04|
|401-0674-00L||Numerical Methods for Partial Differential Equations|
Not meant for BSc/MSc students of mathematics.
|8 KP||4V + 2U + 1A||R. Hiptmair|
|Kurzbeschreibung||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.|
|Lernziel||Main skills to be acquired in this course:|
* Ability to implement advanced 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.
|Inhalt||1 Case Study: A Two-point Boundary Value Problem|
1.2 A model problem
1.3 Variational approach
1.4 Simplified model
1.5.1 Galerkin discretization
1.5.2 Collocation [optional]
1.5.3 Finite differences
2 Second-order Scalar Elliptic Boundary Value Problems
2.1 Equilibrium models
2.1.1 Taut membrane
2.1.2 Electrostatic fields
2.1.3 Quadratic minimization problems
2.2 Sobolev spaces
2.3 Variational formulations
2.4 Equilibrium models: Boundary value problems
3 Finite Element Methods (FEM)
3.1 Galerkin discretization
3.2 Case study: Triangular linear FEM in two dimensions
3.3 Building blocks of general FEM
3.4 Lagrangian FEM
3.4.1 Simplicial Lagrangian FEM
3.4.2 Tensor-product Lagrangian FEM
3.5 Implementation of FEM in C++
3.5.1 Mesh file format (Gmsh)
3.5.2 Mesh data structures (DUNE)
3.5.4 Local computations and quadrature
3.5.5 Incorporation of essential boundary conditions
3.6 Parametric finite elements
3.6.1 Affine equivalence
3.6.2 Example: Quadrilaterial Lagrangian finite elements
3.6.3 Transformation techniques
3.6.4 Boundary approximation
3.7 Linearization [optional]
4 Finite Differences (FD) and Finite Volume Methods (FV) [optional]
4.1 Finite differences
4.2 Finite volume methods (FVM)
5 Convergence and Accuracy
5.1 Galerkin error estimates
5.2 Empirical Convergence of FEM
5.3 Finite element error estimates
5.4 Elliptic regularity theory
5.5 Variational crimes
5.6 Duality techniques [optional]
5.7 Discrete maximum principle [optional]
6 2nd-Order Linear Evolution Problems
6.1 Parabolic initial-boundary value problems
6.1.1 Heat equation
6.1.2 Spatial variational formulation
6.1.3 Method of lines
6.2 Wave equations [optional]
6.2.1 Vibrating membrane
6.2.2 Wave propagation
6.2.3 Method of lines
7 Convection-Diffusion Problems
7.1 Heat conduction in a fluid
7.1.1 Modelling fluid flow
7.1.2 Heat convection and diffusion
7.1.3 Incompressible fluids
7.1.4 Transient heat conduction
7.2 Stationary convection-diffusion problems
7.2.1 Singular perturbation
7.3 Transient convection-diffusion BVP
7.3.1 Method of lines
7.3.2 Transport equation
7.3.3 Lagrangian split-step method
7.3.4 Semi-Lagrangian method
8 Numerical Methods for Conservation Laws
8.1 Conservation laws: Examples
8.2 Scalar conservation laws in 1D
8.3 Conservative finite volume discretization
8.3.1 Semi-discrete conservation form
8.3.2 Discrete conservation property
8.3.3 Numerical flux functions
8.3.4 Montone schemes
8.4.1 Linear stability
8.5 Higher order conservative schemes [optional]
8.5.1 Slope limiting
8.5.2 MUSCL scheme
8.6. FV-schemes for systems of conservation laws [optional]
|Skript||Lecture documents and classroom notes will be made available to the audience as PDF.|
|Literatur||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.
|Voraussetzungen / Besonderes||Mastery 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.
|401-3667-18L||Case Studies Seminar (Spring Semester 2018)||3 KP||2S||V. C. Gradinaru, R. Hiptmair, K. Nipp, M. Reiher|
|Kurzbeschreibung||In the CSE Case Studies Seminar invited speakers from ETH, from other universities as well as from industry give a talk on an applied topic. Beside of attending the scientific talks students are asked to give short presentations (10 minutes) on a published paper out of a list.|
|Inhalt||In the CSE Case Studies Seminar invited speakers from ETH, from other universities as well as from industry give a talk on an applied topic. Beside of attending the scientific talks students are asked to give short presentations (10 minutes) on a published paper out of a list (containing articles from, e.g., Nature, Science, Scientific American, etc.).|
|401-4504-18L||Reading Course: Retarded Potentials and Time Domain Boundary Integral Equations||4 KP||2G||R. Hiptmair|
|Kurzbeschreibung||By studying a current monograph and review articles, this course offers a concise introduction into the mathematical theories and techniques underlying time-domain boundary integral equations with focus on wave equations.|
|Lernziel||Acquire knowledg about the modern mathematical theory of time-domain boundary integral equations for wave equations.|
|Inhalt||Retarded layer potentials.|
Distributions and Laplace transform
Time domains analysis of the single layer potential
|Literatur||F. Sayas, Retarded potentials and time domain boundary integral equations.|
M. Haskell and F.-J. Sayas, Convolution quadrature for wave simulation, https://link.springer.com/chapter/10.1007/978-3-319-32146-2_2
|Voraussetzungen / Besonderes||General knowledge aboput partial differential equations and functional analysis|
|401-5650-00L||Zurich Colloquium in Applied and Computational Mathematics||0 KP||1K||R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, A. Jentzen, S. Mishra, S. Sauter, C. Schwab|