Ralf Hiptmair: Catalogue data in Spring Semester 2016

Name Prof. Dr. Ralf Hiptmair
FieldMathematik
Address
Seminar für Angewandte Mathematik
ETH Zürich, HG G 58.2
Rämistrasse 101
8092 Zürich
SWITZERLAND
Telephone+41 44 632 34 04
Fax+41 44 632 11 04
E-mailralf.hiptmair@sam.math.ethz.ch
URLhttps://www.math.ethz.ch/sam/the-institute/people/ralf-hiptmair.html
DepartmentMathematics
RelationshipFull Professor

NumberTitleECTSHoursLecturers
401-0674-00LNumerical Methods for Partial Differential Equations
Not meant for BSc/MSc students of mathematics.
8 credits4V + 2U + 1AR. Hiptmair
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 Python in one dimension and in C++ in 2D.
Learning objectiveMain 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.
Content1 Case Study: A Two-point Boundary Value Problem
1.1 Introduction
1.2 A model problem
1.3 Variational approach
1.4 Simplified model
1.5 Discretization
1.5.1 Galerkin discretization
1.5.2 Collocation [optional]
1.5.3 Finite differences
1.6 Convergence
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)
3.5.3 Assembly
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
6.1.4 Timestepping
6.1.5 Convergence
6.2 Wave equations [optional]
6.2.1 Vibrating membrane
6.2.2 Wave propagation
6.2.3 Method of lines
6.2.4 Timestepping
6.2.5 CFL-condition
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
7.2.2 Upwinding
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 Timestepping
8.4.1 Linear stability
8.4.2 CFL-condition
8.4.3 Convergence
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]
Lecture notesLecture documents and classroom notes will be made available to the audience as PDF.
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 in MATLAB and C++ are essential.

Homework asssignments involve substantial coding, partly based on a C++ finite element library. The written examination will be computer based and will comprise coding tasks.
401-3640-16LSeminar in Applied Mathematics: The Discontinuous Petrov Galerkin Method Restricted registration - show details
Number of participants limited to 10.
4 credits2SR. Hiptmair
AbstractThe new Discontinuous Petrov Galerkin Method (DPG) is a generalized finite element approach pursuing the idea of choosing approximately optimal test functions in (piecewise polynomial) spaces with relaxed continuity requirements. The benefit is enhanced stability of the discrete variational formulations, which is particularly important for singularly perturbed problems.
Learning objectiveStudying DPG the students should learn about general concepts and numerical analysis techniques relevant for the discretization of boundary value problems for linear PDEs.
ContentThe seminar will comprise presentations based on key scientific publications about the DPG method.
Lecture notesSurvey paper suitable as an introduction to the topic:

L. Demkowicz and J. Gopalakrishnan. An overview of the discontinuous petrov galerkin method. In Xiaobing Feng, Ohannes Karakashian, and Yulong Xing, edi- tors, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, volume 157 of The IMA Volumes in Mathematics and its Applications, pages 149-180. Springer International Publishing, 2014.

See also the ICERM lecture by J. Gopalakrishnan: https://icerm.brown.edu/video_archive/#/play/387
Literature[BGH14] Timaeus Bouma, Jay Gopalakrishnan, and Ammar Harb. Convergence rates of the DPG method with reduced test space degree. Comput. Math. Appl., 68(11):1550- 1561, 2014.

[BTDG13] Tan Bui-Thanh, Leszek Demkowicz, and Omar Ghattas. A unified discontinuous Petrov-Galerkin method and its analysis for Friedrichs' systems. SIAM J. Numer. Anal., 51(4):1933-1958, 2013.

[CDG14] Carsten Carstensen, Leszek Demkowicz, and Jay Gopalakrishnan. A posteriori error control for DPG methods. SIAM J. Numer. Anal., 52(3):1335-1353, 2014.

[CDG15] C. Carstensen, L. Demkowicz, and J. Gopalakrishnan. Breaking spaces and forms for the dpg method and applications including maxwell equations. Numer. Math., 2015.

[CDW12] Albert Cohen, Wolfgang Dahmen, and Gerrit Welper. Adaptivity and variational stabilization for convection-diffusion equations. ESAIM Math. Model. Numer. Anal., 46(5):1247-1273, 2012.

[CEQ14] Jesse Chan, John A. Evans, and Weifeng Qiu. A dual Petrov-Galerkin finite element method for the convection-diffusion equation. Comput. Math. Appl., 68(11):1513- 1529, 2014.

[DG11] L. Demkowicz and J. Gopalakrishnan. Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal., 49(5):1788-1809, 2011.

[DG13] L. Demkowicz and J. Gopalakrishnan. A primal DPG method without a first-order reformulation. Comput. Math. Appl., 66(6):1058-1064, 2013.

[DG14] L. Demkowicz and J. Gopalakrishnan. An overview of the discontinuous petrov galerkin method. In Xiaobing Feng, Ohannes Karakashian, and Yulong Xing, edi- tors, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, volume 157 of The IMA Volumes in Mathematics and its Applications, pages 149-180. Springer International Publishing, 2014.

[DH13] Leszek Demkowicz and Norbert Heuer. Robust DPG method for convection- dominated diffusion problems. SIAM J. Numer. Anal., 51(5):2514-2537, 2013.

[GQ14] J. Gopalakrishnan and W. Qiu. An analysis of the practical DPG method. Math. Comp., 83(286):537-552, 2014.

[RBTD14] Nathan V. Roberts, Tan Bui-Thanh, and Leszek Demkowicz. The DPG method for the Stokes problem. Comput. Math. Appl., 67(4):966-995, 2014.
Prerequisites / NoticeFamiliarity with variational formulations of boundary value problems for (elliptic) PDEs. Participants also should have attended a course on functional analysis. Knowledge about numerical methods for PDEs is certainly beneficial.
401-3667-16LCase Studies Seminar (Spring Semester 2016)3 credits2SV. C. Gradinaru, R. Hiptmair, M. Reiher
AbstractIn 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.
Learning objective
ContentIn 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.).
401-5650-00LZurich Colloquium in Applied and Computational Mathematics Information 0 credits2KR. Abgrall, H. Ammari, P. Grohs, R. Hiptmair, A. Jentzen, S. Mishra, S. Sauter, C. Schwab
AbstractResearch colloquium
Learning objective
406-0663-AALNumerical Methods for CSE
Enrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement.

Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit.
7 credits15RR. Hiptmair
AbstractIntroduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology.
Learning objective* Knowledge of the fundamental algorithms in numerical mathematics
* Knowledge of the essential terms in numerical mathematics and the
techniques used for the analysis of numerical algorithms
* Ability to choose the appropriate numerical method for concrete problems
* Ability to interpret numerical results
* Ability to implement numerical algorithms afficiently
Content1. Direct Methods for linear systems of equations
2. Interpolation
3. Iterative Methods for non-linear systems of equations
4. Krylov methods for linear systems of equations
5. Eigensolvers
6. Least Squares Techniques
7. Filtering Algorithms
8. Approximation of Functions
9. Numerical Quadrature
10. Clustering Techniques
11. Single Step Methods for ODEs
12. Stiff Integrators
13. Structure Preserving Integrators
LiteratureW. Dahmen, A. Reusken "Numerik für Ingenieure und Naturwissenschaftler", Springer 2006.
M. Hanke-Bourgeois "Grundlagen der Numerischen Mathematik und des wissenschaftlichen Rechnens", BG Teubner, 2002
C. Moler, Numerical computing with MATLAB, SIAM, 2004
P. Deuflhard and A. Hohmann, "Numerische Mathematik I", DeGruyter, 2002
Prerequisites / NoticeA course covering the material is taught in German every autumn term
(course unit 401-0663-00L). Exercises and examination are available in English.