Search result: Catalogue data in Spring Semester 2021
Physics Master | ||||||
Electives | ||||||
Electives: Physics and Mathematics | ||||||
Selection: Mathematics | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
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401-0674-00L | Numerical Methods for Partial Differential Equations Not meant for BSc/MSc students of mathematics. | W | 10 credits | 2G + 2U + 2P + 4A | R. Hiptmair | |
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 | 1 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 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. | |||||
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. | |||||
Prerequisites / Notice | 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. | |||||
401-4816-21L | Mathematical Aspects of Classical and Quantum Field Theory | W | 8 credits | 4V | M. Schiavina, University lecturers | |
Abstract | The course will cover foundational topics in classical and quantum field theory from a mathematical standpoint. Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided. | |||||
Objective | The objective of this course is to expose master and graduate students in mathematics and physics to the mathematical foundations of classical and quantum field theory. The course will provide a solid mathematical foundation to essential topics in classical and quantum field theories, both useful to mathematics master and graduate students with an interest but no previous background in QFT, as well as for physics master and graduate students who want to focus on more formal aspects of field theory. | |||||
Content | Abstract (long version) The course will cover foundational topics in classical and quantum field theory from a mathematical standpoint. Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided. The course will feature relevant instances of field theories and sigma models, and it will provide a first introduction the the concepts of quantisation, from mechanics to field theory. Using scalar field theory and quantum electrodynamics as guideline, the course will present an overview of quantum field theory, focusing on its more mathematical aspects, including, if time permits, a modern approach to gauge theories and the renormalisation group. Content The course will start with an overview of geometric concepts that will be used throughout, such as graded differential geometry, as well as fiber and vector bundles. After brief review of classical mechanics, interpreted as a first example of a field theory, a thorough discussion of classical, local, Lagrangian field theory will follow, covering topics such as Noether’s Theorems, local and global symmetries. We will then present and discuss a number of examples from gauge theory. In the second part of the course, quantisation will be discussed. The main examples of the scalar field and electrodynamics will be used as a guideline for more general considerations on the quantisation of more general and involved field theories. In the last part of the course, we plan the discussion of modern approaches to quantisation of field theories with symmetries, renormalisation, and of the open challenges that arise. | |||||
Selection: Electives at the University of Zurich University of Zurich lecturers explicitly recommended the following courses also to physics students at ETH Zurich. Recognition of the corresponding external ECTS credits has to be granted by the Director of Studies. Submit your request to the Study Administration (Link). | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
402-0752-00L | Experimental Astro Particle Physics (University of Zurich) Does not take place this semester. No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: PHY465 Mind the enrolment deadlines at UZH: Link | W | 6 credits | 2V + 2U | University lecturers | |
Abstract | ||||||
Objective | ||||||
402-0770-00L | Physics with Muons: From Atomic to Solid State Physics (University of Zurich) No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: PHY432 Mind the enrolment deadlines at UZH: Link | W | 6 credits | 2V + 1U | University lecturers | |
Abstract | Introduction and overview of muon science. Particularly, the use of polarized muons as microscopic magnetic probes in condensed matter physics will be presented (Muon spin rotation and relaxation techniques, muSR). Examples of recent research results in magnetism, superconductivity, semiconductors, thin film and heterostructures. | |||||
Objective | Basic understanding of the use of muons as microscopic magnetic micro probes of matter. Theory and examples of muon spin precession and relaxation (muSR) in various materials. Selected examples in magnetism, superconductivity, semiconductor physics and investigations of heterostructures. Determination of fundamental constants and atomic spectroscopy with muons. The lecture is a useful introduction for people interested in a Bachelor/Master thesis in muSR research at the Paul Scherrer Institute. | |||||
Content | Introduction: Muon characteristics. Generation of muon beams Particle physics aspects: Muon decay, measurement of the muon magnetic anomaly Hyperfine interaction, muonium spectroscopy Fundamentals of muon spin rotation/relaxation and resonance. Static and dynamic spin relaxation. Applications in magnetism: local magnetic fields, phase transitions, spin-glass dynamics. Applications in superconductivity: determination of magnetic penetration depths and coherence length, phase diagram of HTc superconductors, dynamics of the vortex state Hydrogen states in semiconductors Thin film and surface studies with low energy muons. | |||||
Lecture notes | Lecture notes in English are distributed at the beginning. see also Link | |||||
Literature | Link | |||||
Prerequisites / Notice | Lecture can also be given in English. |
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