401-0674-00L Numerical Methods for Partial Differential Equations
Semester | Spring Semester 2016 |
Lecturers | R. Hiptmair |
Periodicity | yearly recurring course |
Language of instruction | English |
Comment | Not meant for BSc/MSc students of mathematics. |
Courses
Number | Title | Hours | Lecturers | |||||||||||||
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401-0674-00 V | Numerical Methods for Partial Differential Equations The course will adopt a flipped classroom model that is, the students are requested to prepare a new topic, which will then be discussed in class and practiced during plenary tutorial sessions. | 4 hrs |
| R. Hiptmair | ||||||||||||
401-0674-00 U | Numerical Methods for Partial Differential Equations | 2 hrs |
| R. Hiptmair | ||||||||||||
401-0674-00 A | Numerical Methods for Partial Differential Equations Attendance of lectures and tutorials for 401-0674-00 V Numerical Methods for Partial Differential Equations required. All regulations and requirements for that course apply. | 1 hrs | R. Hiptmair |
Catalogue data
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 Python in one dimension and in C++ in 2D. |
Objective | 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. |
Content | 1 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 notes | Lecture documents and classroom notes will be made available to the audience as PDF. |
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 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. |
Performance assessment
Performance assessment information (valid until the course unit is held again) | |
Performance assessment as a semester course | |
In examination block for | Bachelor's Programme in Computational Science and Engineering 2010; Version 01.08.2016 (Examination Block G3) Bachelor's Programme in Computational Science and Engineering 2012; Version 13.12.2016 (Examination Block G3) |
ECTS credits | 8 credits |
Examiners | R. Hiptmair |
Type | session examination |
Language of examination | English |
Repetition | The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit. |
Mode of examination | written 180 minutes |
Additional information on mode of examination | Computer based examination involving coding problems beside theoretical questions. Parts of the lecture slides will be made available as PDF during the examination. A 30-minute mid-term exam and a 30-minute end term exam will be held during the teaching period on dates specified in the beginning of the semester. Points earned in these exams will be taken into account through a bonus of up to 20% of the total points in the final session exam. |
Written aids | Summary of up to 10 pages A4 in the candidates OWN HANDWRITING. No printouts and copies are allowed. |
Online examination | The examination may take place on the computer. |
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
No information on groups available. |
Restrictions
There are no additional restrictions for the registration. |