Introduction to PDEs and Numerical Methods

Introduction to PDEs and Numerical Methods

General Information

Lecturer
Dr. Noemi Friedman


Assistant
Dr. Jaroslav Vondřejc


Tutorials
Stephan Lenz


Schedule

Lecture: Wed 9:45-11:15 in room PK 3.4. (Pockelsstraße 3, Am Okerufer, Hochhaus, 205)

First lecture: 28.10.2015.

Attention: Exercises and Tutorials will start about 30 minutes earlier.

Exercises: Friday 15:30 - 16:15, room 65.4 (Hans SommerStr. 65).

Tutorials: Friday 16:15 - 17:00, room 65.4 (Hans SommerStr. 65).

Start

Prerequisites

Target group

Certificates

Office hours

Exam

17.08.2016. 9:30 - 11:00, in room 812, Mühlenpfordtstr. 23 (Seminarraum WiRe)

The results of the exam will be announced on Monday, 5th September, at 13:00 in Mühlenpfordtstr. 23, room 812 (seminar room), 8th floor.

29.2.2016., 10:30 - 12:00, room ZI 24.1 (Zimmerstraße 24 D (Grotrian), 008)

Information about the exam:

Exam-prerequisits

In the exam it is possible to use one page of size A4 with hand written notes, other than that, only pen and a calculator is needed.

Literature:

A script for the lecture is available HERE.

To download books and papers here: http://www.biblio.tu-bs.de/semapp/, click on Prof. Hermann G. Matthies "Introduction to PDEs and Numerical Methods" and "Numerical methods for PDEs" and then on the corresponding buttons "Volltext anzeigen" (please, use the given password but without the '1').

Lecture slides

  • Lecture 1: Introduction, differential operators, classification of PDEs, introductory examples - the transport equation and the heat equation
  • Lecture 2: Analytitical solution of ODEs and PDEs - from PDEs to ODEs, eigenfunctions, eigenvalues
  • Lecture 3: Analytitical solution of ODEs and PDEs - Fourier series, projection theory
  • Lecture 4: Analytitical solution of ODEs and PDEs, Finite Difference method
  • Lecture 5: Numerical solution of the instationary heat equation: Method of lines, Euler forward, Euler backward and the Theta-method
  • Lecture 6: Convergence, consistence, stability, Von Neumann stability analysis
  • Lecture 7: Solving linear system of equations I.
  • Lecture 8: Solving linear system of equations II.
  • Lecture 9: Weak formulation
  • Lecture 10: Weighted residual methods (Galerkin method and the Finite Element Method)
  • Lecture 11: Repetition, and elementwise integration (element stiffness matrix and assembling the global stiffness matrix)
  • Lecture 11-12: Isoparametric mapping
  • Lecture 13, Gauß quadrature: Higher order basis functions, numerical integration

Homework assignments

To obtain full points explain your solutions thoroughly and self-consistently with all necessary intermediate conclusions and calculation steps as to leave no doubt about the correctness and your understanding. Structure programmes nicely and with comments and argue why you think that it works. Support your reasons with necessary plots, examples, etc. so that it becomes obvious.

  • Reading assignment 1: Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Chapter 1-2, Script 1.1, deadline: 4.11.2015.
  • Homework 1: Differential operators, the heat equation, visualization, deadline: 11.11.2015. Solution
  • Reading assignment 2 :Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Chapter 3, Script 1.2.1, 1.2.2, deadline: 11.11.2015.
  • Homework 2: Classification of PDEs, analytical solution of ODEs and PDEs, deadline: 19.11.2015. Solution
  • Reading assignment 3 :Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Sections 4.1-4.3, 5.1-5.3, deadline: 18.11.2015.
  • Homework 3:Fourier-series, Taylor expansion, finite difference operators, deadline: 26.11.2015.Solution
  • Reading assignment 4 : Script 1.4.1-1.4.4., deadline: 25.11.2015.
  • Homework 4: Numerical solution of the instationary heat equation: Method of lines, Euler forward, Euler backward, deadline: 03.12.2015. Solution.
  • Reading assignment 5: Revise again script 1.4.1-1.4.4., Mark S. Gockenbach: Partial Differential Equations - Chapters 4.4.1 and 4.5.2, deadline: 04.12.2015.
  • Homework 5: There is no homework this week :)
  • Reading assignment 6: Script 1.4.5-1.5.4
  • Homework 6: Consistency, order, and stability of a finite difference scheme. Deadline: 17.12.2015, 9:00. Solution.
  • Reading assignment 7: Script 2.1-2.2, Michael T. Heath: Scientific Computing, Chapter 2. (Systems of Linear Equations), Chapter 1. is also highly recommended
  • Homework 7: Cholesky decomposition, sparsity of matrix, inner product, conjugate gradients. download Matrix A Deadline: 7.1.2016, 9:00. Solution.
  • Reading assignment 8: Script 2.3, Michael T. Heath: Scientific Computing, Chapter 11.4, 11.5. (Direct and Iterative Methods for Linear Systems), Deadline: 6.1.2016.
  • Homework 8: Weak formulation, Lax-Milgram lemma. Deadline: 14.1.2016, 9:00. Solution.
  • Reading assignment 9: Script 2.3, Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Sections 5.4,5.5,5.6, 6.4, 6.5, Script 3.1, 3.2, 3.3 Deadline: 20.1.2016.
  • Homework 9: Finite Element Method, 1D. Deadline: 21.1.2016, 9:00. Solution
  • Reading assignment 10: Mark S. Gockenbach: 8.4. Finite Element Methods in two dimensions. Deadline: 27.1.2016
  • Volunatary homework 10 (for extra points): Finite Element Method - implementation, 1D. Deadline: 21.1.2016, 9:00
    MATLAB implementation     see solution here
    or
    FEniCS implementation of Homework 9
  • Reading assignment 11: Script 3.4, Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Sections 10.1 Deadline: 03. 02.2016.
  • Homework 11: (last obligatory homework!): deadline is only the 13.02.2016. Solution
  • Reading assignment 12: Gauß quadrature, Michael T. Heath: Scientific Computing, Chapter 7. Chapter 8.1-8.3, Deadline: 10.2.2016.

Hint: Please, submit your assignments at the beginning of lecture or into a postbox of Institute für Wissenschaftliches Rechnen in Mühlenpfordtstr. 23, (ground floor at right side from the main entrance); software should be submitted to email address wire.pde(at)gmail.com Always indicate your names and imatriculation number!

Sketch of tutorials and exercises

  • Tutorial 1 - FEniCS - expression sheet (by Stephan Lenz)
  • Tutorial 4 - MATLAB - making a moovie example (by Stephan Lenz)
  • Exercises by Jaroslav. You can find there my notes for tutorial, which can be also editted. If you want to improve a text or just write a questions, you can write directly into a text; I keep all the document history by version control system, so do not be afraid to edit. PDF can be downloaded by clicking on an icon on upper bar.

Fenics (software for finite element modelling):

Additional information can be found in StudIP. The links below work only if you are logged into StudIP.