Numerical Methods for PDEs

Numerical Methods for PDEs

General Information

Lecturers

Dr. Noémi Friedman

Dr. Jaroslav Vondřejc

Lecture: Wed 09:45-11:15 o'clock in room PK 3.4. (Pockelsstraße 3, Am Okerufer, Hochhaus, 205)

Additional lecture and tutorial (due to cancelation of first week): Tuesday 21th and 28th of June at 16:30-18:00 in Mühlenpfordtstr. 23, room 812 (seminar room), 8th floor

Tutorials: Friday 13:25-14:55 o'clock in Mühlenpfordtstr. 23, room 812 (seminar room), 8th floor

13.4.2016 (Cancelled lecture in the first week of the semester will be rescheduled.)

Introduction to PDEs and Numerical Methods

INF-WR

j.vondrejc@tu-bs.de

The 15th of August 2016. 10:45-12:15 (PK 11.2.); requirements for Part I and part III and for part II and IV

For the exam papers will be provided by the Institute, only pen and hand calculator is needed. No additional devices and no hand written notes can be used.

The next exam is planned on Tuesday 21th February, 2017, at 10:00 in the Institute of Scientific Computing, Mühlenpfordstr. 23, 8th floor.

Literature

Literature is available here: http://www.biblio.tu-bs.de/semapp/
There go to: Prof. Hermann G. Matthies: Numerical Methods for PDEs.
The password for downloading material has been given in the lecture.

Lecture Drafts

First block (Lebesgue integration, basics of functional analysis and measure theory, uniqueness and existence of solution of PDEs) :

  • Lectures by Jaroslav are available here (updated: the lecture for 11.5. - Existence results)

Second block (Galerkin projection, piecewise polynomials and the FEM, numerical integration and convergence):

  • Lecture 1: The Galerkin method
  • Lecture 2: Piecewise polynomials and the finite element method
  • Lecture 3: Functional discretization, and a priori error estimates (see draft of Lecture 2 and 3 here)
  • Lecture 4: Variational crimes, numerical integration (See in Gockenbach: Understanding and Implementing the FEM: Chapter 5.5.2), Isoparametric mapping, Gauss-quadrature over rectangular and triangular elements and its accuracy (See in Gockenbach: Understanding and Implementing the FEM: Chapter 7-8)

Third block (Saddle point problems, mixed formulations):

  • Lectures are available here in the part Saddle point problems.
  • This part is complemented with some examples stored on Bitbucket using software Fenics.
  • There is a posibility to use Fenics in our computer room or you can install it on your computer. The guide to install it on windows can be found here. If you want to use your computer, I recommend to try to run examples stored on Bitbucket.

Fourth block (Mesh generation, adaptivity, a posterior error estimates):

  • Mesh generation, adaptivity (See in Gockenbach: Understanding and Implementing the FEM: Chapters 6 and 14)
  • A-posterior error estimators (See in Gockenbach: Understanding and Implementing the FEM: Chapters 15), see draft here.

Homework Assignments & Tests

First block (Lebesgue integration, basics of functional analysis and measure theory, uniqueness and existence of solution of PDEs) :

  • For the first four lectures, there are no assignments. Instead of that, there was a small test during lecture on Wednesday, 25 May 2016 with results here. The requirements are available here. For those students who did not attend to a small test or who want to improve their score, there was another test during a lecture on Wednesday, 29 June 2016 (test with solution).

Second block (Galerkin projection, piecewise polynomials and the FEM, numerical integration and convergence):

  • Reading assignment: Recap: Variational formulation - Gockenbach: Understanding and Implementing the FEM Chapter 1 and 2, and the Galerkin-method: Chapter 3.
  • Reading assignment: Piecewise polynomials and the finite element method (Gockenbach: Understanding and Implementing the FEM: Chapter 4). Recap: Gauß-quadrature, Isoparametric maping, compilation of the stiffness matrix. Due date: 08.06.2016.

Third block (Saddle point problems, mixed formulations) - submit to wire.pde@gmail.com:

  • Assignment 1: Saddle-point linear system (due date: Fri 01.07.2016) solution
  • Assignment 2 (due date: Wed 13.7.2016) The deadline has been prolongated because a file on bitbucket was missing for exercise 1. If you have already submitted, you can add the exercise and resubmit again. The last version will be considered.

Fourth block (Mesh generation, adaptivity, a posterior error estimates):

  • Reading assignment: Chapters 6 and 14 Gockenbach: Understanding and Implementing the FEM (deadline: 08.07.2016.)
  • Assignment : Mesh, data structures, adaptivity. Due date: 20.07.2016.
  • Reading assignment: Chapters 15 Gockenbach: Understanding and Implementing the FEM (deadline: 15.07.2016.)