Numerical Methods for PDEs

Numerical Methods for PDEs

General Information

Lecturer

Dr. Jaroslav Vondřejc

Please register via StudIP

Numerische Methoden für PDEs

j.vondrejc@tu-bs.de

Lecture: Wed 09:45-11:15 o'clock in seminar room of Institute of Scientific Computing, Mühlenpfordstrasse 23.

Tutorials: Friday 13:15-14:45 in seminar or computer room of Institute of Scientific Computing, Mühlenpfordstrasse 23.

Wednesday 4.4.2018

Introduction to PDEs and Numerical Methods

INF-WR

j.vondrejc@tu-bs.de

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Literature and other material:

  • Braess, D. (2007). Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press.
  • Gockenbach, M. S. (2006). Understanding and implementing the finite element method. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics.
  • Langtangen, H.P., Logg, A.: Solving PDEs in Python: The FEniCS Tutorial I, Springer.
  • Bitbucket repository contains some python scripts. See also repository with codes for PDE1.
  • Hints for FEniCS and Python (Installation, tutorials, etc.).

Lectures

  • Lecture 1: Basis informations, motivation.
  • Lecture 2: Lebesgue spaces, norms, scalar product.
  • Lecture 3-4: Sobolev spaces.
  • Lecture 5: Existence of a solution to PDEs
  • Lecture 6: Discretisation of PDEs
  • Lecture 7: Convergence of FEM
  • Lecture 8-9: Choosing approximating basis (Lagrange and Hermite polynomials, different degree basis functtions in 1 and 2D), sparsity and condiiton number of the stiffness matrix, a-priori error estimates, isoparametric-mapping
  • Lecture 9-10: Numerical inegration over quadratic and triangular elements, implementation of the FEM (see in Gockenbach: Understanding and implementing the FEM: Chapter 5.5.2, 7.1-7.2, and tutorial)
  • Lecture 10: Basics of adaptivity and a-posteriori error estimators/indicators (see in Gockenbach: Understanding and implementing the FEM: Chapter 14, Introduction of Chapter 15 and Chapter 15.1)

In case you find some bugs or typos in the lectures, please contact Jaroslav on email or edit the source on Overleaf.

Homework Assignments

Please, submit electronic document (e.g. code) to email address: wire.pde(at)gmail.com

  • Reading assignment: Langtangen, H.P., Logg, A.: Solving PDEs in Python (use link for download above), sections 2.1-2.3, deadline 13.4.
  • Assignment 01: Vector spaces, completeness, FEM norms. Deadline Friday 20/4/2018, before tutorial. Solution can be found in pdf and also in bitbucket repository.
  • Reading assignment (assembling the system matrix): Repeat Assignment 9 from the PDE1 course taught in the winter semester. Read from book of Braess (2007), Chapter II, Paragraph 8 or book of Gockenbach, section 4.6.
  • Assignment 02: Properties of Sobolev functions, assembling local system matrices. Deadline: Friday 4/5/2018. Solution can be found in pdf and as a python script in Bitbucket repository.
  • Reading assingment from Gockenbach, Section 5.
  • Assignment 03: Convergence of FEM, propeties of system matrix. Deadline Wednesday 30/5/2018. Solution is on the bitbucket repository.
  • Assignment 04: Error estimators-indicators, FEM implementation, choosing FEM basis, adaptivity , Deadline: Frieday, 2906.2018, solutions can be accessed here
  • Assignment 05: Saddle point linear systems. Deadline: Wednesday 4.7.2018. Solution can be found on bitbucket repository.
  • Assignment 06: Raviart Thomas elements and mixed formulations. Deadline Wed 18.7.2018. The solution (part) can be found on bitbucket repository.