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# Numerics of partial differential equations

Lecture (4 SWS) with Tutorial (2 SWS) in winter term 2015/2016 at TU Berlin, 10 credit points (ECTS)

Time | Room | Lecturer | |
---|---|---|---|

Lecture | Mon, 14.15 - 15.45 | MA 550 | Dr. Kersten Schmidt |

Lecture | Tue, 10.15 - 11.45 | MA 549 | Dr. Kersten Schmidt |

Tutorial class | Thu, 14.15 - 15.45 | A 052 | Dirk Klindworth (until December 2015), Dr. Anastasia Thöns-Zueva (from January 2016) |

**Instead of the tutorial classes on December 3rd, 10th and 17th there will be consultations from 2.30 p.m. to 4.00 p.m. in MA 366. Please bring your laptop!**- On Thursday, October 15th, 2015 there will be a lecture instead of a tutorial class.
- On Monday, October 19th, 2015 there will be an introductory course on Python instead of a lecture. Please bring your laptop!
- On Tuesday, October 20th, 2015 there will be a tutorial class instead of a lecture.

## Content

The lecture deals with the numerical solution of partial differential equations, especially by the finite element method. We will discuss the theoretical basis as well as the algorithmic concepts.

**Topics:**

- Overview and characterisation of second order PDEs
- Strong formulation of elliptic PDEs and discretisation by finite differences
- Variational formulation of elliptic PDEs
- Sobolev spaces
- Discretisation in discrete subspaces
- Finite element method
- Direct and iterative solution of the linear systems
- Analysis of the variational formulations
- Regularity in Sobolev spaces
- Numerical analysis, especially error estimates

## References

- P. Solin, "Partial Differential Equations and the Finite Element Method", John Wiley & Sons, 2006. (Online verfügbar aus dem Campusnetz der TU Berlin: http://dx.doi.org/10.1002/0471764108)

D. Braess, “Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie”, Springer, 2007. (Online verfügbar aus dem Campusnetz der TU Berlin: http://dx.doi.org/10.1007/978-3-540-72450-6)

P. Deuflhard, M. Weiser „Numerische Mathematik III - Adaptive Lösung partieller Differentialgleichungen“, de Gruyter, 2011.

## Tutorials

The exercises can be solved and handed in in groups of two students. The groups may vary during the semester.

**Programming exercises have to be solved in groups of two students.** In extraordinary cases they can be handed in in groups of three students.

For successfully solving the programming exercises (starting in series 2), it is recommended that in each group there is at least one student with basic knowledge in Python or a comparable programming language such as Matlab.

**The solutions can be handed in in paper form or as PDF document attached to an e-mail to Anastasia Thöns-Zueva.**

- Series 1, to be handed in by October 27th, 2015, 10.15 a.m.
- Series 2, to be handed in by October 27th, 2015, 10.15 a.m.
- Series 3, to be handed in by November 3rd, 2015, 10.15 a.m. (programming exercise by November 10th, 2015, 10.15 a.m.)
- Series 4, to be handed in by November 10th, 2015, 10.15 a.m.
- Series 5, to be handed in by November 17th, 2015, 10.15 a.m.

(programming exercise by November 24th, 2015, 10.15 a.m.) - Series 6, to be handed in by November 24th, 2015, 10.15 a.m.
- Series 7, to be handed in by December 8th, 2015, 10.15 a.m.
- Series 8, to be handed in by January 5th, 2016, 10.15 a.m.
- Series 9, to be handed in by January 18th, 2016, 9.00 a.m.
- Series 10, to be handed in by February 4th, 2016, 9.00 a.m.

**Python introduction**

The Python script myscript.py contains all commands discussed in the Python introduction on October 19th and 20th, 2015.

## Exams

There will be oral exams at the end of the term. For being excepted, active participation in the tutorial classes is required.