# 101.820 VO (2std Vorlesung, Sommersemester 2019)

AKNUM:Nichtlokale Operatoren - Analysis und Numerik

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## Date and location

The lecture will be slightly blocked. Rather than doing 1.5 hours a week, I will give twice a week a 1 hour lecture. This compensates for weeks without lectures.- First lecture: Thursday, 10.10.2019 14:00
Room: Freihaus Sem.R. DA grün 03 B (third floor green)

**Lecture dates: Tuesday, 13:00-14:00, Thursday, 15:00-16:00**Room: Sem.R. DA grün 03C

On Thursday 07.11.2019 different room: Sem.R. DC rot 07

## Aim of the lecture

The aim of this lecture is to understand operator equations with non-local operators analytically and numerically.

## Subject of course

### What are non-local operators?

A non-local operator A (mapping between two function spaces) is characterized by the property that the evaluation of Af(x) does depend on the whole f and not only on values of f in a neighborhood of x. As various classical tools in analysis and numerics are not applicable to this class of operators, we introduce appropriate techniques for such problems.

### Examples of non-local operators

Differential operators such as the Laplacian Δ are local operators. However, integral operators such as the Fourier transform are non-local. In this lecture, we study integral operators of the form## Lectures

# | Date | Topic | Lecturer |
---|---|---|---|

1 | 10.10.2019 | Introduction | Faustmann |

2 | 15.10.2019 | Representation formula | Faustmann |

3 | 17.10.2019 | Function spaces | Faustmann |

22.10.2019 | No lecture | ||

24.10.2019 | No lecture | ||

4 | 29.10.2019 | Conormal derivative | Faustmann |

5 | 31.10.2019 | The Newton potential | Faustmann |

6 | 5.11.2019 | The single layer potential | Faustmann |

7 | 7.11.2019 | The Calderon system | Faustmann |

8 | 12.11.2019 | Integral representations | Faustmann |

9 | 14.11.2019 | Integral representations | Faustmann |

19.11.2019 | No lecture | ||

10 | 21.11.2019 | Jump conditions, ellipticity | Faustmann |

11 | 26.11.2019 | Ellipticity, Galerkin method | Faustmann |

12 | 28.11.2019 | A-priori estimates | Faustmann |

13 | 3.12.2019 | Introduction to fractional operators | Faustmann |

5.12.2019 | No lecture due to distinguished PDE lecture | ||

14 | 10.12.2019 | The Fourier definition | Faustmann |

15 | 12.12.2019 | The Caffarelli Silvestre extension | Faustmann |

16 | 17.12.2019 | FL on bounded domains, localization | Faustmann |

19.12.2019-6.1.2020 | Christmas Holidays | ||

17 | 7.1.2020 | Localization, regularity | Faustmann |

18 | 9.1.2020 | FEM for integral FL | Faustmann |

19 | 14.1.2020 | Weak formulation, regularity spectral FL | Faustmann |

20 | 16.1.2020 | FEM for spectral FL | Faustmann |

21.1.2020 | no lecture - PhD defense of colleague | ||

21 | 23.1.2020 | Dunford-Taylor approach | Faustmann |

## Downloads

In the following you can find the lecture notes for the first part of the lecture.The current iteration of the lecture notes also includes - for sake of completeness - proofs I did not do in the lecture.

I have not done the proofreading yet. Therefore, I would appreciate if you would tell me any mistakes you find!

22.01.2020 | Lecture Notes | [pdf] |

## Literature

W. McLean: Strongly elliptic systems and boundary integral equations Link

S. Sauter, C. Schwab: Boundary Element Methods Link

A. Bonito, J.P. Borthagaray, R.H. Nochetto, E. Otarola, A.J. Salgado: Numerical methods for fractional diffusion Link