Lecture WT 2024/25 (2h)

Time and place

Wed 10-11:30 Small seminar room DA04G10, opens an external URL in a new window
(Freihaus, 4th floor green, inside the institute doors)

Subject of course

In this lecture we deal with non-local operators. Classical examples are integral operators with a singular kernel, which appear in integral equations in reformulations of partial differential equations, and fractional differential operators, i.e., derivatives with non-integer order. A formal definition of a fractional derivative is possible in many ways and in the lecture several definitions are presented and examined for equivalence.

The primary focus of the course is first to create an analytical framework for the treatment of such operator equations and then to present and analyze numerical methods for them. A fundamental difference to local operators, such as classical differential operators, is that discretizations lead to fully populated matrices and thus make an efficient numerical solution difficult.

Date Lecturer Topic
02.10. Markus Faustmann Organization and Introduction
09.10. Markus Faustmann Fractional Laplacian - Definitions
16.10. Markus Faustmann The Caffarelli-Silvestre extension
23.10. Markus Faustmann Fractional Laplacian on bounded domains
30.10. Markus Faustmann Galerkin Approximation for IFL
06.11. Markus Faustmann A-priori error analysis for high regularity
13.11. Markus Faustmann Regularity for the integral fractional Laplacian
20.11. Markus Faustmann Mesh design principles based on regularity
27.11. Markus Faustmann Exponential convergence of hp-FEM
04.12. Paul Dunhofer FFT approximation for fractional PDEs
11.12. Markus Faustmann AFEM for fractional PDEs
18.12. Markus Faustmann Numerical integration
08.01. Markus Faustmann Matrix compression
15.01. Markus Faustmann FEM for the spectral fractional Laplacian
22.01. Markus Faustmann Dunford-Taylor approach
This table gives an overview of all lectures, in the first column is the date of the lecture, in the second column the lecturer, in the third column the topic of the lecture.

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