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B. Bahr:
"Efficient realization of a residual-type error estimator for the fractional Laplacian";
Supervisor: J. Melenk, M. Faustmann; Institut für Analysis und Scientific Computing, 2021; final examination: 2021-09-09.

This thesis focuses on the efficient calculation of a residual error estimator from the work "Quasi-optimal convergence rate for an adaptive method for the integral fractional laplacian"
of M. Faustmann, J. Melenk and D. Praetorius. This estimator is used to steer an adaptive finite element method (FEM) algorithm for the fractional Laplace operator. Calculating the error estimator leads to two problems. First, the function to be calculated contains a singularity and second, the error estimator consists of a double integral. Therefore,the computational effort is quadratic with a bad constant when employing classical quadrature techniques.
The aim of this work is not only to provide fundamental knowledge regarding the fractional Laplace operator and FEM in general, but also to show an upper bound for the error estimator in one dimension that can be calculated in quasi-linear time.