Efficient realization of a residual-type error estimator for the fractional Laplacian

Abstract

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. This is achieved by decomposition of the error estimator into a near-field and a far-field contribution, where the near-field is the integration around the singularity and the far field only contains smooth parts. For the near-field, we prove an analytical upper-bound, which can be computed in constant time. The far-field can be computed in quasi-linear time, employing the technique of hierarchical matrices.This work contains not only the proof of the upper bound for the error estimator but also a pseudocode for a possible implementation. Furthermore, the relevant theorems for the convergence rates of FEM algorithms are given and the associated proofs are sketched. Finally, the theoretical results are confirmed by numerical experiments. In particular, the experiments show that an adaptive FEM algorithm converges with the optimal rate, even tough it uses the upper bound of the error estimator instead of the original one.