In this talk we analyze a classical SOLVE-ESTIMATE-MARK-REFINE adaptive algorithm applied to PDEs involving fractional powers of differential operators, such as the fractional Laplacian. We mainly focus on the derivation of a-posteriori error indicators and present a weighted residual type error estimator that is well-defined for all fractional parameters. Moreover, we show that this leads to an algorithm that converges with optimal algebraic rates.
A drawback of the weighted residual type error estimator is that its computation is very costly. In order to counteract that, we present an upper bound for the error indicators that --using a splitting into near field and far field parts and employing hierarchical matrices - can be computed in logarithmic linear complexity.
This is joint work with Björn Bahr, Jens Markus Melenk and Dirk Praetorius.