The fractional Laplacian has a global character and its solutions have strong boundary layers. Its efficient solution is still an open challenge for the community.
In this talk, we will show a novel a posteriori error estimation method for the spectral fractional Laplacian.
Our method begins with the work of  where a fractional operator is represented by an integral over non-fractional (ie local) parametric operators. The integral and the local operators can then be discretised using a quadrature rule and a standard finite element method, respectively.
We show that the integral representation of  can equally be applied to the construction of an error estimator. A key building block of the method is an efficient hierarchical estimator introduced in .
The estimator has numerous benefits: it is numerically sharp, it allows the measurement of the error in various norms, and it is defined for one, two and three-dimensional problems. It is also fully local and cheap to compute in parallel.
The estimator leads to optimal convergence rates when used to steer adaptive refinement algorithms.