Local a posteriori error estimates for the spectral fractional Laplacian

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 [1] 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 [1] 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 [2].

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.

The implementation is based on FEniCS Error Estimation (FEniCS-EE), a finite element error estimation package for DOLFIN and DOLFINx [1].

References

• [1] A. Bonito and J. E. Pasciak. Numerical Approximation of Fractional Powers of Elliptic Operators, 2015. [DOI: 10.1090/S0025-5718-2015-02937-8]