## Preprint 2022-20

## Olga Barrera, Stéphane P. A. Bordas, Raphaël Bulle, Franz Chouly, Jack S. Hale:

### An a posteriori error estimator for the spectral fractional power of the Laplacian

### Abstract:

We develop a novel a posteriori error estimator for the $L^2$ error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi–discretization scheme using a rational approximation which allows to reformulate the fractional problem into a family of non–fractional parametric problems. The estimator involves applying the implicit Bank–Weiser error estimation strategy to each parametric non–fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy.

This preprint gave rise to the following definitive publication(s):

**Olga BARRERA, Stéphane P. A. BORDAS, Raphaël BULLE, Franz CHOULY, Jack S. HALE: ***An a posteriori error estimator for the spectral fractional power of the Laplacian*. Computer Methods in Applied Mechanics and Engineering, vol. 407, Art. Num. 115943, (2023).