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

Fractional powers of the Laplacian operator are important tools in the modeling and study of non-local phenomena. Several numerical challenges arise from the discretization of these operators due to their non-local nature. For example, a direct discretization via finite element methods can lead to dense and possibly large linear systems. One way to circumvent this density is by using a rational scheme combined with a finite element method.
In this talk we describe a novel local a posteriori estimator for the finite element discretization error measured in the L2 norm that can be used to perform adaptive mesh refinement. This estimator is adapted from the strategy introduced by Bank and Weiser and can be used with any rational approx- imation scheme such as best uniform rational approximations or schemes based on the Dunford–Taylor formula. Especially, our estimator preserves the locality and robustness of the Bank–Weiser estimator and preserves the parallel nature of rational approximations. In addition, oour method can be combined with an estimator for the rational approximation error to obtain a more complete description of the discretization errors.
Finally, we use an implementation in the FEniCSx finite element software to demonstrate the performances of our method on several numerical experiments including three–dimensional prob- lems.