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

TL;DR

This paper introduces a new a posteriori error estimator for the finite element approximation of the fractional Laplacian, enabling effective adaptive mesh refinement through rational approximations and error reconstruction.

Contribution

It develops a novel error estimator leveraging rational approximations and Bank-Weiser strategy, improving adaptive finite element methods for fractional Laplacian problems.

Findings

Estimator effectively guides adaptive mesh refinement.

Accurate error estimation for various fractional powers.

Demonstrated success in 2D and 3D numerical examples.

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 rational approximations which allow 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. In addition we propose an algorithm to adapt both the finite element mesh and the rational scheme in order to balance the discretization errors. 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.