Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D

TL;DR

This paper proves that the hp-FEM method converges exponentially fast for solving the fractional Laplacian problem in one dimension, leveraging weighted analytic regularity of the solution.

Contribution

It establishes weighted analytic regularity for the fractional Poisson problem and demonstrates exponential convergence of hp-FEM on geometric meshes.

Findings

Exponential convergence of hp-FEM for 1D fractional Laplacian.

Weighted analytic regularity of solutions on bounded intervals.

Effective boundary-refined mesh strategies.

Abstract

We prove weighted analytic regularity for the solution of the integral fractional Poisson problem on bounded intervals with analytic right-hand side. Based on this regularity result, we prove exponential convergence of the hp-FEM on geometric boundary-refined meshes.