Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons

TL;DR

This paper proves that hp finite element methods achieve exponential convergence rates for solving the integral fractional Laplacian in polygonal domains, leveraging weighted regularity and geometric mesh refinement.

Contribution

It establishes the exponential convergence of hp FEM for the fractional Laplacian in polygons, combining weighted regularity with anisotropic mesh refinement techniques.

Findings

Exponential convergence in energy norm demonstrated for hp FEM.

Mesh refinement towards boundary is crucial for convergence.

Weighted analytic regularity underpins the theoretical results.

Abstract

We prove exponential convergence in the energy norm of $hp$ finite element discretizations for the integral fractional diffusion operator of order $2s\in (0,2)$ subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains $\Omega\subset \mathbb{R}^2$. Key ingredient in the analysis are the weighted analytic regularity from our previous work and meshes that feature anisotropic geometric refinement towards $\partial\Omega$.