Exponential Convergence of $hp$-FEM for the Integral Fractional Laplacian on cuboids

TL;DR

This paper proves root exponential convergence of $hp$-finite element methods for the integral fractional Laplacian on cuboids, demonstrating rapid error decay for analytic forcing functions and confirming results with numerical experiments.

Contribution

It establishes the first root exponential convergence rates for $hp$-FEM applied to the integral fractional Laplacian on three-dimensional cuboids, using weighted Sobolev space regularity.

Findings

Error bound $oxed{ extstyle ext{exp}(-b N^{1/6})}$ proven for $hp$-FEM

Numerical experiments confirm theoretical convergence rates

Geometric mesh refinement enhances approximation accuracy

Abstract

For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product $hp$-finite element approximations on $(0,1)^3$, for forcing $f$ that is analytic in $[0,1]^3$. Exploiting analytic regularity estimates in weighted Sobolev spaces, we prove for $hp$-GLL interpolation approximations with $N$ degrees of freedom the energy norm error bound $\lesssim \exp(-b\sqrt[6]{N})$. Tensor product mesh families which are geometrically refined towards all sides of $(0,1)^3$ are used. Numerical experiments with $hp$-Galerkin FEM confirm the bound.