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

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.