Exponential Convergence of $hp$-FEM for the Integral Fractional Laplacian on cuboids
This work provides a theoretical foundation for high-order numerical methods in fractional PDEs, which is incremental as it extends prior results to 3D with specific mesh refinements.
The paper tackled the problem of approximating the Dirichlet integral fractional Laplacian on a 3D cube using tensor-product hp-finite element methods, achieving root exponential convergence with an energy norm error bound of exp(-b * sixth root of N) for analytic forcing functions.
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.