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

by B. Bahr and M. Faustmann and C. Marcati and J.M. Melenk and Ch. Schwab

(Report number 2026-08)

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 \cite{FMMS25-regularity3d}, 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. \keywords{Fractional Laplacian \(\cdot\) \(hp\)-FEM \(\cdot\) exponential convergence}}

Keywords:

BibTeX

@Techreport{BFMMS26_1163,
  author = {B. Bahr and M. Faustmann and C. Marcati and J.M. Melenk and Ch. Schwab},
  title = {Exponential Convergence of hp-FEM for the Integral Fractional Laplacian on cuboids},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-08.pdf },
  year = {2026}
}

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