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arXiv · 2603.09449

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

Björn Bahr·Markus Faustmann·Carlo Marcati·Jens Markus Melenk·Christoph Schwab
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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.

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Björn Bahr, Markus Faustmann, Carlo Marcati, Jens Markus Melenk, Christoph Schwab. 2026-03-10. Exponential Convergence of $hp$-FEM for the Integral Fractional Laplacian on cuboids. https://arxiv.org/abs/2603.09449

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