Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons

Markus Faustmann, Carlo Marcati, Jens Markus Melenk, Christoph Schwab

Abstract: We prove exponential convergence in the energy norm of $hp$ finite element discretizations for the integral fractional diffusion operator of order $2s\in (0,2)$ subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains $\Omega\subset \mathbb{R}^2$. Key ingredient in the analysis are the weighted analytic regularity from our previous work and meshes that feature anisotropic geometric refinement towards $\partial\Omega$.