Exponential Convergence of \(hp\)-FEM for the Integral Fractional Laplacian in Polygons

Abstract

.We prove exponential convergence in the energy norm of \(hp\)-finite element discretizations for the integral fractional Laplacian of order \(2s\in (0,2)\) subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains \(\Omega \subset{\mathbb R}^2\). Key ingredients in the analysis are the weighted analytic regularity from [M. Faustmann, C. Marcati, J. M. Melenk, and C. Schwab, SIAM J. Math. Anal., 54 (2022), pp. 6323–6357] and meshes that feature anisotropic geometric refinement towards \(\partial \Omega\).Keywordsfractional Laplaciancorner domainshp-FEMexponential convergenceMSC codes35R1165N1265N30