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

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

(Report number 2022-39)

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 [15] and meshes that feature anisotropic geometric refinement towards \(\partial\Omega\).

Keywords:

@Techreport{FMMS22_1027,
  author = {M. Faustmann and C. Marcati and J. Melenk and Ch. Schwab},
  title = {Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-39.pdf },
  year = {2022}
}

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