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Exponential Convergence of hp FEM for Spectral Fractional Diffusion in Polygons

by L. Banjai and J.M. Melenk and Ch. Schwab

(Report number 2020-67)

Abstract
For the spectral fractional diffusion operator of order \(2s\in (0,2)\) in bounded, curvilinear polygonal domains \(\Omega\subset {\mathbb R}^2\) we prove exponential convergence of two classes of \(hp\) discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm \(\mathbb{H}^s(\Omega)\). The first \(hp\) discretization is based on writing the solution as a co-normal derivative of a \(2+1\)-dimensional local, linear elliptic boundary value problem, to which an \(hp\)-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in \(\Omega\). Leveraging results on robust exponential convergence of \(hp\)-FEM for second order, linear reaction diffusion boundary value problems in \(\Omega\), exponential convergence rates for solutions \(u\in \mathbb{H}^s(\Omega)\) of \({\cal L}^s u = f\) follow. Key ingredient in this \(hp\)-FEM are boundary fitted meshes with geometric mesh refinement towards \(\partial\Omega\).
The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of \({\cal L}^{-s}\) combined with \(hp\)-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in \(\Omega\). The present analysis for either approach extends to (polygonal subsets \(\widetilde{{\cal M}}\) of) analytic, compact \(2\)-manifolds \({\cal M}\), parametrized by a global, analytic chart \(\chi\) with polygonal Euclidean parameter domain \(\Omega\subset {\mathbb R}^2\). Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results.
Exponentially small bounds on Kolmogoroff \(n\)-widths of solutions sets for spectral fractional diffusion in polygons are deduced.

Keywords: Fractional diffusion, nonlocal operators, Dunford-Taylor calculus, anisotropic hp-refinement, geometric corner refinement, exponential convergence, n-widths.

@Techreport{BMS20_940,
  author = {L. Banjai and J.M. Melenk and Ch. Schwab},
  title = {Exponential Convergence of hp FEM for Spectral Fractional Diffusion in Polygons},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-67},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-67.pdf },
  year = {2020}
}

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