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Tensor FEM for spectral fractional diffusion

by L. Banjai and J. Melenk and R. Nochetto and E. Otarola and A. Salgado and Ch. Schwab

(Report number 2017-36)

Abstract
We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains \(\Omega \subset \mathbb R^d\) with \(d=1,2\). For the solution to the extension problem, we establish analytic regularity with respect to the extended variable \(y\in (0,\infty)\). We prove that the solution belongs to countably normed, power--exponentially weighted Bochner spaces of analytic functions with respect to \(y\), taking values in corner-weighted Kondat'ev type Sobolev spaces in \(\Omega\). In \(\Omega\subset \mathbb R^2\), we discretize with continuous, piecewise linear, Lagrangian FEM (\(P_1\)-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data \(f\in \mathbb{H}^{1-s}(\Omega)\). We also prove that tensorization of a \(P_1\)-FEM in \(\Omega\) with a suitable \(hp\)-FEM in the extended variable achieves log-linear complexity with respect to \(\cal N_\Omega\), the number of degrees of freedom in the domain \(\Omega\). In addition, we propose a novel, sparse tensor product FEM based on a multilevel \(P_1\)-FEM in \(\Omega\) and on a \(P_1\)-FEM on radical--geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to \(\cal N_\Omega\). Finally, under the stronger assumption that the data be analytic in \(\overline{\Omega}\), and without compatibility at \(\partial \Omega\), we establish exponential rates of convergence of \(hp\)-FEM for spectral, fractional diffusion operators in energy norm. This is achieved by a combined tensor product \(hp\)-FEM for the Caffarelli-Silvestre extension in the truncated cylinder \(\Omega \times (0,\cal Y)\) with anisotropic geometric meshes that are refined towards \(\partial\Omega\). We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods to other problem classes and to other boundary conditions on \(\partial \Omega\).

Keywords: Fractional diffusion, nonlocal operators, weighted Sobolev spaces, regularity estimates, finite elements, anisotropic $hp$--refinement, corner refinement, sparse grids, exponential convergence

@Techreport{BMNOSS17_732,
  author = {L. Banjai and J. Melenk and R. Nochetto and E. Otarola and A. Salgado and Ch. Schwab},
  title = {Tensor FEM for spectral fractional diffusion},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-36},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-36.pdf },
  year = {2017}
}

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