Tensor FEM for spectral fractional diffusion

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 $\Omega \subset {\mathbb{R}}^d$, 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 ${\mathcal{N}}_{\Omega }$, 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 ${\mathcal{N}}_{\Omega }$. Finally, under the stronger assumption that the data is analytic in $\overline{\Omega }$, and without compatibility at $\partial \Omega$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.

References in corpus (5)

• An extension problem related to the fractional Laplacian
• A fractional Laplace equation: regularity of solutions and Finite Element approximations
• Numerical Methods for Fractional Diffusion
• Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra
• $hp$-Finite Elements for Fractional Diffusion

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• Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons
• Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization
• A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator
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• Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D
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