arXiv2017
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 $Ω\subset \mathbb{R}^d$ 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 $Ω$. In $Ω\subset \mathbb{R}^d$, 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 $f\in \mathbb{H}^{1-s}(Ω)$. We also prove that tensorization of a $P_1$-FEM in $Ω$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_Ω$, the number of degrees of freedom in the domain $Ω$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $Ω$ 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 $\mathcal{N}_Ω$. Finally, under the stronger assumption that the data is analytic in $\overlineΩ$, and without compatibility at $\partial Ω$, 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.