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L. Banjai, J. Melenk, R. Nochetto, E. Otarola, A. Salgado, C. Schwab:
"Tensor FEM for spectral fractional diffusion";
in: "ASC Report 13/2017", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2017, ISBN: 978-3-902627-10-0, 1 - 44.

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 ⊂ Rd with d = 1, 2. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable y ∈ (0, ∞). 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 ⊂ R2, we discretize with continuous, piecewise linear, Lagrangian FEM (P1-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data f ∈ H1−s(Ω).

Fractional diffusion, nonlocal operators, weighted Sobolev spaces, regularity esti-mates, finite elements, anisotropic hp-refinement, corner refinement, sparse grids, exponential con-vergence.

http://www.asc.tuwien.ac.at/preprint/2017/asc13x2017.pdf