Contributions to Books:
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,
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.
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.