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
$$\varOmega \subset {\mathbb {R}}^d$$
with
$$d=1,2$$
. For the solution to the Caffarelli–Silvestre extension, we establish analytic regularity with respect to the extended variable
$$y\in (0,\infty )$$
. Specifically, the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to *y*, taking values in corner-weighted Kondrat’ev-type Sobolev spaces in
$$\varOmega $$
. In
$$\varOmega \subset {\mathbb {R}}^2$$
, we discretize with continuous, piecewise linear, Lagrangian FEM (
$$P_1$$
-FEM) with mesh refinement near corners and prove that the first-order convergence rate is attained for compatible data
$$f\in \mathbb {H}^{1-s}(\varOmega )$$
with
$$0<s<1$$
denoting the fractional power. We also prove that tensorization of a
$$P_1$$
-FEM in
$$\varOmega $$
with a suitable *hp*-FEM in the extended variable achieves log-linear complexity with respect to
$${\mathscr {N}}_\varOmega $$
, the number of degrees of freedom in the domain
$$\varOmega $$
. In addition, we propose a novel, *sparse tensor product FEM* based on a multilevel
$$P_1$$
-FEM in
$$\varOmega $$
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
$${\mathscr {N}}_\varOmega $$
. Finally, under the stronger assumption that the data be analytic in
$$\overline{\varOmega }$$
, and *without compatibility at*
$$\partial \varOmega $$
, 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
with anisotropic geometric meshes that are refined toward
$$\partial \varOmega $$
. 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
$$\partial \varOmega $$
.