  We present a finite element (FE) scheme for the numerical approximation of
the solution to a non-local Poisson equation involving the one-dimensional
fractional Laplacian $(-d_x^2)^s$ on the interval $(-L,L)$. In particular, we
include the complete computations for obtaining the stiffness matrix, starting
from the variational formulation of the problem. The problem being
one-dimensional, the values of the matrix can be explicitly calculated, without
need of any numerical integration, thus obtaining an algorithm which is very
efficient in terms of the computational cost. As an application, we analyze the
corresponding parabolic equation from the point of view of controllability
properties, employing the penalized Hilbert Uniqueness Method (HUM) for
computing the numerical approximation of the null-control, acting from an open
subset $\omega\subset(-L,L)$. In accordance to the theory, our numerical
simulations show: (1) that the method solves the elliptic equation with an
acceptable approximation in the natural functional setting, (2) that the
parabolic problem is null-controllable for $s>1/2$ and (3) that for $s\leq 1/2$
we only have approximate controllability.
