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We present a finite element (FE) scheme for the numerical approximation ofthe solution to a non-local Poisson equation involving the one-dimensionalfractional Laplacian $(-d_x^2)^s$ on the interval $(-L,L)$. In particular, weinclude the complete computations for obtaining the stiffness matrix, startingfrom the variational formulation of the problem. The problem beingone-dimensional, the values of the matrix can be explicitly calculated, withoutneed of any numerical integration, thus obtaining an algorithm which is veryefficient in terms of the computational cost. As an application, we analyze thecorresponding parabolic equation from the point of view of controllabilityproperties, employing the penalized Hilbert Uniqueness Method (HUM) forcomputing the numerical approximation of the null-control, acting from an opensubset $\omega\subset(-L,L)$. In accordance to the theory, our numericalsimulations show: (1) that the method solves the elliptic equation with anacceptable approximation in the natural functional setting, (2) that theparabolic problem is null-controllable for $s>1/2$ and (3) that for $s\leq 1/2$we only have approximate controllability.

A Finite Element approximation of the one-dimensional fractional Poisson equation with applications to numerical control