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

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.