  We propose a new nonconforming finite element algorithm to approximate the
solution to the elliptic problem involving the fractional Laplacian. We first
derive an integral representation of the bilinear form corresponding to the
variational problem. The numerical approximation of the action of the
corresponding stiffness matrix consists of three steps: (i) apply a sinc
quadrature scheme to approximate the integral representation by a finite sum
where each term involves the solution of an elliptic partial differential
equation defined on the entire space, (ii) truncate each elliptic problem to a
bounded domain, (iii) use the finite element method for the space approximation
on each truncated domain. The consistency error analysis for the three steps is
discussed together with the numerical implementation of the entire algorithm.
The results of computations are given illustrating the error behavior in terms
of the mesh size of the physical domain, the domain truncation parameter and
the quadrature spacing parameter.

Numerical Analysis