Numerical Approximation of the Integral Fractional Laplacian

Understand

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.