Ludwig Striet - Approximation of fractional operators and fractional PDEs using a sinc-basis

We introduce a spectral method to approximate PDEs involving the fractional Laplacian with zero exterior condition. Our approach is based on interpolation by tensor products of sinc-functions, which combine a simple representation in Fourier-space with fast enough decay to suitably approximate the bounded support of solutions to the Dirichlet problem. This yields a numerical complexity of O(NlogN) for the application of the operator to a discretization with N degrees of freedom. Iterative methods can then be employed to solve the fractional partial differential equations with exterior Dirichlet condition. We show a number of example applications  and establish rates of convergence that are in line with rates for finite element based approaches.