A sinc-function based numerical method for the Dirichlet problem with fractional Laplacian

Patrick Dondl (Albert-Ludwigs-Universität Freiburg) #

We introduce a spectral method to approximate 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 \(\mathcal{O}(N \log N)\) 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 prove a convergence rate that is in line with rates for finite element based approaches. This is joint work with Ludwig Striet (Freiburg) and Harbir Antil (Fairfax, VA).