Mathematics

We discuss numerical approximations of two kinds of fractional differential operators: spectral fractional powers of an elliptic operator and the integral fractional Laplacian. We approximate the Dunford-Taylor integral representation of the former with an exponential convergent sinc quadrature scheme and discretize the integrand (a boundary value problem for the reaction-diffusion equation) at each quadrature point using the finite element method. In terms of the integral fractional Laplacian, we apply the Dunford-Taylor integral representation to the corresponding variational formulation. Unlike the spectral case, the integrand is a solution of reaction-diffusion equation defined in the whole space. We approximate the integrand problem on a truncated domain together with the finite element method. For both fractional operators, we provide error estimates between solutions and their final approximations. Some extensions and numerical simulations are also provided in this talk.