A Fractional Stokes Equation and Its Spectral Approximation

Abstract

In this paper, we study the well-posedness of a fractional Stokes equation and its numerical solution. We first establish the well-posedness of the weak problem by suitably define the fractional Laplacian operator and associated functional spaces. The existence and uniqueness of the weak solution is proved by using the classical saddle-point theory. Then, based on the proposed variational framework, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: 1) a theoretical framework for the variational solutions of the fractional Stokes equation; 2) an efficient spectral method for solving the weak problem, together with a detailed numerical analysis providing useful error estimates for the approximative solution; 3) a fast implementation technique for the proposed method and investigation of the discrete system. Finally, some numerical experiments are carried out to confirm the theoretical results.