Fast spectral Petrov-Galerkin method for fractional elliptic equations

Abstract

In this work, we revisit the spectral Petrov-Galerkin method for fractional elliptic equations with the general fractional operators. To prove the optimal convergence of the method, we first present the ultra-weak formulation and establish its well-posedness. Then, based on such a novel formulation, we are able to prove the discrete counterpart and obtain the optimal convergence of the spectral method in the weighted L2-norm. For simple and easy implementation of the method, we also describe the fast solver with linear storage and quasilinear complexity. To support our theory, we carry out the numerical experiments and provide several numerical results to show the accuracy and efficiency of our method.