Fast spectral Petrov-Galerkin method for fractional elliptic equations

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