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 $L^2$-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.

在这项工作中，我们使用通用分数算子重新讨论了分数椭圆方程的谱Petrov-Galerkin方法。为了证明该方法的最佳收敛性，我们首先介绍了超弱公式并建立了它的适定性。然后，基于这种新颖的公式，我们能够证明离散的对应项，并获得加权法中频谱方法的最优收敛性。${\mathrm{大号}}^2$-规范。为了简单方便地实现该方法，我们还描述了具有线性存储和拟线性复杂度的快速求解器。为了支持我们的理论，我们进行了数值实验并提供了一些数值结果，以证明我们方法的准确性和效率。