Though the finite element method has been widely used in solving fractional differential equations, the effects of the Gaussian quadrature rule on the numerical results have rarely been considered. Since the fractional derivatives of the basis functions are not polynomials with integer power and always have weak singularities on some elements, the Gaussian quadrature rule (Gauss-Legendre quadrature rule) may not be suitable in assembling the fractional stiffness matrix. By splitting the integrand of the inner products into a weak singularity part and a smooth part and utilizing the Gauss-Jacobi quadrature rule for the weak singularity part, we present a modified algorithm to assemble the fractional stiffness matrix. The numerical results, conducted on 1D and 2D domains, show that our method can significantly improve the accuracy of the stiffness matrix as well as the accuracy of the numerical solution with much fewer Gaussian points.

尽管有限元法在求解分数阶微分方程中得到了广泛的应用，但很少考虑高斯求积法则对数值结果的影响。由于基函数的分数阶导数不是具有整数次幂的多项式，并且在某些元素上总是具有弱奇异性，因此高斯求积规则（Gauss-Legendre quadrature rule）可能不适用于组装分数刚度矩阵。通过将内积的被积函数拆分为弱奇异部分和平滑部分，并利用弱奇异部分的高斯-雅可比求积法则，我们提出了一种改进的算法来组装分数刚度矩阵。在一维和二维域上进行的数值结果，