Using Gauss-Jacobi quadrature rule to improve the accuracy of FEM for spatial fractional problems

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.