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

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Abstract

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.

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Funding

This research was supported by the National Natural Science Foundation of China (11971386), the National Key R&D Program of China (2020YFA0713603), the Natural Science Foundation of Shaanxi Province (2020JM-132), and the Fundamental Research Funds for the Central Universities (310201911cx025).

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Authors and Affiliations

  1. Research Center for Computational Science, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, China

    Zongze Yang, Jungang Wang, Zhanbin Yuan & Yufeng Nie

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  1. Zongze Yang
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  2. Jungang Wang
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Correspondence to Yufeng Nie.

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Yang, Z., Wang, J., Yuan, Z. et al. Using Gauss-Jacobi quadrature rule to improve the accuracy of FEM for spatial fractional problems. Numer Algor 89, 1389–1411 (2022). https://doi.org/10.1007/s11075-021-01158-x

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  • DOI: https://doi.org/10.1007/s11075-021-01158-x

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