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An improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term

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Abstract

In this paper, we propose an improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term. The second-order backward differentiation formula (BDF2) and weighted and shifted Grünwald-Letnikov difference (WSGD) formula are employed to discretize first-order time derivative and the time fractional-order derivative, respectively. Moreover, a linearized difference scheme is proposed to approximate the nonlinear source term. Together with the Galerkin finite element method in the space direction, we present a fully discrete scheme for the fractional Rayleigh–Stokes problem with a nonlinear source term. Based on a novel analytical technique, the stability and the convergence accuracy in \(L^{2}\)-norm with \(O(\tau ^{2}+h^{k+1})\) are derived in detail, and this convergence order is higher than the previous work. Finally, some numerical examples are presented to validate our theoretical results.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 11671321, 11971387).

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  1. School of Science, Northwestern Polytechnical University, Xi’an, 710129, China

    Zhen Guan, Xiaodong Wang & Jie Ouyang

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  1. Zhen Guan
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  2. Xiaodong Wang
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  3. Jie Ouyang
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Correspondence to Xiaodong Wang.

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Guan, Z., Wang, X. & Ouyang, J. An improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term. J. Appl. Math. Comput. 65, 451–479 (2021). https://doi.org/10.1007/s12190-020-01399-4

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