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Convergence and Asymptotic Stability of the BDF Schemes for the Nonlocal Partial Differential Equations with Delay

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Abstract

In this paper, we present the first and second order backward differentiation formulas (BDF1, vBDF1, BDF2 and vBDF2) with finite element method for a class of non-autonomous nonlocal partial differential equations with delay. First, the unique solvability of the numerical schemes is proved. The \(L^2\) upper bound of the numerical solution is given by energy estimation. The convergence analysis is fulfilled for nonlinear delay term, with \(\mathcal {O}(h^q+\Delta t)\) order for BDF1 and \(\mathcal {O}(h^{q}+\Delta t^2)\) order for BDF2. The convergence order with respect to the spatial discretization is recovered to an optimal \(\mathcal {O}(h^{q+1})\) order for a kind of specific delay term \(\psi =\nabla \cdot A(x,t)\nabla u\). Asymptotic stability analysis is also given, and it is proved that the numerical solution will approach the stationary solution at an exponential speed. Finally, a few numerical experiments are presented to show the convergence order, asymptotic behavior, and multiple stationary solutions of the numerical solutions.

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Acknowledgements

Chen thanks the Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, for the support.

Funding

Chen is supported by the National Natural Science Foundation of China (NSFC 12071090 and NSFC 12241101).

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Shuxun Shi

  2. School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Wenbin Chen

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  1. Shuxun Shi
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Shi, S., Chen, W. Convergence and Asymptotic Stability of the BDF Schemes for the Nonlocal Partial Differential Equations with Delay. J Sci Comput 95, 88 (2023). https://doi.org/10.1007/s10915-023-02214-5

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  • DOI: https://doi.org/10.1007/s10915-023-02214-5

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