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Error Analysis of a Unconditionally Stable BDF2 Finite Element Scheme for the Incompressible Flows with Variable Density

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Abstract

Based on an equivalent form of the variable density flows, we propose and study a second-order linearized finite element scheme for the approximation of the three-dimensional incompressible Navier–Stokes equations with variable density, where the two-step backward differentiation formula is used in the discretization of time derivative. It is shown that the proposed finite element scheme is unconditionally stable in the sense that the discrete energy inequalities hold without any condition on the time step size and mesh size. By a rigorous error analysis, the optimal second-order convergence rate is proved in \(L^2\)-norm. Finally, numerical results are provided to confirm our theoretical analysis.

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Data availibility statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 11771337) and Natural Science Foundation of Zhejiang Province (No. LY23A010002).

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  1. College of Mathematics and Physics, Wenzhou University, Wenzhou, People’s Republic of China

    Yuan Li & Rong An

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  1. Yuan Li
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  2. Rong An
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Correspondence to Rong An.

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Li, Y., An, R. Error Analysis of a Unconditionally Stable BDF2 Finite Element Scheme for the Incompressible Flows with Variable Density. J Sci Comput 95, 73 (2023). https://doi.org/10.1007/s10915-023-02205-6

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