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Optimal Convergence of the Scalar Auxiliary Variable Finite Element Method for the Natural Convection Equations

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Abstract

This paper concerns an efficient finite element method for the natural convection equations with the scalar auxiliary variable approach. The linearly extrapolated Crank-Nicolson techniques are used to discretize nonlinear terms in the Navier–Stokes equations and the heat equation. The induced scalar auxiliary equation is an univariate ordinary equation. With the benefit of the new defined system, the redefined stability of the proposed method is obtained under the fully explicit scheme for the nonlinear terms without the requirement of skew-symmetric trilinear forms. The optimal convergence rates in space for all the variables are proved. The second order convergence rates in time are also obtained. Finally, numerical experiments are provided to support the theoretical analysis and demonstrate the efficiency of the proposed scheme.

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Acknowledgements

This work was partially supported by the Science Challenge Project (No. TZ2018001), the National Natural Science Foundation of China (Nos. 12071261, 11831010, 11871068 and 12001325), the National Key R &D Program of China (No. 2018YFA0703900) and the State Key Program of National Natural Science Foundation of China Grant No. 12131014.

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  1. School of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

    Yaping Li, Weidong Zhao & Wenju Zhao

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  1. Yaping Li
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Correspondence to Wenju Zhao.

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Li, Y., Zhao, W. & Zhao, W. Optimal Convergence of the Scalar Auxiliary Variable Finite Element Method for the Natural Convection Equations. J Sci Comput 93, 39 (2022). https://doi.org/10.1007/s10915-022-01981-x

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