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Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

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Abstract

In this paper, we consider the Shliomis ferrofluid model and study its numerical approximation. We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics (SFHD) equations. First, we establish the well-posedness and some regularity results for the solution of the SFHD model. Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the \({\varvec{L}}^2 \text -{\varvec{H}}^1\) error estimates for the time-discrete solution. Moreover, certain regularity results for the time-discrete solution are proved rigorously. With the help of these regularity results, we prove the unconditional \({\varvec{L}}^2 \text -{\varvec{H}}^1\) error estimates for the finite element solution of the SFHD model. Finally, some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 12271514, 11871467, 12161141017) and the National Key Research and Development Program of China (2023YFC3705701).

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Authors and Affiliations

  1. LSEC and ICMSEC, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Beijing, 100190, China

    Shipeng Mao & Jiaao Sun

  2. Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China

    Shipeng Mao & Jiaao Sun

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Mao, S., Sun, J. Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-023-00347-w

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