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Volume 17, Issue 2
Convergence Analysis of a BDF Finite Element Method for the Resistive Magnetohydrodynamic Equations

Lina Ma, Cheng Wang & Zeyu Xia

Adv. Appl. Math. Mech., 17 (2025), pp. 633-662.

Published online: 2024-12

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  • Abstract

In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation (BDF) and the finite element method (FEM) to solve the incompressible resistive magnetohydrodynamic (MHD) equations. In the discrete scheme, the pressure variable in the fluid field equation is computed through a Poisson equation, and a linear and decoupled method is adopted to separate both the magnetic and the fluid field functions from the original system. As a result, the original system is divided into several sub-systems for which the numerical solutions can be obtained efficiently. We prove the unique solvability, the unconditional energy stability, and particularly optimal error estimates for the proposed scheme. Numerical results are presented to validate the theory of the scheme.

  • AMS Subject Headings

65M60, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-17-633, author = {Ma , LinaWang , Cheng and Xia , Zeyu}, title = {Convergence Analysis of a BDF Finite Element Method for the Resistive Magnetohydrodynamic Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2024}, volume = {17}, number = {2}, pages = {633--662}, abstract = {

In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation (BDF) and the finite element method (FEM) to solve the incompressible resistive magnetohydrodynamic (MHD) equations. In the discrete scheme, the pressure variable in the fluid field equation is computed through a Poisson equation, and a linear and decoupled method is adopted to separate both the magnetic and the fluid field functions from the original system. As a result, the original system is divided into several sub-systems for which the numerical solutions can be obtained efficiently. We prove the unique solvability, the unconditional energy stability, and particularly optimal error estimates for the proposed scheme. Numerical results are presented to validate the theory of the scheme.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0118}, url = {http://global-sci.org/intro/article_detail/aamm/23737.html} }
TY - JOUR T1 - Convergence Analysis of a BDF Finite Element Method for the Resistive Magnetohydrodynamic Equations AU - Ma , Lina AU - Wang , Cheng AU - Xia , Zeyu JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 633 EP - 662 PY - 2024 DA - 2024/12 SN - 17 DO - http://doi.org/10.4208/aamm.OA-2023-0118 UR - https://global-sci.org/intro/article_detail/aamm/23737.html KW - Resistive MHD equations, finite element methods, BDF decoupled scheme, unconditional energy stability, optimal error estimates. AB -

In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation (BDF) and the finite element method (FEM) to solve the incompressible resistive magnetohydrodynamic (MHD) equations. In the discrete scheme, the pressure variable in the fluid field equation is computed through a Poisson equation, and a linear and decoupled method is adopted to separate both the magnetic and the fluid field functions from the original system. As a result, the original system is divided into several sub-systems for which the numerical solutions can be obtained efficiently. We prove the unique solvability, the unconditional energy stability, and particularly optimal error estimates for the proposed scheme. Numerical results are presented to validate the theory of the scheme.

Ma , LinaWang , Cheng and Xia , Zeyu. (2024). Convergence Analysis of a BDF Finite Element Method for the Resistive Magnetohydrodynamic Equations. Advances in Applied Mathematics and Mechanics. 17 (2). 633-662. doi:10.4208/aamm.OA-2023-0118
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