Abstract
In this article, we mainly consider a first-order decoupling penalty method for the 2D/3D time-dependent incompressible magnetohydrodynamic (MHD) equations in a convex domain. This method applies a penalty term to the constraint “divu = 0,” which allows us to transform the saddle point problem into two small problems to solve. The time discretization is based on the backward Euler scheme. Moreover, we derive the optimal error estimate for the penalty method under semi-discretization with the relationship 𝜖 = O(Δt). Finally, we give abundant of numerical tests to verify the theoretical result and the spatial discretization is based on Lagrange finite element.











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Acknowledgements
The authors are very much indebted to the referees for their constructive suggestions and insightful comments, which greatly improved the original manuscript of this paper.
Funding
This work is partly supported by the NSF of China (No. 12126361, 12126372, 12061076), Tianshan Youth Project of Xinjiang Province (No. 2017Q079), Scientific Research Plan of Universities in the Autonomous Region (No. XJEDU2020I 001), and Key Laboratory Open Project of Xinjiang Province (No. 2020D04002).
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Shi, K., Feng, X. & Su, H. Optimal error estimate of the penalty method for the 2D/3D time-dependent MHD equations. Numer Algor 93, 1337–1371 (2023). https://doi.org/10.1007/s11075-022-01470-0
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DOI: https://doi.org/10.1007/s11075-022-01470-0