Volume 56, Number 3, May-June 2022
|Page(s)||767 - 789|
|Published online||25 April 2022|
Optimal error estimates of a Crank–Nicolson finite element projection method for magnetohydrodynamic equations
Mathematics Department, University of Massachusetts, North Dartmouth, MA 02747, USA
2 Beijing Computational Science Research Center, Beijing 100193, P.R. China
3 School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P.R. China
* Corresponding author: firstname.lastname@example.org.
Accepted: 14 February 2022
In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank–Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the Van Kan type [Van Kan, SIAM J. Sci. Statist. Comput. 7 (1986) 870–891] in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Error estimates are proved in the discrete L∞(0, T; L2) norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results.
Mathematics Subject Classification: 35K20 / 65M12 / 65M60 / 76D05
Key words: Magnetohydrodynamic equations / modified Crank–Nicolson scheme / finite element / unique solvability / unconditional energy stability / error estimates
© The authors. Published by EDP Sciences, SMAI 2022
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