Error estimate of a fully discrete defect correction finite element method for unsteady incompressible Magnetohydrodynamics equations

ABSTRACT

In this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuous work of our formal paper [Math Method Appl Sci. 2017. DOI:10.1002/mma.4296]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [see Math Method Appl Sci. 2017. DOI:10.1002/mma.4296] show that this method is highly efficient for the unsteady incompressible MHD problems.

KEYWORDS:

• Unsteady incompressible MHD equations
• finite element method
• fully discrete defect correction method
• stability analysis
• error analysis

AMS SUBJECT CLASSIFICATIONS:

• 76D05
• 35Q30
• 65M60
• 65N30

Acknowledgements

The authors would like to thank the editor and the referees for their valuable comments, which led to the improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the author was supported in part by the National Natural Science Foundation of China [grant number 11401177].