A subgrid stabilization finite element method for incompressible magnetohydrodynamics

Abstract

This paper studies a numerical scheme for approximating solutions of incompressible magnetohydrodynamic (MHD) equations that uses eddy viscosity stabilization only on the small scales of the fluid flow. This stabilization scheme for MHD equations uses a Galerkin finite element spatial discretization with Scott-Vogelius mixed finite elements and semi-implicit backward Euler temporal discretization. We prove its unconditional stability and prove how the coarse mesh can be chosen so that optimal convergence can be achieved. We also provide numerical experiments to confirm the theory and demonstrate the effectiveness of the scheme on a test problem for MHD channel flow.

Keywords:

• finite element method
• MHD
• subgrid stabilization
• stability analysis
• convergence analysis
• Scott-Vogelius elements

2010 AMS Subject Classifications :

• 76W05
• 65M60
• 65M12

Acknowledgements

L.G. Rebholz and N.E. Wilson are partially supported by NSF grants DMS 0914478 and DMS 1112593.