This paper proposes a stabilization method to approximate analytical solutions of magnetohydrodynamics (MHD) equations. The method adds two modular grad-div steps into fully-discrete finite element MHD solver. The main idea in these intrusive steps is to penalize the divergence of the velocity/magnetic fields both in L2 and H1-norms. The paper confirms the optimal convergence of the method, and gives numerical experiments which reveal positive effect of the method as in the usual grad-div stabilization.

1.
P. A.
Davidson
,
An introduction to magnetohydrodynamics,
(
Cambridge Texts in Applied Mathematics, Cambridge University Press
,
Cambridge
,
2001
.)
2.
K.
Galvin
and
A.
Linke
and
L.
Rebholz
and
N.
Wilson
,
Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput
.
Methods Appl. Mech. Engrg
,
237–240
,
166
176
(
2012
).
3.
M.
Olshanskii
and
A.
Reusken
,
Grad-div stabilization for Stokes equations
,
Math. Comp.
,
73
(
248
),
1699
1718
(
2004
).
4.
A.
Cıbık
,
The effect of a sparse graddiv stabilization on control of stationary NavierStokes equations
,
J. Math. Anal. Appl.
,
437
(
1
),
613
628
(
2016
).
5.
J. A
Fiordilino
,
W.
Layton
, and
Y.
Rong
,
An efficient and modular grad-div stabilization, Comput
.
Methods Appl. Mech. Engrg.
,
335
,
327
346
(
2018
).
6.
M.
Belenli
,
S. Kaya, L. Rebholz, and N. Wilson
,
A subgrid stabilization finite element method for incom-pressible magnetohydrodynamics
,
International Journal of Computer Mathematics
,
90
(
7
),
1506
1523
(
2013
).
This content is only available via PDF.
You do not currently have access to this content.