Abstract: This talk is focused on several important issues arising in numerical solutions of incompressible

magnetohydrodynamic (MHD) equations. 
1. Conservations: how to conserve the mass, the magnetic induction, and the electric charges in

designing mixed finite element methods?
2. Preconditioners: robust preconditioners for solving the fully-coupled discrete problem.
3. Interface problem: extended finite element method (XFEM) for 3D interface problem or free

surface problem on unstructured meshes.

I will briefly talk about some MHD models, numerical methods, and block preconditioners both in Hilbert

space setting and in finite element setting. Numerical experiments will be presented to verify the theories.