Edward Phillips
Block Preconditioners for Fully Coupled MHD with Lagrange Multipliers

Mathematics Building
University of Maryland
College Park
MD 20742
egphillips@math.umd.edu
Howard Elman
Eric Cyr
John Shadid

Magnetohydrodynamics (MHD) models the flow of electrically conducting fluids by coupling the Navier-Stokes to Maxwell's equations. In this setting Maxwell's equations are often posed in mixed form to explicitly enforce the solenoidal condition $\nabla\cdot\vec{B} = 0$ , which requires adding a Lagrange multiplier $r$ to the induction equation governing the magnetic field. The resulting equations form a set of two coupled saddle point problems. Discretizing the fully coupled system results in a sequence of linear systems with the block form

$$\left(\begin{array}{cccc} A & D^t & -Z^t & 0 \\ D & 0 & 0 & 0 \\ ... ...n{array}{c}\mathbf{g}\\ \mathbf{0}\\ \mathbf{f}\\ \mathbf{0}\end{array}\right).$$

If we define the blocks

$$\mathcal{M} = \left(\begin{array}{cc} A & D^t \\ D & 0 \end{array}\right), \quad \mathcal{Z} = \left(\begin{array}{cc} Z & 0 \end{array}\right),$$

then a block LU decomposition of the discrete MHD system suggests a preconditioner of the form

$$\mathcal{P} = \left(\begin{array}{ccc} \hat{\mathcal{M}} & -\mathcal{Z}^t & 0 \\ 0 & \hat{X} & B^t \\ 0 & 0 & \hat{Y} \end{array}\right),$$

where $\hat{\mathcal{M}}$ is an approximation to $\mathcal{M}$ , and $\hat{X}$ and $\hat{Y}$ are approximations to the Schur complements

$$X = F + \mathcal{Z}\mathcal{M}^{-1}\mathcal{Z}^t,\quad Y = -BX^{-1}B^t.$$

We consider expressions for $\hat{\mathcal{M}}$ proposed for Maxwell's equations in mixed form and use these to develop the approximations $\hat{X}$ and $\hat{Y}$ . We test the performance of the resulting preconditioners for both stable and stabilized finite element discretizations, demonstrating their parallel scalability as well as their robustness over physical parameters on a set of two- and three-dimensional test problems.