A Parallel, Adaptive, First-Order System Least-Squares (FOSLS)
Algorithm for Incompressible, Resistive Magnetohydrodynamics
Tom Manteuffel
University of Colorado, Boulder

Abstract.  Magnetohydrodynamics (MHD) is a fluid theory that describes
Plasma Physics by treating the plasma as a fluid of charged particles.
Hence, the equations that describe the plasma form a nonlinear system
that couples Navier-Stokes with Maxwell's equations. We describe how
the FOSLS method can be applied to incompressible resistive MHD to
yield a well-posed, H$^1$-equivalent functional minimization.

To solve this system of PDEs, a nested-iteration-Newton-FOSLS-AMG-AMR
approach is taken. Much of the work is done on relatively coarse
grids, including most of the linearizations.  We show that at most one
Newton step and a few V-cycles are all that is needed on the finest
grid. Estimates of the local error and of relevant problem parameters
that are established while ascending through the sequence of nested
grids are used to direct local adaptive mesh refinement (AMR), with
the goal of obtaining an optimal grid at a minimal computational
cost. An algebraic multigrid solver is used to solve the linearization
steps.

A parallel implementation is described that uses a binning
strategy. We show that once the solution is sufficiently resolved,
refinement becomes uniform which essentially eliminates load balancing
on the finest grids.

The ultimate goal is to resolve as much physics as possible with the
least amount of computational work. We show that this is achieved in
the equivalent of a few dozen work units on the finest grid. (A work
unit equals a fine grid residual evaluation).

Numerical results are presented for two instabilities in a large
aspect-ratio tokamak, the tearing mode and the island coalescence
mode.