  Highly accurate simulation of plasma transport is needed for the successful
design and operation of magnetically confined fusion reactors. Unfortunately,
the extreme anisotropy present in magnetized plasmas results in thin boundary
layers that are expensive to resolve. This work investigates how mesh
refinement strategies might reduce that expense to allow for more efficient
simulation. It is first verified that higher order discretization only realizes
the proper rate of convergence once the mesh resolves the thin boundary layer,
motivating the focusing of refinement on the boundary layer. Three mesh
refinement strategies are investigated: one that focuses the refinement across
the layer by using rectangular elements with a ratio equal to the boundary
layer width, one that allows for exponential growth in mesh spacing away from
the layer, and one adaptive strategy utilizing the established Zienkiewicz and
Zhu error estimator. Across 4 two-dimensional test cases with high anisotropy,
the adaptive mesh refinement strategy consistently achieves the same accuracy
as uniform refinement using orders of magnitude less degrees of freedom. In the
test case where the magnetic field is aligned with the mesh, the other
refinement strategies also show substantial improvement in efficiency. This
work also includes a discussion generalizing the results to larger magnetic
anisotropy ratios and to three-dimensional problems. It is shown that isotropic
mesh refinement requires degrees of freedom on the order of either the layer
width (2D) or the square of the layer width (3D), whereas anisotropic
refinement requires a number on the order of the log of layer width for all
dimensions. It is also shown that the number of conjugate gradient iterations
scales as a power of layer width when preconditioned with algebraic multigrid,
whereas the number is independent of layer width when preconditioned with ILU.
