his paper examines two established preconditioners which were developed
to accelerate the solution of discontinuous Galerkin nite element method (DG-
FEM) discretisations of the elliptic neutron di usion equation. They are each
presented here as a potential way to accelerate the solution of the Modi ed In-
terior Penalty (MIP) form of the discontinuous di usion equation, for use as a
di usion synthetic acceleration (DSA) of DG-FEM discretisations of the neutron
transport equation. The preconditioners are both two-level schemes, di ering
in the low-level space utilised. Once projected to the low-level space a selection
of algebraic multigrid (AMG) preconditioners are utilised to obtain a further
correction step, these are therefore \hybrid" preconditioners. The rst precon-
ditioning scheme utilises a continuous piece-wise linear nite element method
(FEM) space, while the second uses a discontinuous piece-wise constant space.
Both projections are used alongside an element-wise block Jacobi smoother in
order to create a symmetric preconditioning scheme which may be used along-
side a conjugate gradient algorithm. An eigenvalue analysis reveals that both
should aid convergence but the piece-wise constant based method struggles with
some of the smoother error modes. Both are applied to a range of problems in-
cluding some which are strongly heterogeneous. In terms of conjugate gradient
(CG) iterations needed to reach convergence and computational time required,
both methods perform well. However, the piece-wise linear continuous scheme
appears to be the more e ective of the two. An analysis of computer memory
usage found that that the discontinuous piece-wise constant method had the
lowest memory requirements.