**
Thái Anh Nhan
and Niall Madden**

### Abstract

We investigate the solution of linear systems of equations that arise
when a two-dimensional singularly perturbed reaction-diffusion problem
is solved using a standard finite difference method on a layer adapted,
piecewise uniform (``Shishkin'') mesh. It is known that
%, due to the discretization on layer adapted mesh,
there are difficulties in solving such systems by direct methods
when the perturbation parameter, $\varepsilon$, is small [MacLachlan
and Madden, 2013].
Therefore, iterative methods are natural choices. However, we show
that the condition number of the coefficient matrix grows
unboundedly when $\varepsilon$ tends to zero, and so unpreconditioned
iterative schemes, such as the conjugate gradient algorithm, perform
poorly with respect to $\varepsilon$. We provide a careful analysis of
diagonal and incomplete Cholesky preconditionings, and show that the
condition number of the preconditioned linear system is independent
of perturbation parameter. We demonstrate numerically the surprising
fact that these schemes are more efficient when $\varepsilon$ is small,
than when $\varepsilon$ is $\mathcal{O}(1)$. Furthermore, our analysis
shows that when the singularly perturbed problem features no corner
layers, the incomplete Cholesky preconditioner performs extremely
well when $\varepsilon\ll 1$.