  We consider the solution of large linear systems of equations that arise when
two-dimensional singularly perturbed reaction-diffusion equations are
discretized. Standard methods for these problems, such as central finite
differences, lead to system matrices that are positive definite. The direct
solvers of choice for such systems are based on Cholesky factorisation.
However, as observed by MacLachlan and Madden (SIAM J. Sci. Comput. 35-5
(2013), pp. A2225-A2254), these solvers may exhibit poor performance for
singularly perturbed problems. We provide an analysis of the distribution of
entries in the factors based on their magnitude that explains this phenomenon,
and give bounds on the ranges of the perturbation and discretization parameters
where poor performance is to be expected.
