Thái Anh Nhan and Niall
Madden,
Abstract
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 (2013), 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.