Cholesky Factorisation of Linear Systems Coming from Finite Difference Approximations of Singularly Perturbed Problems
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 in MacLachlan and Madden (SIAM J Sci Comput 35:A2225–A2254, 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.
Proceedings of BAIL 2014–Boundary and Interior Layers–Computational and Asymptotic Methods
Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 108)
Nhan, T. A., & Madden, N. (2015). Cholesky Factorisation of Linear Systems Coming from Finite Difference Approximations of Singularly Perturbed Problems. In P. Knobloch (Ed.), Boundary and Interior Layers, Computational and Asymptotic Methods—BAIL 2014 (pp. 209–220). Springer International Publishing. https://doi.org/10.1007/978-3-319-25727-3_16