Boundary layer preconditioners for finite-element
discretizations of singularly perturbed reaction-diffusion
problems
Thai Anh Nhan, Scott MacLachlan and
Niall Madden
July 2017.
Abstract
We consider the iterative solution of linear systems of equations
arising from the discretization of singularly perturbed
reaction-diffusion differential equations by finite-element
methods on boundary-fitted meshes. The equations feature a perturbation parameter, which may be
arbitrarily small, and, correspondingly,
their solutions feature layers: regions where
the solution changes rapidly. Therefore, numerical solutions are
computed on specially designed, highly anisotropic layer-adapted
meshes.
Usually, the resulting linear systems are ill-conditioned,
and, so, careful
design of suitable preconditioners is necessary in order to solve them
in a way that is robust, with respect to the
perturbation parameter, and efficient. We propose a boundary
layer preconditioner, in the style of that introduced by MacLachlan
and Madden (2013) for a
finite-difference method. We prove the optimality
of this preconditioner and establish a suitable stopping criterion
for one-dimensional problems. Numerical results are presented
which demonstrate that the ideas extend to problems in two
dimensions.
Full text:
https://www.math.mun.ca/~smaclachlan/research/FEM_BLPCG.pdf