An efficient parameter uniform spline-based technique for singularly
perturbed weakly coupled reaction-diffusion systems
Abstract
A parameter-uniform numerical scheme for a system of weakly coupled
singularly perturbed reaction-diffusion equations of arbitrary size with
appropriate boundary conditions is investigated. More precisely,
quadratic $B$-spline basis functions with an exponentially graded mesh
are used to solve a
$\ell\times\ell$ system
whose solution exhibits parabolic (or exponential) boundary layers at
both endpoints of the domain. A suitable mesh generating function is
used to generate the exponentially graded mesh. The decomposition of the
solution into regular and singular components is obtained to provide
error estimates. A convergence analysis is addressed, which shows a
uniform convergence of the second order. To validate the theoretical
findings, two test problems are solved numerically.