Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear reaction-diffusion system

摘要

A singularly perturbed linear system of second order ordinary differentialequations of reaction-diffusion type with given boundary conditions isconsidered. The leading term of each equation is multiplied by a small positiveparameter. These singular perturbation parameters are assumed to be distinct.The components of the solution exhibit overlapping layers. Shishkinpiecewise-uniform meshes are introduced, which are used in conjunction with aclassical finite difference discretisation, to construct a numerical method forsolving this problem. It is proved that the numerical approximations obtainedwith this method is essentially second order convergent uniformly with respectto all of the parameters.