Parameter-uniform numerical methods for a class of parameterized singular perturbation problems

In this article, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed from the backward Euler scheme to a monotone hybrid scheme. Three kinds of nonuniform grids are considered: a standard Shishkin mesh, a Bakhavalov–Shishkin mesh, and an adaptive grid. The methods are shown to be uniformly convergent with respect to the perturbation parameter for all three types of meshes. The rate of convergence is of first order for the backward Euler scheme and of second order for the monotone hybrid scheme. Furthermore, the proposed method is extended to a parameterized problem with mixed type boundary conditions and is shown to be uniformly convergent. Numerical experiments are carried out to show the efficiency of the proposed schemes, which indicate that the estimates are optimal.