A Hybrid Difference Scheme for a Second-Order Singularly Perturbed Reaction-Diffusion Problem with Non-smooth Data

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. In Miller et al. (J Appl Numer Math 35(4):323–337, 2000) the authors discussed problems that arises naturally in the context of models of simple semiconductor devices. Due to the discontinuity, interior layers appear in the solution. The problem is solved using a hybrid difference scheme on a Shishkin mesh. We prove that the method is second order convergent in the maximum norm, independently of the diffusion parameter. Numerical experiments support these theoretical results and indicate that the estimates are sharp.