Singularly perturbed elliptic problems with interior characteristic layers

Singularly perturbed elliptic problems, of convection-diffusion type, with a non-smooth forcing term are examined.This forcing term introduces an interior characteristic layer into the solution, due to fact that the solution of the reduced problem is discontinuous. A decomposition of the solution into a sum of regular, boundary layer and interior layer components is constructed. Parameter-explicit pointwise bounds on the partial derivatives of each of these components are established. Based on these bounds, an appropriate Shishkin mesh is identified and this is combined with a simple upwinded finite difference operator to form a numerical method. A global pointwise error bound is established, indicating almost first order convergence.Numerical results are presented to illustrate the performance of the method.​