A parameter robust Petrov-Galerkin scheme for advection-diffusion-reaction equations

In this paper, a singularly perturbed convection diffusion boundary value problem, with discontinuous diffusion coefficient is examined. In addition to the presence of boundary layers, strong and weak interior layers can also be present due to the discontinuities in the diffusion coefficient. A priori layer adapted piecewise uniform meshes are used to resolve any layers present in the solution. Using a Petrov-Galerkin finite element formulation, a fitted finite difference operator is shown to produce numerical approximations on this fitted mesh, which are uniformly second order (up to logarithmic terms) globally convergent in the pointwise maximum norm.