Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

A numerical scheme for the convection–diffusion–reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving CDR problems particularly designed to treat the most interesting case of small diffusion. We use the subgrid nodes in the Link-cutting bubble (LCB) strategy [5] to construct a numerical algorithm that can easily be extended to the higher dimensions. The method adapts very well to all regimes with continuous transitions from one regime to another. We also compare the performance of the present method with the Streamline-upwind Petrov–Galerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method.