A stabilizing augmented grid for rectangular discretizations of the convection–diffusion–reaction problems

We propose a numerical method for approximate solution of the convection–diffusion–reaction problems in the case of small diffusion. The method is based on the standard Galerkin finite element method on an extended space defined on the original grid plus a subgrid, where the original grid consists of rectangular elements. On each rectangular elements, we construct a subgrid with few points whose locations are critical for the stabilization of the problem, therefore they are chosen specially depending on some specific conditions that depend on the problem data. The resulting subgrid is combined with the initial coarse mesh, eventually, to solve the problem in the framework of Galerkin method on the augmented grid. The results of the numerical experiments confirm that the proposed method shows similar stability features with the well-known stabilized methods for the critical range of problem parameters.