An analysis of the bilinear finite volume method for the singularly-perturbed convection-diffusion problems on Shishkin mesh

In this paper, we study the bilinear finite volume element method for solving the singularly perturbed convection-diffusion problem on the Shishkin mesh. We first prove that the finite volume element scheme is ϵ-uniformly stable. Then, based on new expression of the finite volume bilinear form and some detailed integral calculations, an ϵ-uniform error estimation is derived in the ϵ-weighted gradient norm, including the $L_2$-norm. This error estimate is better than the known result. Moreover, we also give the $L_{\infty }$-error estimate near the boundary layer regions. At last, numerical experiments show the effectiveness of our method.