Optimal weighted upwind finite volume method for convection–diffusion equations in 2D

Abstract This paper develops and analyzes the optimal weighted upwind finite volume method (OWUFVM) for convection–diffusion equations on nonuniform rectangular meshes. The dual meshes in this method are associated with the local Peclet numbers, which are determined by the convection and diffusion coefficients and the mesh size. We first prove the stability and H 1 -norm error estimate. Then we prove a weak error estimate, which plays a key role in deriving the optimal second-order error estimate in the L 2 norm. The L 2 -norm error estimate is the most important result of this paper. We also briefly examine the discrete maximum principle. Numerical experiments are presented to illustrate the theoretical analysis.